Tensor Spherical Harmonics and Tensor Spherical ... - CiteSeerX

W. Freeden, T. Gervens, M. Schreiner: Tensor Spherical Harmonics and Tensor Spherical Splines

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Tensor Spherical Harmonics and Tensor Spherical Splines by

W. Freeden , T. Gervens , M. Schreiner

University of Kaiserslautern

Laboratory of Technomathematics 67663 Kaiserslautern Germany

Bayer AG AV{IM Applied Mathematics, B407 51368 Leverkusen Germany

(accepted for publication in Manuscripta Geodaetica)

Abstract. In this paper, we deal with the problem of spherical interpolation of discretely given data of tensorial type. To this end, spherical tensor elds are investigated and a decomposition formula is described. It is pointed out that the decomposition formula is of importance for the spectral analysis of the gravitational tensor in (spaceborne) gradiometry. Tensor spherical harmonics are introduced as eigenfunctions of a tensorial analogue to the Beltrami operator and discussed in detail. Based on these preliminaries, a spline interpolation process is described and error estimates are presented. Furthermore, some relations between the spline basis functions and the theory of radial basis functions are developed.

Introduction In the last decade, the interpolation problem of discretely given data by spherical splines has been solved in numerous papers. Both theoretical and numerical aspects of spherical splines have been analyzed in depth, for scalar{valued as well as for vector{valued data (cf. Freeden (1981), Freeden (1982), Freeden (1990), Freeden and Gervens (1993), Freeden and Hermann (1986), Wahba (1981), Wahba (1984), and the references therein). It turns out that the spherical splines are natural generalizations of the spherical harmonics respectively the vector spherical harmonics, having quite similar properties as the well{known one{dimensional polynomial splines. Spherical splines are well{suited for various interpolation and best approximationproblems preferably in the geosciences. In particular, they have been used for the macro{ and micro{modelling of the earth's gravitational eld from discretely given data (cf. e.g. Euler et al. (1985), Euler et al. (1986), Freeden (1990), v. Gysen and Merry (1990), Kling et al. (1987)). Moreover, based on the ideas of the scalar theory, a successfull

application of spherical splines has been given to the analysis of meteorological data (Wahba 1982). In this paper, we develop the general theory of tensor spherical splines for the interpolation of spherical tensor elds, its components given in discrete points on the sphere. The main idea hereby is the decomposition of the tensor elds in nine scalar functions and to use the basic results from the theory of scalar spherical splines. This decomposition is motivated by the theory of tensor spherical harmonics, as described in Backus (1966) and Backus (1967). (Other approaches for the de nition of tensor spherical harmonics are described e.g. in Zerilli (1970), James (1976) and Jones (1980). A review article about tensor spherical harmonics used in the general relativity literature is due to Thorne (1980).) We extend this theory and show that the tensor spherical harmonics are the eigenfunctions of the tensorial analogue of the well{known Beltrami operator. This theory is the main tool for the establishment of tensor spherical splines which is in fact a generalization of these harmonics. As in the scalar theory, the interpolation problem is a well{posed problem of minimizing a semi{norm of a suitable Sobolev space under interpolatory constraints. Tensor spherical splines are useful, for example, in multipole expansions of gravitational radiation (cf. e.g. Regge and Wheeler (1957), Zerilli (1970)), the deformation analysis of the earth's surface (cf. e.g. Grafarend (1986), Georgiadou and Grafarend (1986), Mochizuki (1988), McClung (1989)), in the investigation of the in uence of rotation on the free elastic{gravitational oscillations of the earth (cf. e.g. Backus and Gilbert (1961), Dahlen and Smith (1975), etc.). Furthermore, they oer possibilities to solve geodetic problems in gradiometry, namely the determination of the earth's gravitational eld from the gravitational tensor of second derivatives (Hesse matrix). Terrestrial gradiometer measurements are very much aected by density variation and topographic fea-


tures in the immediate vicinity of the observation point. Thus, in spite of the very laborious work, they are very suitable for exploration geophysics (cf. Jung (1961)). Gravity gradiometry measurements, taken at certain altitude above the earth's surface, are highly sensitive to the ne structure of the earth's gravity eld by observing in principle the relative motion of neighbouring, free{falling test masses. Spaceborne gradiometry, as planned for the next decade, will improve the "microscopic structure\ of the gravity eld. This will lead us to a better understanding of the composition and dynamics of the solid earth (cf. e.g. Rummel (1986), Rummel and Colombo (1984), Sacerdote and Sanso (1989), Schwarz and Krynski (1977)). The mathematical characteristics of gradiometry are commonly explained employing spectral analysis as applied to spherical harmonic expansions (cf. Rummel (1975), Rummel and van Gelderen (1992)). To discuss the gravity tensor of second derivatives tensor spherical harmonics are needed similar to the application of vector spherical harmonics to the expansion of the de ections of the vertical (cf. Meissl (1971a), Groten and Moritz (1964)). To obtain gravity information from discrete gradiometer data tensor spherical splines as natural generalizations of "tensor spherical polynomials\ i.e. tensor spherical harmonics, seem to be adequate structures thereby avoiding speci c diculties (e.g. oscillation phenomena, convergence problems, determination of higher order terms, etc.) in interpolation and approximation. Moreover, spline techniques become available for smoothing noisy data (cf. e.g. Freeden and Witte (1982), Wahba (1984), Girard (1991), Utreras (1990)) thereby using control parameters for the trade o between "approximation to the data\ and "smoothness\ of the solution. Some emphasis is drawn on the choice of the norm. Here, we develop tools to combine the spherical spline theory with generalizations of the concept of radial basis functions (cf. Micchelli (1986), Madych and Nelson (1988) and many others). The paper is organized as follows: In Section 1 some preliminaries are given. The theory of tensor spherical harmonics is established in Section 2. In detail, we consider spherical tensor elds, their decomposition in normal and tangential parts to get nine scalar functions. Tensor spherical harmonics are recognized as eigenfunctions of a tensorial analogue of the Beltrami operator. To de ne the tensor splines in the third section, we rst have a look at adequate Sobolev spaces. After that we consider the interpolation algorithm and prove the convergence of our method. Appendix A illustrates the explicit decomposition of the gravitational tensor (Hesse matrix) e.g. for single poles and multipoles (i.e. solid spherical harmonics). Finally, Appendix B gives a mathematical overview about possible choices of norms and relations to the theory of radial basis functions.

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1 Preliminaries We begin by introducing some notations that will be used throughout the paper.

1.1 De nitions and Notations

Let us use x; y; : : : to represent the elements of Euclidean space R3. For all x 2 R3, x = (x1 ; x2; x3)T , dierent from the origin, we have

q

r = jxj = x21 + x22 + x23 ;

x = r ;

where = (1 ; 2; 3)T is the uniquely determined directional unit vector of x 2 R3. The unit sphere in R3 will be denoted by . If the vectors "1 ; "2; "3 form the canonical orthonormal basis in R3, we may represent the points 2 in polar coordinates by

p

= t "3 + 1 ? t2 (cos ' "1 + sin ' "2 ); ?1 t 1 ; 0 ' < 2 ; t = cos :

(1.1)

As usual, inner{, and vector{, and dyadic (tensor) product of two vectors x; y 2 R3, respectively, are de ned by xy = x^y = x y =

3 X

xi yi ; i=1 T (x 0 2xy3y? xx3yy2; x3xy1y?1x1y3; x1y2 ? x2y1) ; @ x12y11 x12y22 x12y33 A : (1.2) x3y1 x3 y2 x3 y3

In terms of the polar coordinates (1.1) the gradient

r in R3 reads

@ + 1 r ; rx = @r r

where r is the surface gradient of the unit sphere . Moreover, the Laplace operator = r r in R3 has the representation @ )2 + 2 @ + 1 ; x = ( @r r @r r2 where = r r is the Beltrami operator of the unit sphere (for explicit representations in terms of polar coordinates see e.g. Meissl (1971a)). A function F : ! R (resp. f : ! R3) 1 possessing k continuous derivatives on the unit sphere

is said to be of class C (k) ( ) (resp. c(k)( )). C (0)( ) (resp. c(0)( )) is the class of real continuous scalar valued (resp. vector valued) functions on . 1 Scalar valued (resp. vector valued, tensor valued) functions are denoted by capital (resp. small, small bold) letters throughout this paper.


C (0)( ) is a complete normed space endowed with kF kC = sup j F() j : (0)

2

In C (0) ( ) we have the inner product (; )L corresponding to the norm 2

kF kL =

sZ

2

j F() j2 d!();

(d! denotes the surface element). In connection with (; )L , C (0) ( ) is a pre{Hilbert space. For each F 2 C (0)( ) we have the inequality p kF kL 4kF kC : By L2 ( ) we denote the space of (Lebesgue) square{ integrable scalar functions on : L2( ) is a Hilbert space with respect to the inner product (; )L . L2( ) is the completion of C (0) ( ) with respect to the norm k kL . Since the operators r and are of particular interest throughout this paper, we list some of their properties. Especially, for 2 , we have r (r F()) = F(); r ( ^ r F()) = 0; r ( ^ ( ^ r F())) = ? F(); r (F()f()) = (r F()) f() +F()(r f()): By the theorems of Gau and Stokes we obtain 2

2

(0)

2

2

Z

Z

Z

Z

r f()d!() = 0;

r (f() ^ )d!() = 0;

f() r F()d!() =

r F() r G()d!() =

Z

F()r f()d!();

Z

?

Z

F() G()d!()

= ? G() F()d!();

where r f() (resp.

r (f()^) ) is the surface diver

gence (resp. surface curl) of the vector eld f at 2 .

Note that the surface curl 7! r (f() ^ ); 2 , de nes a scalar{valued function on the unit sphere R3 (that is the reason why we do not use the notation r ^ f() instead of r (f() ^ )). Furthermore, Green's identities read 1 Z F()d!() F() = 4 ? Z 1 + 4 r 2 ln j ?2 j r F() d!();

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Z 1 F() = 4 F()d!() ? Z 1 2 ln j ?2 j F() d!(); ? 4

provided that F : ! R resp. f : ! R3 are suciently often continuously dierentiable. Moreover, for H 2 C (2) [?1; 1], we have

r H( ) = H 0 ( )( ? ( )); H( ) = ?2( )H 0 ( ) +(1 ? ( )2 )H 00( ): In particular,

r ( ) = ? ( ); ( ) = ?2( ):

1.2 Scalar Spherical Harmonics

As usual, the spherical harmonics Yn of order n are de ned as the everywhere on in nitely dierentiable eigenfunctions of the Beltrami operator corresponding to the eigenvalues ?n(n + 1); n = 0; 1; 2; : : :, i.e., (? ? n ) Yn () = 0; n = n(n + 1); 2 : The functions Hn : R3 ! R de ned by Hn(x) = rnYn (); r = jxj; are homogeneous polynomials of degree n in rectangular coordinates which satisfy the Laplace equation x Hn(x) = 0; x 2 R3. Conversely, every homogeneous harmonic polynomial of degree n when restricted to the unit sphere is a spherical harmonic of order n. (An ecient algorithm for the exact generation of homogeneous harmonic polynomials in R3 can be found in Freeden and Reuter (1990).) The Legendre polynomials Pn : [?1; 1] ! [?1; 1]; n = 0; 1; : : : given by Pn(t) =

[n] X 2

s=0

? 2s)! n?2s (?1)s 2n(n ?(2n2s)!(n ? s)!s! t ;

are the only everywhere on the interval [?1; 1] in nitely dierentiable eigenfunctions of the Legendre operator, i.e.,

?(1 ? t2 )

d 2 dt

!

d ? P (t) = 0 + 2t dt n n

for t 2 [?1; 1], which satisfy Pn(1) = 1. Apart from a constant factor, the functions Pn("3 ) : ! R, 7! Pn("3 ); 2 , are the only spherical harmonics which are invariant under orthogonal transformations which leave "3 xed. As is well-known, jPn(t)j Pn(1) = 1 for all t 2 [?1; +1] and all n. The linear space Harmn of all spherical harmonics of order n is of dimension 2n + 1. Thus, there


exist 2n + 1 linearly independent spherical harmonics Yn;1; :::; Yn;2n+1. We assume this system to be orthonormalized in the sense of the L2( ){inner product (Yn;j ; Ym;k )L =

Z

Yn;j () Ym;k () d!() = j;k n;m :

2

Throughout the paper, for convenience, the capital letter Y followed by double indices (for example Yn;j ) denotes a member of a L2 ( ){orthonormal system fYn;1; : : :; Yn;2n+1g. For any pair ; 2 , the sum Fn (; ) =

2X n+1

j =1

Yn;j ()Yn;j ()

is invariant under all orthogonal transformations t, i.e., Fn(t; t) = Fn(; ) . For a xed vector 2

; Fn(; ) : ! R is a spherical harmonic of order n. Fn (; ) is symmetric in and and depends only on the inner product of and . Hence, we have apart from a multiplicative constant an Fn(; ) = an Pn( ) : For the evaluation of an we set = . This gives Fn(; ) = an Pn(1) = an : Integration over yields 2n+1 = 4 an . Therefore we nally obtain the addition theorem of scalar spherical harmonics (cf. Muller (1966), Freeden (1978)) n+1 2n + 1 P ( ) = 2X Yn;j ()Yn;j () : 4 n j =1 2X n+1

j =1

and sup jYn;j ()j

(Yn;j ())2

= 2n4+ 1 ;

2n + 1

where

^ = 2 H(n)

Z1 ?1

1 2

H(t) Pn(t) dt:

rential equation

( + n)G( + n ; ; ) = ? 2n4+ 1 Pn( )

for all 2 nf g. (ii) for every 2 , the function G( + n ; ; ) ? 1=(4) ln(1 ? ) is continuously dierentiable for all 2 (characteristic singularity). (iii) G( +n ; ; ) depends only on the inner product of and (spherical symmetry), i.e. G( + n ; t; t) = G( + n; ; ) for all orthogonal transformations t. (iv) the normalizing condition

; j = 1; : : :; 2n + 1 : 4 Let H be a function of class L1[?1; 1] and Yn be a spherical harmonic of order n. Then Hecke's formula states Z ^ n () ; H( ) Yn() d!() = H(n)Y (1.3) 2

that of isotropic operators, e.g. developed by P. Meissl (1971b). Since the kernel H depends only on the inner product , or equivalently on the distance between and , the spherical harmonics Yn are the eigenfunctions of the integral operator de ned by the left hand side of (1.3) corresponding to the eigenva^ lues H(n). Therefore, Hecke's formula greatly simpli es most manipulations with spherical harmonics, in particular when spectral analysis is pursued. Frequently, we will make use of the Green function G( + n ; ; ) with respect to the operator + n . The de ning properties of G( + n ; ; ) read as follows: (i) for every 2 , G( + n ; ; ) is in nitely dierentiable for all 2 nf g, and we have the die-

Z

For = we get the formula

(1.4)

This formula establishes the close connection between the orthogonal invariance of the sphere and the addition theorem. The principle of the Hecke formula is

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G( + n ; ; )Pn( )d!() = 0

holds. G( + n ; ; ) is uniquely determined by the conditions (i){(iv). Especially, for ; 2 ; 6= , the Green functions are given by 1 ln(1 ? ) + 1 ? 1 ln 2 ; (1.5) G( ; ; ) = 4 4 4 and for n = 1; 2; : : : by 1 P ( ) ln(1 ? ) G( + n ; ; ) = 4 n nX ?1 (1.6) ? 21 2k?+1 Pk ( ) n k=0 k 2n + 1 Z +1 1 2 ? 4 2 Pn (t) ln(1 ? t)dt Pn( ): ?1 The spherical harmonics of order n, i.e. the eigenfunctions of the Beltrami operator with respect to the


eigenvalue ?n , are eigenfunctions of Green's function with respect to the operator + k in the sense ? k Z G( + ; ; )Y ()d!(); k 6= n: Yn() = n 4 k n

Let G(( + 0 ) : : : ( + m ); ; ), for m = 1; 2; : : :, be de ned by convolution as follows G((Z + 0 ) : : : ( + m ); ; ) = G(( + 0 ) : : : ( + m?1 ); ; )

G( + m ; ; )d!(): Then G(( + 0 ) : : : ( + m ); ; ) is the Green function with respect to the operator ( + 0 ) : : : ( + m ). For m > 0 and all ; 2 , G(( + 0 ) : : : ( + m ); ; ) admits a representation in terms of the bilinear expansion 1 2k + 1 X 1 Pk ( ) k=m+1 4 (k ? 0 ) : : : (k ? m ) which converges absolutely and uniformly both in and . The kernel (null space) of the operator ( + 0 ) : : : ( + m ) is the space Harm0;:::;m of all spherical harmonics of order m or less. Harm0;:::;m has the dimension M=

m X

n=0

(2n + 1) = (m + 1)2 :

With respect to the inner product (; )L we have the orthogonal decomposition Harm0;:::;m = Harm0 : : : Harmm : Furthermore, the space L2( ) resp. C (0)( ) is the completion of Harm0;:::;1 = span(Yn;j ) with respect to the L2( ){topology resp. C (0)( ){topology. A set XM = f1; :::; M g of M points 1 ; :::; M on is called Harm0;:::;m {unisolvent if the rank of the (M; M)-matrix 2

m 2n + 1 X

n=0

!

4 Pn(j k )

j=1;:::;M k=1;:::;M

(1.7)

is equal to M. The following statements are equivalent: (i) XM = f1; :::; M g is Harm0;:::;m {unisolvent, ? (ii) rank Yn;j (k ) n=0;:::;m;j =1;:::;2n+1;k=1;:::;M = M; (iii) the functions m 2n + 1 X

m X

2n + 1 P ( ) Pn(1); : : :; n M 4 n=0 4 n=0 are linearly independent.

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The equivalence "(i) , (ii)" follows from the addition theorem, while "(ii) , (iii)" is obtained using the fact that Z Pn(j k ) = 2n4+ 1 Pn(j )Pn (k )d!( ) :

Hence, (1.7) can be regarded as a matrix of Gram type. If XM is a Harm0;:::;m {unisolvent set, then we are able to interpolate given real numbers w1; :::; wM by a unique Y 2 Harm0;:::;m , i.e. Y (k ) = wk ; k = 1; :::; M.

1.3 Vector Spherical Harmonics

A function f : ! R3 is called spherical vector eld. By l2 ( ) we denote the space of (Lebesgue) square integrable vector elds on , i.e.

Z

l2 ( ) = f : ! R3 f() f()d!() < 1 :

l2( ) is a Hilbert space equipped with the inner product (f; g)l = 2

Z

f() g()d!()

1=2

:

c(k)( ) denotes the space of vector elds with k{times continously dierentiable components on . For a given vector eld f : ! R3 the eld fnorm : 7! fnorm () = ( f()); 2

is called the normal part of f, while ftan : 7! ftan () = f() ? fnorm (); 2

is called the tangential part of f. A vector eld f is called tangential (resp. normal), if f() = ftan() (resp. f() = fnorm ()) for all 2 . The study of vector elds on the sphere can be greatly simpli ed by the following decomposition theorem for continously dierentiable vector elds f : !

R3

where and

f() = fnorm () + ftan(); 2 ; (1) () + f (2) (); 2 ; ftan() = ftan tan (1) () r ftan (1) () ^ ) r (ftan (2) () r ftan (2) () ^ ) r (ftan

= r ftan (); = 0; = 0; = r (f() ^ ): Due to Backus (1966) (see also Meissl (1971a)) there exist uniquely determined functions Fi 2 C (2) ( ); i = 1; 2 satisfying

Z

Fi()d!() = 0; i = 1; 2;


such that

f (1) ()

tan (2) () ftan

= r F1(); 2 ; = ^ r F2(); 2 :

Freeden and Gervens (1993) give the explicit representations of Fi; i = 1; 2, in terms of Green's function G(; ; ), namely F1() =

Z

G(; ; )r (f() ? ( f()))d!()

Z

F2() = ?

G(; ; )r ( ^ f())d!()

Using the operators O(i) : c(1)( ) ! C (0)( ); i = 1; 2; 3, de ned by O(1)f() = f() O(2)f() = ?r (f() ? ( f())) O(3)f() = r ( ^ f()) we are able to rewrite the decomposition formula as follows fnorm () = O(1)f()

Z

f (1) ()

= ?r

f (2) ()

= ? ^ r

tan tan

G(; ; )O(2)f()d!()

Z

G(; ; )O(3)f()d!():

Corresponding to the operators O(i) we introduce operators o(i) : C (1) ( ) ! c(0) ( ); i = 1; 2; 3, by setting o(1) F()

= F() o(2) F() = r F() o(3) F() = ^ r F(): Then it is not dicult to show that for F 2 C (2) ( ) O(j ) (o(i) F()) = 0; j 6= i: Moreover, for i = j, we get O(1) (o(1)F()) = F() O(2) (o(2)F()) = ? F() O(3) (o(3)F()) = ? F(): By de nition, let 0 for i = 1 oi = 1 for i = 2; 3: Now, let Yn be a scalar spherical harmonic of order n. Then it follows that O(i) (o(i) Yn()) = n(i) Yn ()

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for all 2 and i = 1; 2; 3, and for n = oi ; oi + 1; : : :, where 1 for i = 1 ( i ) n = n for i = 2; 3: The vector elds yn(i) () = o(i)Yn (); 2 ; n = oi ; oi + 1; : : : are called vector spherical harmonics of order n and kind i. yn(1) describes a normal eld, while yn(2) ; yn(3) are tangential elds of order n. Obviously, according to our construction (cf. Freeden and Gervens (1993)), ^ yn(1) () yn(2) () yn(3) () ^ yn(2) () and

= = = =

0; 0; r ( ^ yn(2) ()) = 0; 0; r yn(3) () = 0; yn(3) ()

yn(i) () yn(j ) () = 0; i 6= j: (i) = ((i) )?1=2o(i) Y (), n = The vector elds yn;j n n;j oi; oi + 1; : : :, form a l2 {orthonormal system (fYn;j g being always assumed to be L2{orthonormal); more explicitly

Z

(i) () y(j ) ()d!() = : yn;k i;j n;m k;l m;l

(1.8)

The vector spherical harmonics of order n are eigenfunctions of the vectorial analogue to the Beltrami operator corresponding to the eigenvalues ?n = ?n(n + 1), that is (? ? n )yn(i) () = 0; 2 ;

(1.9)

where is de ned by

f() = f() ? 2( ^ r ) ^ f() ? 2f() (1.10) for f 2 c(2)( ). Consider the kernel

n+1 2n + 1 p(i;k)(; ) = 2X (i) () y(i) (); ; 2 : yn;j n n;j 4 j =1

In can be deduced that for every vector spherical harmonic yn(i) of order n and kind i the reproducing property 2n + 1 Z p(i;i)(; )y(i) ()d!() = y(i) (); 2 ; n n 4 n (1.11) is valid. Let t be an orthogonal transformation. Then it follows that pn(i;k)(t; t) = tpn(i;k)(; )tT for any pair of unit vectors , and i = 1; 2; 3. Therefore, pn(i;k)(; ) is invariant under orthogonal transformations. By straightforward calculations and observing the


structure of the tensor product (1.3) we obtain the following vectorial analogue of the addition theorem:

p(1n ;1)(; ) = Pn( ) ; p(1n ;2)(; ) = p 1 Pn0 ( ) [ ? ( )];

tr(p(3n ;3)(; )) = Pn( ) tr(p(1n ;3)(; )) = tr(p(3n ;1)(; )) = 0 tr(p(3n ;2)(; )) = tr(p(2n ;3)(; )) = 0: For all ; 2 and i; k; j 2 f1; 2; 3g, j pn(i;k)(; )"j j 1: In particular,

7

n(n + 1) p(1n ;3)(; ) = p 1 Pn0 ( ) ^ ; n(n + 1) 2X n+1 p(2n ;1)(; ) = p 1 Pn0 ( )[ ? ( )] ; yn;j (i) () 2 = 2n + 1 ; 2

n(n + 1) 4 1 j =1 (3 ; 1) 0 pn (; ) = p Pn( ) ^ ; n(n + 1) so that p(2n ;2)(; ) = n(n1+ 1) (i) 2n + 1 1=2 ; j = 1; : : :2n + 1: sup yn;j () 4 fPn00( )[ ? ( )] [ ? ( )] 2

+Pn0 ( )[i ? ? [ ? ( )] ]g; A vectorial analogue of the Hecke formula reads as follows: p(3n ;3)(; ) = n(n1+ 1) fPn00( ) ^ ^ Z 0 +Pn( )(( )i ? )g; H( )p(1n ;1)(; )d!() =

;2) p(3n ;2)(; ) = n(n1+ 1) fPn00( ) ^ [ ? ( )] H^ (1;1)(n)p(1n ;1)(; ) + H^ (1;2)(n)p(1 n (; ); Z +Pn0 ( )[( )i ? ]g; H( )p(2n ;2)(; )d!() =

1 p(2n ;3)(; ) = n(n + 1) fPn00( )[ ? ( )] ^ ;2) H^ (2;1)(n)p(2n ;1)(; ) + H^ (2;2)(n)p(2 n (; ); Z +Pn0 ( )(?j() ? ^ )g; H( )p(3n ;3)(; )d!() =

for all ; 2 , where the identity matrix i and the H^ (3;3)(n)p(3n ;3)(; ); surface rotation matrix j are given by 01 0 01 where the numbers H^ (i;j ) are given as follows i = @0 1 0A; (1.12) n + 1 H(n ^ + 1) + n H(n ^ ? 1); H^ (1;1)(n) = 2n 0 0 1 + 1 2n + 1 0 0 ? "3 "2 1 pn(n + 1) ^ ^ + 1) ? H(n ^ ? 1)); 3 1 @ A j() = " 0 ? " : (1.13) H(1;2)(n) = ? 2n + 1 (H(n 2 1 p ? " " 0 n(n + 1) (H(n ^ ^ + 1) ? H(n ^ ? 1)); H(2;1)(n) = ? 2n Several simple results can be derived directly from +1 n+1 ^ +2 ^ the addition theorem of vector spherical harmonics. H^ (2;2)(n) = 3n The identities 2n + 1 H(n + 1) + 2n + 1 H(n ? 1) ^ H^ (3;3)(n) = H(n); ( ^ ) ( ^ ) = ?(1 ? ( )2 ); ^ is de ned by (1.4). ( ? ( )) ( ? ( )) = ?( )(1 ? ( )2 ) and H(n) The cartesian components of vector spherical harenable us to calculate the trace of pn(i;k)(; ), ; 2 : monics of order n and kind 1 and 2 can be shown to be linear combinations of scalar spherical harmonics of tr(p(1n ;1)(; )) = Pn( ) order n ? 1 and n + 1, while the cartesian components 1 0 2 (1 ; 2) of a vector spherical harmonic of order n and kind 3 P ( )(1 ? ( ) ) tr(pn (; )) = p is a linear combination of scalar spherical harmonics of n(n + 1) n order n. We obtain two consequences, namely tr(pn(2;1)(; )) = p 1 Pn0 ( )(1 ? ( )2) n(n + 1) (1) (?) = (?1)n+1 y(1) (); 2 ; yn;j n;j 1 (2) (2) (); 2 ; n +1 ((1 ? ( )2 )Pn0 ( ) tr(pn(2;2)(; )) = p yn;j (?) = (?1) yn;j n(n + 1) (3) (?) = (?1)n y(3) (); 2 ; +n(n + 1)( )Pn( )) yn;j n;j


and

8

2.1 Tensor Fields

Z Z

Z

(1) ()d!() = 0 if jl ? nj 6= 1 Yl ()yn;j (2) ()d!() = 0 if jl ? nj 6= 1 Yl ()yn;j (3) ()d!() = 0 if l 6= n Yl ()yn;j

provided that Yl is a scalar spherical harmonic of order l. We conclude our excursion into vector spherical harmonic theory with a third signi cant consequence. Suppose that f is a continuous vector eld on . Then, of course, for any given " > 0, there exists a linear combination sN =

3 X N 2X n+1 X

l=1 n=0 j =1

l) Y "l d(n;j n;j

(1.14)

such that jjf ? sN jjc ", i.e. the system fYn;j "l j n = 0; 1; : : :; j = 1; : : :; 2n + 1; l = 1; 2; 3g is closed in c( ) (with respect to jj jjc). But we know already that for l = 1; 2; 3 Yn;j "l

=

3 nX +1 2X r+1 X

s=1 r=n?1 t=1

(s) y(s) : cr;t r;t

(1.15)

(i) g is closed in c( ) with This shows that the system fyn;j respect to jjjjc. Because of the norm estimate jjf jjl p (i) g in c( ) 4jjf jjc for f 2 c( ), the closure of fyn;j (i) g in with respect to jj jjc implies the closure of fyn;j c( ) with respect to jj jjl . Since the space c( ) is dense in l2( ) (with respect to jjjjl ), we nally obtain (i) g in l2( ) with respect to jj jj , the closure of fyn;j l which, by a functional analytic argument (cf. e.g. Davis (i) g in (1963)), is equivalent to the completeness of fyn;j (i) ) = 0 l2( ), i.e. for a given function f 2 l2 ( ), (f; yn;j for n = oi ; oi + 1; : : :; j = 1; : : :; 2n + 1, i = 1; 2; 3, implies f = 0. But this means, in particular, that 2

2

2

2

3 N 2n+1 X X X (i) ) y(i) = 0 (1.16) (f; yn;j lim f ? l n;j N !1 i=1 n=o j =1 2

i

l2

As usual (cf. e.g. Gurtin (1972)), a second order tensor f is understood to be a linear mapping that assigns each x 2 R3 a vector y 2 R3. The (cartesian) components Fij of f are de ned by Fij = "i (f "j ) = ("i )T (f "j ); (2.1) so that y = f x is equivalent to 3 X i y " = Fij (x "j ): j =1

We write f T for the transpose of f ; it is the unique tensor satisfying (f y) x = y (f T x) for all x; y 2 R3. Moreover, we write tr(f ) for the trace and det(f ) for the determinant of f . The dyadic (tensor) product x y of two elements x; y 2 R3 (cf. (1.2)) is the tensor that assigns to each u 2 R3 the vector (y u)x: (x y)u = (y u)x (2.2) for every u 2 R3. The inner product f g of two second order tensors f ; g is de ned by

f g = tr(f T g) =

3 X

i;j =1

Fij Gij ;

while

jf j = (f f )1=2 is called the norm of f . Given any tensor f and any pair x and y, x (f y) = f (x y):

(2.3) (2.4) (2.5)

This identity implies that ("i "j ) ("k "l ) = ik jl ; (2.6) so that the nine tensors "i "j are orthonormal. Moreover 3 X

(Fij "i "j )x =

i;j =1

3 X

i;j =1

Fij (x "j )"i = f x

(2.7)

holds for all elds f 2 l2 ( ). For more details about vector spherical harmonics and splines the interested reader is referred to Freeden and Gervens (1993) and the references therein.

and thus f 2 R33 with

2 Tensor Spherical Harmonics

In particular, the identity tensor i (cf. (1.12)) is given P 3 i by i = i=1 " "i . By a spherical second order tensor eld we mean a function f that assigns to each point 2 a second order tensor f () 2 R33.

Before we turn to the introduction of tensor spherical harmonics we deal with some basic settings about tensor elds.

f=

3 X

i;j =1

Fij "i "j :

(2.8)


By l2( ) we denote the space of all quadratic integrable tensor elds, i.e.,

Z 2 3 3 l ( ) = f : ! R f () f ()d!() < 1 :

With respect to the scalar product

Z

(f ; g)l = 2

f () g()d!()

i;j =1

the space l2( ) forms a Hilbert space. With c(k)( ) we denote the space of all tensor elds f with k continuously dierentiable components Fi;j . c(0) ( ) is the class of continuous tensor elds (i.e. continuous components). Each tensor eld f : ! R33 can be written as sum of elementary dyadic elds 3 X

i;j =1

Fi;j ()"i "j ; 2 :

(2.9)

For later use we introduce some abbreviations. Let

f 2 l2( ), g 2 c(1)( ), and F 2 C (2)( ), then, by de nition, we let

^ f () = ^

3 ? X

i=1

f ()"i "i

!

3 ? ? X = ^ f ()"i "i ; i=1 ! 3 X i ? T i

f () ^ =

=

r g() = =

" f ()"

i=1 3 X ? ? "i ^ f T ()"i ; i=1 ! 3 X i i g()(" " ) r i=1 3 ? X r (g()"i ) "i ; i=1

and

r r F() = 3 X

i=1

r ("i r F()) "i ;

r ^ r F() = 3 X

r ("i ( ^ r F())) "i ;

i=1 ^ r r F() =

i=1

^ r ("i r F()) "i ;

^ r ^ r F() = 3 X

i=1

1 Z 0X 3 @ Fi;j ()Gi;j ()A d!() =

f : 7! f () =

3 X

9

^ r ("i ( ^ r F())) "i :

r g() is the surface divergence of the tensor eld g at 2 , while r (g() ^ ) is the surface curl of the tensor eld g at 2 . For a given spherical tensor eld f , the eld 7! fnor () = ( T f ()) ; 2

is called the normal part of f , while for 2

7! fnor;tan() = (f T () ? ( T f ())); 7! ftan;nor() = (f () ? ( T f ())) ; respectively, is called the left normal/right tangential and left tangential/right normal part of f . For a given tensor eld f , the eld 7! ftan() = f () ? fnor () ? fnor;tan() ? ftan;nor (); 2 ; is said to be the tangential part of f (cf. Backus (1966)). A tensor eld is called tangential (resp. normal) if it coincides with its tangential (resp. normal) part. Obviously, ftan() = 0; T ftan() = 0; 2 : (2.10) Due to Backus (1966) a general tensor eld can be represented in a fashion analogous to the aforementioned vector decomposition, where nine scalar functions are involved in the representation. From these nine scalar functions, four functions are used for the representation of the tangential part. The following investigations present, in addition, the explicit structure of these scalar elds by using the concept of (scalar) Green's functions on with respect to (iterated) Beltrami derivatives. We identify fourth{order tensors with linear transformations on the space of all (second{order) tensors. Thus, a fourth{order tensor C : R33 ! R33 is a linear mapping of type C : g 7! f = Cg; f ; g 2 R33; where the components are de ned by Ci;j;k;l = "i C("k "l )"j = ("i "j ) C("k "l ): The mapping relation can be reformulated in its components as follows Fi;j =

3 X

k;l=1

Ci;j;k;lGk;l :


If Cî;j;k;l are the components of C with respect to a second orthonormal basis "^1 ; "^2; "^3 in R3 given by "î = t"i (t orthogonal), then it is not dicult to see that Cî;j;k;l =

3 X

m;n;p;q=1

tm;i tn;j tp;k tq;l Cm;n;p;q :

T ) of C is de ned by The transpose CT = (Ci;j;k;l T Ci;j;k;l = Ck;l;i;j ;

and we get the following property

?

?

tr f (Cg)T = tr (CT f )gT : As usual (cf. e.g. Gurtin (1972)), we de ne the trace of a fourth-order tensor C by tr(C) = tr(Ci;j;k;l) =

3 X

i;j =1

Ci;j;i;j :

Obviously,

tr(CT ) = tr(C): It is clear that the trace is invariant with respect to orthonormal basis transformations, i.e. tr(C) = tr(C^ ), where C^ is de ned as before If f ; g are second{order tensors we understand by the dyadic fourth-order tensor B of f and g

B = f g; Bi;j;k;l = Fi;j Gk;l ; the mapping assigning to each second{order tensor h the second{order tensor

Bh = tr(ghT )f = (g h)f : The transpose and trace of B are then given by BT = g f ; tr(B) = tr(fgT ) = f g: In analogy to (2.9), each fourth{order tensor possesses the following representation

B=

3 X

i;j;k;l=1

? ? Bi;j;k;l "i "j "k "l :

(2.11)

2.2 Decomposition Formula

Let us consider a tensor eld f 2 c(1) ( ). Of course, f can be decomposed by using the nine elementary dyadic tensors "i "j ; i; j = 1; 2; 3:

f () =

3 X

i;j =1

Fi;j ()"i "j =

3 X

i=1

fi () "i ;

(2.12)

10

where Fi;j : ! R are dierentiable P functions with Fi;j () = "i (f ()"j ) and fi () = 3j =1 Fi;j ()"j . Formula (2.12) can be used to reduce tensorial dierential or integral equations to scalar ones, but it has the drawback that essential properties (e.g. spherical symmetry) of the tensor elds are ignored. This diculty can be overcome by the already mentioned decomposition formula which will be established now. Starting point is the sum f = fnor + fnor;tan + ftan;nor + ftan (2.13) where, according to our construction, ^ fnor () = 0; fnor () ^ = 0; ^ fnor;tan() = 0; fnor;tan() = 0; (2.14) ftan;nor() ^ = 0; T ftan;nor() = 0; T () = 0; ftan ftan() = 0 for all 2 . It is easily seen that the decomposition is unique. The orthogonal projection operator P mapping a spherical tensor eld to its tangential part is introduced by P : f 7! P(f )() = ftan(); 2 : Explicitly written out we have Lemma 2.1 Let f 2 c(1)( ) be a tensor eld. There exist uniquely determined tangent vector functions

v1; v2 2 c(1) ( ) and a function F 2 C (1)( ) such that f () ? P(f )() = F() + v1 () + v2 () ; 2 ; (2.15) where

F() = T f (); v1 () = f T () ? F(); v2 () = f () ? F(): For each f 2 c(1) ( ), Lemma 2.1 enables us to express the projection operator P as follows: P(f )() = (i ? )T f ()(i ? ) = f () ? (f T ()) ?(f ()) + ( T f ()) : The spherical tangential identity operator itan is given by P(i)() = itan(); 2 : For any tangent vector eld f the relations f T ()itan() = f T (); itan()f() = f() T itan() = 0; itan() = 0 hold. Setting f = i we obtain P(i)() = itan() = i ? ; 2 :


Furthermore, it is easy to see that (2.13) and (2.15) represent orthogonal decompositions with respect to the scalar product (; )l . The calculation of the surface divergences 2

r (F()itan()) = r F() ? 2F(); r (F()itan() ^ ) = ^ r F(); resp. the surface curls

r (F()itan() ^ ) = ^ r F(); r (F()(itan() ^ ) ^ ) = 2F() ? r F() can be performed by elementary operations provided that F is continuously dierentiable on . According to our construction we have

Lemma 2.2 (fnor ; ftan)l = 0; (fnor ; fnor;tan)l = 0; (fnor;tan; ftan)l = 0; (fnor ; ftan;nor )l = 0; (ftan;nor ; ftan)l = 0; (fnor;tan; ftan;nor )l = 0: 2

2

2

2

2

2

From the vector theory we know that the vector elds v1 and v2 occuring in (2.15) can be decomposed in their surface divergence and surface curl parts, i.e., there exist functions Fi 2 C (2)( ); i = 1; : : :; 4, such that v1 () = r F1() + ^ r F2(); (2.16) v2 () = r F3() + ^ r F4(); where Fi ; i = 1; :::; 4 can be represented as described in the vector theory as follows F1 ()Z = G(; ; )r (v1 () ? (v1 () ))d!();

F2 () =Z ? G( ; ; )r ( ^ v1 ()))d!();

F3 ()Z = G(; ; )r (v2 () ? (v2 () ))d!();

F4 () =Z ? G( ; ; )r ( ^ v2 ()))d!();

and G( ; ; ) denotes the Green function de ned in (1.5). Combining (2.16) with (2.15) we nd

Theorem 2.1 Suppose that f 2 c(1)( ). Then there exist uniquely determined functions Fi 2 C (2)( ); i = 1; : : :; 5, with

Z

Fi ()d!() = 0; i = 2; :::; 5

11

such that for all 2

f () ? P(f )() = F1() + r F2() + ^ r F3() +r F4() + ^ r F5() ;

where

F1() = T f (); F2()Z = G(; ; )r (f T () ? F1())d!();

F3() =Z ? G(; ; )r ( ^ (f T ()))d!();

F4()Z = G(; ; )r (f () ? F1())d!();

F5() =Z ? G(; ; )r ( ^ (f ()))d!():

Next we want to decompose the tangential part P(f )(). It is not dicult to show (cf. Backus (1966)) that there exist scalar functions G1 ; :::; G4 2 C 2( ) such that (1) () + f (2) () + f (3) () + f (4) (); P(f )() = ftan tan tan tan where (1) () = G ()i (); ftan 1 tan (2) ftan() = G2()itan() ^ ; (3) () = 2P(r r G )() ftan 3 ? G3 ()itan(); (4) () = P((r ^ r G )() + ftan 4 ^ r r G4()); and (i) ; f (j ) ) = 0 for i 6= j: (ftan tan l Thereby, the scalar functions G3; G4 are orthogonal to the spherical hamonics of order 0 and 1. With this condition G1 ; : : :; G4 are uniquely determined by ftan. (3) In order to determine the tangential projections of ftan (4) we use and ftan Lemma 2.3 Assume that F 2 C (2)( ). Then r r F() = (r r F())T ? r F() + r F(); ^ r ^ r F() = ( ^ r ^ r F())T ? i ^ ( ^ r F()); r ^ r F() = ( ^ r r F())T ? i ^ r F(); ^ r r F() = (r ^ r F())T ? i ^ r F(): 2


Proof: Simple calculations in cartesian components

show

rx rxF(x) = (rx rx F(x))T ; (x ^ rx) (x ^ rx)F(x) = (x ^ rx x ^ rxF(x))T ? i ^ (x ^ rx F(x)); rx x ^ rxF(x) = (x ^ rx rxF(x))T ? i ^ rx F(x) for every F 2 C (2)( ). Thus, the usual process of se-

parating radial and angular parts provides us with the required result. Observe that the last equation can be obtained by forming the transpose of the matrices occuring in the third equation.

satisfy the orthogonal relations

0 = 0 = 0 = 0 =

Z

Z

such that (1) () + f (2) () + f (3) () + f (4) (); P(f )() = ftan tan tan tan

where (1) () = F ()i (); ftan 1 tan (2) ftan() = F2()itan() ^ ; (3) () = 2r r F () ftan 3 +2r F3 ? F3 ()itan(); (4) () = r ^ r F () ftan 4 + ^ r r F4() + 2 ^ r F4()

Z

Z

Z

(i) () f (j ) ()d!(); i 6= j; ftan tan (j ) ()d!(); j = 1; : : :; 4; fnor () ftan (j ) ()d!(); j = 1; : : :; 4; fnor;tan() ftan (j ) ()d!(); j = 1; : : :; 4: ftan;nor() ftan

o(1 ;1)F() = F() ; o(1 ;2)F() = r F(); o(1 ;3)F() = ^ r F(); o(2 ;1)F() = r F() ; o(3 ;1)F() = ^ r F() ; (2.17) (2 ; 2) o F() = F()(i ? ); o(3 ;3)F() = F()(i ^ ); o(2 ;3)F() = 2r r F() +2r F() ? F()(i ? ); o(3 ;2)F() = r ^ r F() + ^ r r F() + 2 ^ r F() : Theorem 2.3 Let f 2 c(1)( ). Then there exist uniquely determined functions Fi;j 2 C (2)( ); i; j = R 1; 2; 3; with Fi;j ()d!() =R 0 if (i; j) 2 f(1; 2); (2; 1); (1;3);(3; 1)g, resp. Fi;j ()d!() = 0 if (i; j) 2 f(2; 3); (3; 2)g, such that f=

Fi()d!() = 0; i = 3; 4;

Fi()d!() = 0; i = 3; 4;

Z

Theorem 2.1 and Theorem 2.2 allow the following reformulation by using the operators o(i;k) : C (2)( ) ! c(0)( ), introduced as follows

By right hand side multiplication with of the formulae written down in Lemma 2.3 we obtain (r r F()) = ?r F(); ( ^ r ^ r F()) = r F(); (r ^ r F()) = ? ^ r F(); ( ^ r r F()) = ? ^ r F(): Therefore we get from (2.16) P(r r F)() = r r F() + r F() and P(r ^ r F + ^ r r F)() = r ^ r F() + ^ r r F() +2 ^ r F() : Summarizing our results we obtain Theorem 2.2 Suppose that f 2 c(1)( ). Then there exist uniquely determined functions Fi 2 C (2)( ); i = 1; : : :; 4, with

12

and

3 X

i;j =1

o(i;j)Fi;j

o(i;j)Fi;j; o(k;l) Fk;l l = 0; (i; j) 6= (k; l): 2

By elementary calculations we nd

Lemma 2.4 The surface divergence of the tangential tensor elds o(i;j )F , (i; j) 2 f(2; 2); (3; 3); (2; 3); (3;2)g, respectively, reads as follows

r (o(2 ;2)F()) r (o(3 ;3)F()) r (o(2 ;3)F()) r (o(3 ;2)F())

= = = =

r F() ? 2F(); ^ r F(); r ( F() + 2F()); ^ r ( F() + 2F());


whereas the surface curl, respectively, is given by

r ( ^ o(2 ;2)F()) r ( ^ o(3 ;3)F()) r ( ^ o(2 ;3)F()) r ( ^ o(3 ;2)F())

= = = =

? ^ r F(); r F() ? 2F(); ^ r ( F() + 2F()); ?r ( F() + 2F());

provided that F is suciently often dierentiable.

In particular, we obtain as immediate consequence ;2) (r (o(2 F())) ;3) (r ( ^ o(3 F())) r r (o(2 ;3)F()) r r ( ^ o(3 ;2)F())

= = = =

?2F(); ?2F();

( + 2)F(); ? ( + 2)F();

for all 2 . These identities give rise to introduce the operators O(i;j ) : c2 ( ) ! C (0) ( ) by O(1;1)f () = T f (); O(1;2)f () = ?r (f T () ? O(1;1)f ()); O(1;3)f () = r ( ^ (f T ())); O(2;1)f () = ?r (f () ? O(1;1)f ()); O(3;1)f () = r ( ^ (f ())); (2.18) ? (2 ; 2) O f () = ? r (P(f )()) ; ? O(3;3)f () = ? r ( ^ P(f )()) ; O(2;3)f () = ? 2r r (P(f ) ? 12 itanO(2;2)f ()) ; O(3;2)f () = ? ?2r r ( ^ P(f ) ? 21 itanO(2;2)( ^ P(f ) :

The operators O(i;k) assign to each tensor f nine functions Fi;k determined by Theorem 2.3. It is easy to see that for F 2 C (2)( ) O(i;j )(o(r;s) F()) = 0;

if (i; j) 6= (r; s). Moreover, for (i; j) = (r; s), we get ;1) O(1;1)(o(1 F()) ;2) O(1;2)(o(1 F()) ;3) O(1;3)(o(1 F()) ;1) O(2;1)(o(2 F()) ;1) O(3;1)(o(3 F()) ;2) O(2;2)(o(2 F())

= = = = = =

F(); ? F(); ? F(); ? F(); ? F(); 2F();

13

O(3;3)(o(3 ;3)F()) = 2F(); O(2;3)(o(2 ;3)F()) = 2( ( + 1 ))F(); O(3;2)(o(3 ;2)F()) = 2( ( + 1 ))F(): In particular, we have for every scalar spherical harmonic Yn of order n O(i;j )(o(i;j )Yn ) = n(i;j )Yn ; n oi;j ;

(2.19)

where we have used the abbreviations oi;j =8

< 0 for (i; j) 2 f(1; 1); (2; 2); (3; 3)g (i; j) 2 f(1; 2); (2; 1); (1; 3);(3; 1)g ; : 12 for for (i; j) 2 f(2; 3); (3; 2)g

and ) n(i;j 8=

> > < > > :

1 2 n

for (i; j) = (1; 1) for (i; j) 2 f(2; 2); (3; 3)g for (i; j) 2 f(1; 2); (2; 1); : (1; 3); (3; 1)g 2n(n ? 1 ) for (i; j) 2 f(2; 3); (3; 2)g

We are able to generalize the de nition of O(i;j ) to fourth{order tensors in the following way. Let C be a fourth{order tensor. Then we set for 2

?

(2.20) O(i;j )C() = O(i;j )(Cr;s;k;l())r;s k;l : If C is a fourth{order tensor of the dyadic form C() = s() t() we have O(i;j )C() = (O(i;j )s())t(): Using (2.18) in Theorem 2.3 we see that every twice continuously dierentiable tensor eld de ned on can be expressed by its O(i;j ){components in the following way: Theorem 2.4 Let f 2 c( ). Then there exist uniquely determined scalar functions Fi;j 2 C 2( ); i; j = R 1; 2; 3, with Fi;j ()d!() =R 0 if (i; j) 2 f(1; 2); (2; 1); (1;3);(3; 1)g, resp. Fi;j ()d!() = 0 if (i; j) 2 f(2; 3); (3; 2)g, such that

f=

3 X

i;j =1

and

F1;1() = O(1;1)f (); F2;2() = 21 O(2;2)f (); F3;3() = 12 O(3;3)f ();

o(i;j)Fi;j


F1;2() = ? F2;1() = ? F1;3() = ?

Z Z

Z

Z

for n = oi;k ; oi;k + 1; : : :, forms a l2( ){orthonormal system, i.e.,

G(; ; )O(1;2)f ()d!();

Z

G(; ; )O(2;1)f ()d!();

G(; ; )O(1;3)f ()d!();

F3;1() = ? G(; ; )O(3;1)f ()d!(); Z

1 F3;2() = 2 G(( + 1 ); ; )O(3;2)f ()d!(); Z

1 F2;3() = 2 G(( + 1 ); ; )O(2;3)f ()d!():

A geodetically relevant example of this decomposition formula is given in Appendix A.

2.3 De nition of Tensor Spherical Harmonics

Let Yn be a scalar spherical harmonic of order n. For i; k = 1; 2; 3 and n = oi;k ; oi;k + 1; : : :, the tensor elds

yn(i;k)() = o(i;k)Yn (); 2

are called tensor spherical harmonics of order n and kind (i; k). yn(1;1) describes a normal tensor eld on

, while yn(1;2); yn(2;1); yn(1;3) and yn(3;1) are tensor elds consisting of combined normal and tangential components. Finally, according to Theorem 2.2, the elds yn(2;2); yn(2;3); yn(3;2) and yn(3;3) are tangential elds. Obviously, ^ yn(1;k)() ^ yn(2;1)() y(2;2)() ^

= = = n (i;k) T yn () =

0;

for k = 1; 2; 3;

n yn(i;k)() = 0;

if i; k = 2; 3:

yn(3;1)(); y(3;3)();

We denote by harmn(i;k) the space of all tensor spherical harmonics of order n and kind (i; k). harmn is de ned by

harm0 = harm(10 ;1) harm(20 ;2) harm(30 ;3); harm1 =

3 M

i;k=1 i;k)6=(2;3);(3;2)

harm1(i;k);

(2.21)

(

harmn =

3 M

i;k=1

harmn(i;k), if n 2:

Equipped with the scalar product (; )l the space harmn(i;k) is a (2nn+ 1){dimensional Hilbert space. o (i;k) : ! ( i;k ) The system yn;j of tensor elds yn;j R33, de ned by (i;k)() = ((i;k))?1=2o(i;k)Y () yn;j n;j n 2

14

(r;s) () y(u;t)()d!() = : yn;j r;u s;t n;m j;k m;k

In analogy to the scalar and vectorial theory it can be proved that the space harmn(i;k) is invariant in the sense that y 2 harmn(i;k) implies tT y(t)t 2 harmn(i;k) for all orthogonal transformations t. Furthermore, known from the theory of scalar and vector spherical harmonics, harmn(i;k) is irreducible, i.e. there exists no real invariant subspace of harmn(i;k). From this fact it follows that all results known from the theory of scalar or vector spherical harmonics which are based on these algebraic facts, carry over to the tensorial case. Given an element xn(i;k) 2 harmn(i;k), we consider the space w(i;k) of all linear combinations of elements tT xn(i;k)(t)t. Since w(i;k) is an invariant subspace of harmn(i;k), it follows that w(i;k) = harmn(i;k). Therefore we obtain

Lemma 2.5 Let xn(i;k) be an element of harmn(i;k). Then there exist 2n + 1 orthogonal transformations t1; : : :; t2n+1 such that every tensor spherical harmonic yn(i;k) of order n and kind (i; k) can be expressed in the form

yn(i;k) =

2X n+1

j =1

cj tTj xn(i;k) (tj )tj ;

where c1; : : :; c2n+1 are real numbers.

The rôle of the Legendre function will be played by the tensor eld (i;k) (; ) = o(i;k)P (); 2 ; pLeg;n n

(2.22)

which represents uniquely (apart from a multiplicative factor) that orthogonal invariant tensor eld satisfying (i;k) (t; ) = tT p(i;k) (; )t; pLeg;n Leg;n for all orthogonal transformations t having 2 un-

changed. In connection with Lemma 2.5 each tensor spherical harmonic can be expressed by means of Legendre functions. We obtain

Lemma 2.6 There exist 2n + 1 points 1; : : :; 2n+1 2

such that every tensor spherical harmonic yn(i;k) 2 harmn(i;k) can be written as yn(i;k) =

2X n+1

j =1

(i;k) (; ) cj pLeg;n j

with real numbers c1; : : :; c2n+1.


Next, we consider the fourth{order tensor n+1 2n + 1 P(i;k)(; ) = 2X (i;k)()) (y(i;k)()) (yn;j n;j 4 n j =1 (2.23) for ; 2 . Then, for every xed a 2 R33 and 2 , the tensor eld Pn(i;k)(; )a is an element of harmn(i;k), and for every element yn(i;k) 2 tn(i;k) we have using (2.23) the reproducing property 2n + 1 Z P(i;k)(; )(y(i;k)())T d!() n 4 n = =

2X n+1

j =1 2X n+1

Z (i;k)() tr?y(i;k)()(y(i;k)())T d!() yn;j n n;j

(y(i;k); y(i;k))

n;j j =1 yn(i;k)();

n

l

2

y(i;k)() n;j

= 2 : More explicitly, Z 2n + 1 ( i;k ) Pn(i;k)(; )(yn(i;k)())T d!(): yn () = 4

The determination of (2.23) in terms of scalar Legendre functions leads us to the addition theorem for tensor spherical harmonics. As an example, we just state the following formula. We have for ; 2 , and n 0 2X n+1 (2;2)() y(2;2)() = 2n + 1 P(2;2)(; ); (2.24) yn;j n;j 4 n j =1

;2) where the fourth order tensor P(2 n (; ) is de ned by Pn(2;2)(; ) = 21 Pn(; )(i ? ) (i ? ): Of special interest is the trace of Pn(i;k)(; ). For that purpose we rst show the following invariance property: Lemma 2.7 Let ; 2 . Then, for each orthogonal

transformation t, we have

tr(Pn(i;k)(; )) = tr(Pn(i;k)(t; t)): Proof: From the orthogonal invariance of harmn(i;k) (i;k)(t)t can be written as we know that tT yn;j (i;k)(t)t = tT yn;j

Now, we have

2X n+1

r=1

(i;k): cj;r yn;r

Z (i;k)()t)(tT y(i;k)()t)T d!() tr (tT yn;r n;s

2X n+1 2X n+1 Z (i;k)()y(i;k)()d!() = tr cr;l cs;l0 yn;l n;l0 l=1 l0 =1

2X n+1 =

l=1

cr;l cs;l :

15

On the other hand,

Z (i;k)()t)(tT y(i;k)()t)T d!() = tr (tT yn;r n;s

Z

(i;k)()(y(i;k) ())T d!() = r;s : tr yn;r n;s

Combining these relations, we get that the coecients cr;l are the components of an orthogonal matrix 2X n+1

l=1

cr;l cs;l = r;s :

This fact can be used to complete the proof: tr

2n + 1 4

= = = =

2X n+1

j =1 2X n+1

P(i;k)(t; t) n

(i;k)(t))(y(i;k)(t))T tr (yn;j n;j

(i;k)(t)t)(tT y(i;k)(t)t)T tr (tT yn;j n;j

j =1 2X n+1 2X n+1 2X n+1

(i;k)())(y(i;k)())T cj;l cj;l0 tr (yn;l n;l0

j =1 l=1 l0 =1 2n + 1 ( i;k ) tr 4 Pn (; ) :

Lemma 2.7 means that the scalar function tr(Pn(i;k)(; )) depends only on the scalar product . Therefore, the scalar function 7! tr(Pn(i;k)(; )) is constant. Hence, there exists a number n 2 R such that tr(Pn(i;k)(; )) = n for all 2 , i.e., 2X n+1

j =1

(i;k)())(y(i;k)())T ) = ; 2 : tr((yn;j n n;j

Integrating both sides over the unit sphere we get

Lemma 2.8

2X n+1

j =1

(i;k)() y(i;k)() = 2n + 1 : yn;j n;j 4

Now, let

yn(i;k)() =

2X n+1

j =1

(i;k)(); a = (y(i;k); y(i;k)) : aj yn;j j n n;j l 2


The application of the Cauchy-Schwarz inequality yields

j yn(i;k)() j2

2X n+1

j =1

a2j

2n4+ 1

P

2X n+1

j =1 2X n+1 j =1

(i;k)() j2 j yn;j

a2j :

n+1 a2 = (y(i;k); y(i;k)) . Thus we are Note that 2j =1 n n l j able to derive the following estimate for tensor spherical harmonics: 2

Lemma 2.9

r

1=2 sup j yn(i;k)() j 2n4+ 1 yn(i;k); yn(i;k) l : 2

2

We know already (cf. Section 1.3) that each vector spherical harmonic of order n can be characterized componentwisely by certain linear combinations of scalar spherical harmonics. The operator (resp. r ; ^ r ) transforms a spherical harmonic of order n in those of order n ? 1 and n + 1 (resp. n ? 1 and n + 1, n). If we apply one of the operators ; r ; ^ r once again to construct tensor spherical harmonics we see that for (i; k) 2 f(1; 1); (1; 2); (2; 1); (2;2); (2;3)g the tensor eld yn(i;k) is a linear combination of scalar spherical harmonics of order n ? 2; n and n + 2 in each component while for (i; k) 2 f(1; 3); (3; 1); (3; 3); (3;2)g the tensor eld yn(i;k) componentwisely forms a linear combination of scalar spherical harmonics of order n ? 1 and n + 1. From this fact we are able to obtain Lemma 2.10 Let 2 . Then yn(i;k)(?) = (?1)n yn(i;k)() if (i; k) 2 f(1; 1); (1; 2); (2;1);(2; 2); (2;3)g, and yn(i;k)(?) = (?1)n+1 yn(i;k)() if (i; k) 2 f(1; 3); (3; 1); (3;3);(3; 2)g. Furthermore, we have Lemma 2.11 Let Ym be a scalar spherical harmonic of order m. Then

Z

Ym ()yn(i;k)()d!() = 0;

Finally, if we apply O(i;k)O(i;k) to the fourth{order tensor Pn(i;k)(; ), we get for all ; 2

O(i;k)(O(i;k)Pn(i;k)(; )) = n(i;k)Pn( ) for n = oi;k ; oi;k + 1; : : :. A tensorial analogue to the well known Hecke formula can be derived, if both sides of the usual Hecke formula (1.3) are multiplied with the operator O(i;k). We obtain Lemma 2.12 Let H be a function of class L1[?1; +1]. Then, for all ; 2 ,

Z

(i;k) (; )d!() = H(n) (i;k) (; ): ^ pLeg;n H( )pLeg;n

In what follows we study closure and completeness of the system of tensor spherical harmonics in the topology of the spaces c( ) and l2( ), respectively. First we mention n (i;k)o is closed in c( ). Lemma 2.13 The system yn;j

Proof: Let f be of class c( ). Then from the closure of scalar spherical harmonics it is clear that the tensorial system Yn;j "i "k forms a closed system in c( ) in the sense that for every " > 0 there exist N 2 N and (i;k) 2 R such that dn;j

3 N 2n+1

f ? X X X d(i;k)Y "i "k

": (2.26)

i;k=1 n=o j=1 n;j n;j

c

i;k

From Lemma 2.11 it can be deduced that each Yn;j "i

"k can be represented by nite linear combinations of tensor spherical harmonics of order n ? 2; : : :; n+2, i.e., Yn;j "i "k =

3 nX +2 X r X

o;p=1 r=n?2 m=1

o;p) y(o;p) : c(r;m r;m

(2.27)

Combining (2.26) and (2.27), we have the required result. As an immediate consequence we obtain

o

n

(i;k) is closed and conLemma 2.14 The system yn;j sequently complete in l2( ) with respect to k kl . Proof: According to standard arguments in Functio2

if (i; k) 2 f(1; 1); (1; 2); (2; 1); (2;2); (2;3)g and m 6= n ? 2; n; n + 2, or if (i; k) 2 f(1; 3); (3; 1); (3; 3);(3; 2)g and m 6= n ? 1; n + 1.

Applying the operator O(i;k) to tensor spherical harmonics we get using (2.19) O(k;l) yn(k;l) () = (nk;l) Yn ; n oi;k :

16

(2.25)

nal Analysis (cf. e.g. Davis (1963)), Lemman 2.14 ois (i;k) , equivalent to the completeness of the system yn;j (i;k)) = 0 for all n = i.e., f 2 l2( ) with (f ; yn;j l oi;k ; oi;k +1; : : :; j = 1; : : :; 2n+1; i; k = 1; 2; 3, implies that f = 0. 2

Lemma 2.14 is equivalent to the fact that every tensor


eld f 2 l2( ) can be written as orthogonal expansion in terms of tensor spherical harmonics. More explicitly,

3 3 N 2n+1

X X X X

(i;k)) y(i;k)

= 0:

f ? lim ( f ; y n;j l n;j

N !1 i=1 k=1 n=o j =1 i;k

2

l

2

At the end of this section, we want to de ne the tensorial analogue of the Beltrami operator . So we look for a dierential operator n (i;k)o for spherical tensor elds represents the eigenfuncsuch that the system yn;j tions of this operator corresponding to the spectrum f?n jn = oi;k ; oi;k + 1; : : :g. For the de nition of this ^ , we use the (usual) Belnew operator, denoted by trami operator and its vectorial counterpart . ^ for the normal part of a We start by de ning (2) tensor eld f 2 c ( ), decomposed as in (2.13). We set ^ (1) nor f () = T fnor () ;

(2) T () ; ^ nor;tan f () = T fnor ^ (3) tan;nor f () = fnor () :

Then it is clear, that the sum of these three opera^ nor , ^ nor;tan , and ^ tan;nor, mapping c(2)( ) tors (i;k), into c(0)( ), has the tensor spherical harmonics yn;j (i; k) 2 f(1; 1); (1; 2); (1;3); (2; 1);(3; 1)g, as its eigenfunctions to the eigenvalue ?n = ?n(n + 1), n = oi;k ; oi;k + 1; : : :. Since the tangential tensor eld ftan can be written as a nite sum of tensor elds in dyadic form, it is ^ tan of ^ for enough to de ne the tangential part tangential tensor elds of dyadic form. Hence, assume that ftan is given by ftan() = u() v(), where u and v are tangential vector elds. Then we de ne

^ tan

^ tan to the eigenvalue ?n and that the de nition of is independent on the decomposition of the tangential tensor eld in dyadic form. Hence, the de nition ^ := ^ nor + ^ nor;tan + ^ tan;nor + ^ tan provides the tensorialnanalogue o of the Beltrami operator. Since the (i;k) is closed in l2( ), it is clear that all system yn;j eigenfunctions are found.

3 Tensor Spherical Splines The theory of tensor spherical harmonics will be used now to develop the tensor spherical spline theory.

3.1 Sobolev Spaces

Let A denote the linear space of all sequences fAng of all real numbers An, n = 0; 1; : : :. De nition 3.1 Let fAng ; fBng 2 A. The sequence fAn g is called fBn g{summable, if An 6= 0 for n = 0; 1; : : : and 1 2n + 1 B X n < 1: 4 An 2 n=0

Of special interest are sequences fAn g that are f(n + 1=2) g{summable for a real number . For i; k = 1; 2; 3 we consider the series expansion

f (i;k)() =

^ tan ftan() f () =

where

tan (u() v()) = u() v() + u() v() + ? 2 ( ^ u() ^ v()) + 2 r u() + u() T ?r v() + v() ; and where r u() is de ned analogously to (2.10) by r u() =

3 X ? r "i u() "i :

i=1

It can be proved that the tensor spherical harmo(i;k), i; k = 2; 3 are the eigenfunctions of ^ tan nics yn;j

n=oi;k j =1

j f (i;k)() j

(i;k)) y(i;k) A?n 1(^f (i;k); yn;j l n;j 2

1 2n + 1 X 1=2 j An j?2

n=oi;k 4 n+1 1 2X X (i;k))2 1=2 < 1; (^f (i;k); yn;j 2 l n=oi;k j =1

^ tan (u() v()) ;

^

n+1 1 2X X

for 2 , corresponding to a function ^f (i;k) 2 l2( ). Then, by virtue of the Cauchy-Schwarz inequality, we get for all 2 ;

=

17

provided that fAng is f(n + 1=2) g{summable ( 0). In other words, h and h(i;k); i; k = 1; 2; 3; de ned by

n

h(i;k) = f (i;k) j f (i;k) = n+1 1 2X X

n=oi;k j =1

o (i;k)) y(i;k); ^f (i;k) 2 l2( ) ; A?n 1 (^f (i); yn;j l n;j 2

and

h=

n

f jf =

3 X

i;k=1

o

f (i;k); f (i;k) 2 h(i;k); i; k = 1; 2; 3 ;


are linear subspaces of the space c(0) ( ) of all continuous tensor elds, on which we are able to de ne the structure of a separable Hilbert space by introducing the inner product for elements f (i;k) and g(i;k) de ned by ^f (i;k) 2 l2( ) resp. g^ (i;k) 2 l2( ) as n+1 1 2X X

f (i;k) = g(i;k)

n=oi;k j =1 n+1 1 2X X

=

n=oi;k j =1

(i;k)) y(i;k); A?n 1 (^f (i;k); yn;j l n;j

(

n=oi;k j =1 3 X

(f ; g)h =

n;j

l

n;j

2

)

)

)

n+1 1 2X X

(i;k)()) (A?1 y(i;k)()): (A?n 1 yn;j n n;j

(

)

Thus, a neccessary and sucient condition that h(i;k) have a reproducing kernel is ful lled (cf. Davis (1963)).

o

Theorem 3.1 Let fAng be (n + 1=2)2oi;k { summable. Then the space h(i;k) is a Hilbert subspace of c(oi;k ) ( ). Each space h(i;k); i; k = 1; 2; 3 possesses a uniquely determined reproducing kernel

Kh i;k (; ) = =

)

1 X

n=oi;k

j An j?2 2n4+ 1 Pn(i;k)(; )

n+1 1 2X X

n=oi;k j =1

(i;k)() y(i;k)(); j An j?2 yn;j n;j

for ; 2 in the following sense:

)

(

)

i;k=1

Kh i;k (; ); ; 2 ; (

)

in the following sense: (i) For all 2

)

(ii) For every 2

)

n

3 X

(

Suppose now, in addition, that for i; k 2 f1; 2; 3g, fAn g is f(n + 1=2)2oi;k g{summable. Then the tensor functions O(i;k)Kh i;k (; ) are uniformly bounded. More explicitly, 1 X j O(i;k)Kh i;k (; ) j 2n4+ 1 n(i;k)A?n 2 : (3.1) n=0

(

Kh (; ) =

O(i;k)Kh (; ) = O(i;k)Kh i;k (; ) 2 h

)

(

(

O(i;k)Kh r;s (; ) = 0; (i; k) 6= (r; s):

The addition theorem (2.23) yields 1 X A?n 2 2n4+ 1 Pn(i;k)(; ): Kh i;k (; ) = n=oi;k (

ducing kernel

2

(

?

(i;k)() y(i;k): j An j?2 O(i;k)yn;j n;j

Corollary 3.1 Let fAng be f(n + 1=2) g{summable with 4. Then the space h is a Hilbert subspace of c(0) ( ). h possesses the uniquely determined repro-

2

(f (i;k); g(i;k))h i;k = (^f ; g^ )l : (

)

?

Furthermore, consider the kernel Kh i;k : ! by Kh i;k (; ) = n=oi;k j =1

(

1 2X n+1 X

O(i;k)f (i;k)() = O(i;k)Kh i;k (; ); f (i;k) h i;k (3.2) for i; k = 1; 2; 3.

)

R3 de ned (

O(i;k)Kh i;k (; ) =

)

(ii) For every f (i;k) 2 h(i;k) and all 2 the reproducing property holds

A?1 (^g(i;k); y(i;k)) 2 y(i;k);

(i;k)) (^ (i;k) (^f (i;k); yn;j l g(i;k); yn;j )l

i;k=1

(

n=oi;k j =1

by setting (f (i;k); g(i;k))h i;k = n+1 1 2X X

(i) For any xed 2 , O(i;k)Kh i;k (; ) 2 h(i;k), where

2

n

18

(

(iii) For every f 2 h and all 2

?

)

O(i;k)f () = O(i;k)Kh (; ); f h ; i; k = 1; 2; 3: ) The space hm(i;k) = harmo(i;k i;k ;:::;m ; m oi;k ; i; k = 1; 2; 3; of all tensor spherical harmonics of order m and kind (i; k) is a nite{dimensional Hilbert space with the semi{inner product (; )hmi;k corresponding to the norm (

kf (i;k)khmi;k = (

)

n+1 m 2X X n=oi;k j =1

)

(i;k))2 j An j2 (f (i;k); yn;j l

2

1 2

:

hm(i;k) has the reproducing kernel Khmi;k (; ) = (

)

n+1 m 2X X

n=oi;k j =1

(i;k)() y(i;k)(): j An j?2 yn;j n;j

Let us denote by hm(i;k)? the orthogonal complement of hm(i;k) in h(i;k). Then the linear space hm(i;k)?


is a Hilbert space with the inner product (; )hmi;k ? corresponding to the norm kf (i;k)khmi;k ? = (

(

)

1 2X n+1 X n=m+1 j =1

h(i;k)? m

(i;k))2 j An j2 (f (i;k); yn;j l2

1 2

)

:

(

)

(

h(i;k)?

)

(

)

(

(

with

(

)

(

)

(

)

)

)

)

(

)

(

)

(

)

(

3 X

(f (i;k); g(i;k))hmi;k ? (

i;k=1 ( i;k ( i;k ) ( i;k ) for f ; g 2 hm )?, i; k = 1; 2; 3, and 3 X

Kh?m (; ) =

i;k=1

)

Khmi;k ? (; ): (

)

3.2 De nition of Tensor Spherical Splines ) The linear space harmo(i;j i;j ;:::;m of all tensor spherical harmonics of order m and kind (i; j) has the dimension M ? oi;j . Consequently, the linear space

harm0;:::;m =

m M n=0

harmn ; m 2;

[3

XM =

i;j =1

XM(i;j ):

Therefore, XM is a subset of consisting of at least M and of at most 9M ? 12 points. For i; j 2 f1; 2; 3g, the system XM(i;j ) is called harmo(i;ji;j);:::;m{unisolvent, if the matrix m 2n + 1 X Pn ((i;j ) (i;j )) n=oi;j

k

4

l

l;k

is regular. The following statements are equivalent: ) (i) XM(i;j ) is harmo(i;j i;j ;:::;m {unisolvent, (ii) the matrix

m X

)? 2n + 1 p(i;j ) ((i;j ); (i;j )) O(i;j i;j 4 Leg;n k l l;k n=oi;j l

)

3 < = X (i;k); f (i;k) 2 h(i;k)?; i; k = 1; 2; 3 f f j f = m : ; i;k=1

(f ; g)h?m =

9(2n + 1) = 9M ? 12; m 2:

o

)

)

)

(

(

(

)

(

n=2

We set

Hence, m and (; )h i;k = (; )hmi;k + (; )hmi;k ? : For i; k 2 f1; 2; 3g; we denote by projhmi;k , resp. projhmi;k ? , the projection operators in h(i;k) onto hm(i;k), resp. hm(i;k)?, projhmi;k : h(i;k) ! hm(i;k); f 7! projhmi;k (f ); projhmi;k ? : h(i;k) ! hm(i;k)?; f 7! projhmi;k ? (f ): The operators are linear and bounded. The kernels projh?m1i;k (0) and projh?m1i;k ? (0) are as h(i;k){subspaces orthogonal to hm(i;k) and hm(i;k)?, respectively. hm(i;k) is a closed linear subspace of h(i;k), and it is clear that hm(i;k) = (hm(i;k)?)?: Using the reproducing property of the kernels we nd for f (i;k) 2 h(i;k); 2 : projhmi;k (f (i;k))() =(Khmi;k (; ); f (i;k))hmi;k ; projhmi;k ? (f (i;k))() =(Khmi;k ? (; ); f (i;k))hmi;k ? : We let h?m =8 9 (

m X

n

(i;k)() y(i;k)(): j An j?2 yn;j n;j

n=m+1 j =1 h(i;k) = hm(i;k)

3+73+

(i;j ) ) XM(i;j ) = o(i;j i;j ; : : :; M ?1 ; i; j = 1; 2; 3:

)

n+1 1 2X X

of all tensor spherical harmonics of order m or less (cf. (2.21) is of dimension

Now, for i; j 2 f1; 2; 3g consider the subset XM(i;j ) of M ? oi;j points on the unit sphere

has the reproducing kernel Khmi;k ? (; ) = (

19

is regular, (iii) the matrix

? O(i;ji;j) O(i;ji;j) 2n4+ 1 Pn(i;j )(k(i;j ); l(i;j )) l;k l n=oi;j k

m X

(

)

(

)

is regular. The system XM is called harm0;:::;m -unisolvent, if the nine matrices m 2n + 1 X (i;j ) (i;j ) ) ; i; j = 1; 2; 3 P ( n k l l;k n=oi;j 4 are regular. Assuming that XM is harm0;:::;m { unisolvent, any tensor spherical harmonic y of order m can be written as

y() =

3 M X X?1

i;j =1 k=oi;j

for 2 .

ck(i;j )

m 2n + 1 X (i;j ) (i;j ) 4 pLeg;n(; k )

n=oi;j


Now, we start with the precise treatment of tensor spline interpolation. For i; j 2 f1; 2; 3g; a set XN(i;j )

; N M; given by

Lemma 3.1 If f (i;j) 2 IN(i;j)(w(i;j)); s(i;j) 2 Sm (XN(i;j ) ), then

is called

(

oi;j ;:::;m {admissible,

if it contains a oi;j ;:::;m {unisolvent subset. In the following, we ) assume that XM(i;j ) XN(i;j ) is harmo(i;j i;j ;:::;m {unisolvent. This is always achievable by reordering. Therefore it makes sense to call a set

harm(i;j)

XN =

[3

i;j =1

XN(i;j ) ; XN(i;j ) ; N M;

harm0;:::;m {admissible, if it contains XM as harm0;:::;m {unisolvent subset. De nition 3.2 Given a harmo(i;ji;j);:::;m{admissible system XN(i;j ), then any function s(i;j ) 2 h(i;j ); i; j = 1; 2; 3; of the form

s(i;j) = p(i;j) +

NX ?1 k=oi;j

ak(i;j )O(i;ji;j) Khmi;j ? (k(i;j ); ) (

k

)

(

)

with p(i;j ) 2 hm(i;j ) is called tensor spherical spline in h(i;j) relative to XN(i;j) , if the coecients ak(i;j) satisfy the linear system

PN ?1 a(i;j)O(i;j) y(i;j)((i;j)) = 0; k=oi;j k ki;j n;l k (

)

n = oi;j ; : : :; m; l = 1; :::; 2n+ 1: If XN is a harm0;:::;m {admissible system, then, s() = P3i;j=1 s(i;j)() is called tensor spherical spline in h relative to XN . The space of all tensor spherical splines in h(i;j ) relative to XN(i;j ) is denoted by Sm (XN(i;j ) ), whereas Sm (XN ) denotes the set of all tensor spherical splines in h relative to XN . Now let there be given the data points ( i;j (k ); wk(i;j )) 2 XN(i;j ) R; i; j = 1; 2; 3; k = ) oi;j ; : : :; N ? 1; corresponding to a harmo(i;j i;j ;:::;m { admissible system XN(i;j ) . For xed i; j 2 f1; 2; 3g, we consider the spline problem of nding the "hm(i;j )?{ smallest interpolant" kf (i;j )khmi;j ? ; ks(i;j )khmi;j ? = i;j inf i;j i;j (

where

f

)

(

)

2IN( ) (w( ) )

(

)

n

IN(i;j )(w(i;j )) = f 2 h(i;j ) j o O(i;ji;j) f (k(i;j )) = wk(i;j ); k = oi;j ; : : :; N ? 1 : (

k

)

First we prove some lemmata. We start with

NX ?1

(s(i;j ); f (i;j ))hmi;j ? =

(i;j ) ) XN(i;j ) = fo(i;j i;j ; : : :; N ?1g

harm(i;j)

20

)

k=oi;j

ak(i;j )wk(i;j ):

Proof: It follows from (3.2) that NX ?1

k=oi;j

ak(i;j )O(i;ji;j) f (i;j )(k(i;j ) ) = (

NX ?1

k

)

?

(

k=oi;j NX ?1

+

ak(i;j ) O(i;ji;j) Khmi;j (k(i;j ); ); f (i;j ) h i;j k

?

(

)

)

(

)

ak(i;j ) O(i;ji;j) Khmi;j ? (k(i;j ); ); f (i;j ))h i;j : k k=oi;j (

(

)

)

(

)

Now the coecients ak(i;j ) satisfy the linear system, where PN ?1 a(i;j)O(i;j) y(i;j)((i;j)) = 0; k=oi;j k ki;j n;l k n = oi;j ; : : :; m; l = 1; : : :; 2n + 1: Hence, (s(i;j ); f (i;j ))hmi;j ? = (

NX ?1

(

)

)

?

ak(i;j ) O(i;ji;j) Khmi;j ? (k(i;j ) ; ); f (i;j) hmi;j ? : k k=oi;j This is the desired result. (

(

)

)

(

)

Lemma 3.2 Let the data points (k(i;j) ; wk(i;j)) 2 XN(i;j ) R; i; j = 1; 2; 3 corresponding to a harmo(i;ji;j);:::;m{admissible system XN(i;j) be given. Then Sm (XN(i;j ) ) \ IN(i;j )(w(i;j )) consists of only one element, denoted brie y by sN(i;j ): If N = M , then Sm (XN(i;j )) \ (i;j ) (i;j )) . ) IN(i;j )(w(i;j )) = harmo(i;j i;j ;:::;m \ IN (w Proof: The conditons O(i;j)sN(i;j)(k(i;j) ) = wk(i;j), k = oi;j ; : : :; N ? 1 give us the linear equations wk(i;j ) =

m 2X n+1 X

(i;j )O(i;j ) y(i;j )((i;j )) cn;l i;j n;l k (

)

(3.3)

k n=o(i;j) l=1 NX ?1 ) ? (i;j) (i;j ); (i;j )) : ak(i;j ) O(i;j + (i;j ) O (i;j ) Kh(i;j )? (l k m l k l=oi;j

The linear equations PN ?1 a(i;j)O(i;j) y(i;j)((i;j)) = 0; k=oi;j k ki;j n;l k n = oi;j ; : : :; m; l = 1; : : :; 2n + 1; (

)

(3.4) (3.5)


provide M ? oi;j further equations. We have to show that the whole linear system is non{singular. This will be established, if it can be shown that the corresponding homogeneous system possesses only the trivial solution in which all coecients vanish. To prove this, let ^s(i;j ) 2 Sm (XN(i;j ) ) \ IN(i;j )(0). From Lemma 3.1 we get (i;j ) ) k^s(i;j )khmi;j = 0, i.e. ^s(i;j ) 2 harmo(i;j i;j ;:::;m \ IN (0). With our assumptions imposed on XN(i;j ) we get ^s(i;j ) = 0; i; j = 1; 2; 3; as required. (

)

As immediate consequences we nd Lemma 3.3 (First minimum property) If f (i;j) 2 IN(i;j )(w(i;j )), then

kf (i;j )k2hmi;j ? = ksN(i;j )k2hmi;j ? + ksN(i;j ) ? f (i;j )khmi;j ? : (

)

(

(

)

(

(

)

Theorem 3.2 The spline interpolation problem kf (i;j )khmi;j ? ksN(i;j )khmi;j ? = i;j inf f 2INi;j (w i;j ) (

)

(

)

(

(

)

N

=

3 m X X

n=oi;j l=1 NX ?1

+

k=oi;j

n;l

(

)

(

)

(i;j ); a(i;j ) satisfy the linear equawhere the coecients cn;l k tions (3.3) and (3.4). Corollary 3.2 sN 2 h given by

sN =

3 X

i;j =1

n=oi;j

(

)

) Then every y(i;j ) 2 harmo(i;j i;j ;:::;m can be represented as follows ( 2 )

y(i;j)() =

MX ?1 ? k=oi;j

O(i;ji;j) y(i;j )(k(i;j )) lk(i;j )(): (

k

)

Therefore, for every f (i;j ) 2 h(i;j ), the unique harmo(i;ji;j);:::;m{interpolant phmi;j f (i;j) of f (i;j) for the harmo(i;ji;j);:::;m{unisolvent set XM(i;j) is given by the "Lagrange form" (

phmi;j f (i;j ) = (

)

M ? X k=oi;j

)

O(i;ji;j) f (i;j )(k(i;j )) lk(i;j ): (

)

k

The mapping phmi;j : h(i;j ) ! hm(i;j ) is a continuous linear projector of h(i;j ) c(0) ( ) onto hm(i;j ). Hence, phmi;j determines the decomposition of h(i;j ) as direct sum h(i;j ) = hm(i;j ) h_ m(i;j )?; where (

)

)

n

ak(i;j )O(i;ji;j) Khmi;j ? (k(i;j ); ); k

m 2n + 1 X (i;j ) (i;j ) 4 pLeg;n(; r ) (3.6)

k

(

c(i;j )y(i;j ) n;l

r=oi;j

(i;j ) ck;r

r;k = O(i;ji;j) lr(i;j )(k(i;j )):

)

is well{posed in the sense that its solution exists, is unique and depends continuously on the data wk(i;j ); k = oi;j ; : : :; N ? 1. The uniquely determined solution sN(i;j ) is given in the explicit form

s(i;j)

MX ?1

for k = oi;j ; : : :; M ? 1, satisfying the conditions

)

Lemma 3.3 and Lemma 3.4 follow by straightforward calculation. Summarizing our results we therefore obtain )

lk(i;j) =

)

(

(

Numerical experiences have shown that the linear systems stated above tend to be ill{conditioned unless m and N are both small, and this may cause diculties if the attempt is made to solve these systems directly. Therefore we are now interested in giving an alternate formulation of the solution sN(i;j ) that is more suitable for computational purposes. To this end we remember that, given (i; j), there exists in hm(i;j ) a unique basis lo(i;ji;j); : : :; lM(i;j?)1 of the form

)

Lemma 3.4 (Second minimum property) If f (i;j) 2 IN(i;j )(w(i;j )); s(i;j ) 2 Sm (XN(i;j ) ); then ks(i;j ) ? f (i;j )khmi;j ? = ksN(i;j ) ? f (i;j )k2hmi;j ? + ks(i;j ) ? sN(i;j )khmi;j ? :

21

sN(i;j)

is the unique solution of the interpolation problem

ksN kh?m = inf kf kh?m ; f P where f = 3i;j =1 f (i;j ) and f (i;j ) 2 IN(i;j )(w(i;j )).

h_ m(i;j)? = f_ (i;j) 2 h(i;j) j (3.7) o O(i;ji;j) f_ (i;j )(k(i;j )) = 0; k = oi;j ; : : :; M ? 1 : (

)

k

Consequently, any f (i;j ) 2 h(i;j ) admits the unique representation f (i;j) = ph i;j f (i;j) + f_ (i;j); f_ (i;j) 2 h_ m(i;j)?: (3.8) (

m

)

h_ m(i;j)? de ned by (3.7) and equipped with the scalar product (; )hmi;j ? is a Hilbert function subspace of c(0)( ). Furthermore it can be readily seen that for 2 (cf. Freeden and Hermann (1986)) (

)

?f_ (i;j); O(i;j)K

(i;j ) (i;j ) h_ mi;j ? (; ) hmi;j ? = O f_ (); (

)

(

)

(3.9)


3.3 Convergence

where the tensor Kh_ mi;j ? (; ); is given by (

)

harm0;:::;m {admissible system XN = SFor3 any ( i;j ) i;j =1 XN we de ne the XN {width by

Kh_ mi;j ? (; ) = Khmi;j ? (; ) (

)

?

(

MX ?1

lr(i;j)() ghmi;j ? (r(i;j); )

r=oi;j MX ?1

?

(

r=oi;j k=oi;j

where

2XN(

Ghmi;j ? (r(i;j ); k(i;j ))lk(i;j )() lr(i;j )() (

)

(

)

n=m+1 r=1 (

)

(

n=m+1 r=1

?

?

(i;j )() O(i;j )y(i;j )()); j An j?2 O(i;j )yn;r n;r

for i; j = 1; 2; 3. By standard arguments of the theory of Hilbert spaces (cf. Freeden (1990)) we therefore nd

hmi;j ? = f i;j 2Iinf i;j N (w i;j ) (

)

(

(

)

)

(

)

sN(i;j)() =

k=oi;j NX ?1

(

k=M

wl(i;j ) ?

)

?

MX ?1

k=oi;j

with l = M; : : :; N ? 1.

(

)

((i;j ); );

(

)

wk(i;j )O(i;ji;j) lk(i;j )(l(i;j )) (

l

)

kf (i;j )khmi;j ? ; (

)

sup j O(i;j )f () ? O(i;j )sN(i;j )() j 2

(i;j ) (i;j ) (i;j ) 4Fm;; fAn g (N ) kf khmi;j ? ; )

P1 2n+1 ?2 ? n 2+oi;j n=m+1 4 An (2n + 1) 2 ? +2 n +1 C (i;j ) + (C (i;j ))2 n

and

dk(i;j )O(i;ji;j) O(i;ji;j) Kh_ mi;j ? (k(i;j ) ; l(i;j )) = k l k=M (

(

Furthermore, suppose that 2 [0; 1], and let fAn g be f(n+1=2) g{summable with 1+2(2 +oi;j ). Then

2

where the coecients dk(i;j ); k = M; : : :; N ? 1 are the solutions of the linear equations NX ?1

)

(i;j ) Fm;; fAn g =

(i;j )? k k(i;j) Kh_ m

k

(

)

k

where

wk(i;j )lk(i;j )() +

d(i;j )O(i;j )

)

(

kf (i;j )khm(i;j)?

is given in the explicit form MX ?1

(

n

of the spline interpolation problem N

inf

f i;j 2INi;j (w i;j )

IN(i;j )(w(i;j )) = f 2 h(i;j ) j o O(i;ji;j) f (k(i;j )) = wk(i;j ); k = oi;j ; : : :; N ? 1 :

Theorem 3.3 The uniquely determined solution sN(i;j) ks(i;j )k

)

where

)

n+1 1 2X X

Theorem 3.4 Let f (i;j ) be of class h(i;j ); i; j 2 f1; 2; 3g: Assume that ) the system XN(i;j ) is harmo(i;j i;j ;:::;m {admissible. Denote by sN(i;j ) 2 h(i;j ) the uniquely determined solution of the ksN(i;j )khmi;j ? =

(i;j )()y(i;j )(); j An j?2 ?O(i;j )yn;r n;r

Ghmi;j ? (; ) =

)

interpolation problem

ghmi;j ? (; ) = O(i;j)Khmi;j ? (; ) = 1 2X n+1 X

N(i;j ) = max ( mini;j j ? j): 2

)

with (

N = i;jmax ((i;j ) ); =1;2;3 N

)

ghmi;j ? (r(i;j); ) lr(i;j)()

r=oi;j ?1 MX ?1 MX

+

)

(

22

)

The coecient matrix in the unknowns dk(i;j ) is a Gram matrix, thus, it is symmetric and positive de nite. Therefore, the linear system has a unique solution and the solution process can be carried out by standard algorithms, for which powerful routines are available.

C (i;j ) = M;

MX?1 MX?1 k=oi;j l=oi;j

M;

M;

j c(i;j ) j l;k

2

1 2

m 2n + 1 X n n=0

4

2

:

(i;j ) are the constiHere, as usual, n = n(n + 1) and cl;k tuting coecients of lk(i;j ) (cf. (3.6)).

Proof: It is clear that for 2 there exists a point k(i;j ) 2 XN(i;j ) (i = 1; 2; 3) with j ? k(i;j ) j N(i;j ). Observing the fact that for k = oi;j ; : : :; N ? 1 O(i;ji;j) sN(i;j )(k(i;j )) = O(i;ji;j) f (i;j )(k(i;j ) ) = wk(i;j ); (

k

)

(

k

)

we obtain using the triangle inequality and the representation (3.8) j O(i;j )sN(i;j )() ? O(i;j )f (i;j )() j j O(i;j )s_ N(i;j )() ? O(i;ji;j) s_ N(i;j )(k(i;j ) ) j k ( i;j ) ( i;j ) _ + j O f () ? O(i;ji;j) f_ (i;j )(k(i;j )) j : (

)

k(

)


As an interpolating spline, sN(i;j ) is the smoothest interpolant, i.e.,

From (3.9) it can be deduced that O(i;j )s_ N () ? O(i;ji;j) s_ N(i;j )(k(i;j )) = (

)

k

(O(i;j )Kh_ mi;j ? (; ) ? O(i;ji;j) Kh_ mi;j ? (k(i;j ); ); s_ N(i;j ))hmi;j ? (

and

k

)

(

)

(

)

(

)

ks_ N(i;j )khmi;j ? kf_ (i;j )khmi;j ? : (

k

(

k

)

)

(

)

(

1 2

(

k

1 2

(

(

)

(

)

)

1 2

m

where (i;j )(; k(i;j )) = O(i;j )O(i;j )Khmi;j ? (; ) ?2O(i;ji;j) O(i;j )Khmi;j ? (; k(i;j )) (

(

k

)

+O(i;j ) O(i;j )

(

)

)

Theorem 3.4 allows the following reformulation.

Corollary 3.3 The uniquely determined solution sN 2 h of the spline interpolation problem ksN kh?m = inf kf kh?m ; (3.10) f 2IN (w) where

IN (w) = ff 2 h j O(i;ji;j) f (k(i;j )) = wk(i;j ); k k = oi;j ; : : :; N ? 1; i; j = 1; 2; 3g (

(i;j ); (i;j )) (i;j )? (

Using the inequalities (Freeden and Hermann 1986) n j Pn ( ) ? Pn( ) j 2 2 j ? j and j Pn( ) ? 2Pn( ) + Pn ( ) j 2 4(2n + 1) 2n j ? j for any 2 [0; 1] and all ; ; 2 we obtain after elementary calculations (i;j ) 2 (i;j ) 2 j (i;j )(; k(i;j )) j 4(Fm;; fAn g ) j ? k j :

)

This proves Theorem 3.4.

)

k k k(i;j) k(i;j) Khm MX ?1 ? ) (i;j )((i;j )) ?2 O(i;j )lr(i;j )() ? O(i;j (i;j ) lr k k r=oi;j ?O(i;j)O(i;j) K (i;j)? ((i;j); ) r r(i;j) hm ( i;j ) ( i;j ) ( i;j ?O(i;j) Or(i;j) Khm(i;j)? (r ); k(i;j )) k ?1 ? MX ?1 MX ) (i;j )((i;j )) O(i;j )lr(i;j )() ? O(i;j + (i;j ) lr k k s=oi;j r=oi;j ?O(i;j)l(i;j)() ? O(i;j) l(i;j)((i;j)) s k k(i;j) s ( i;j ) ( i;j ) Os(i;j) Or(i;j) Khm(i;j)? (r(i;j ); s(i;j )) :

)

(

)

((i;j )(; k(i;j ))) ks_ N(i;j )khmi;j ? ; j O(i;j )f_ (i;j )() ? O(i;ji;j) f_ (i;j )(k(i;j ) ) j k ( i;j ) ( i;j ) ( (; k )) kf_ (i;j )kh i;j ? ;

(

(

)

The Cauchy-Schwarz inequality yields j O(i;j )s_ N(i;j )() ? O(i;ji;j) s_ N(i;j )(k(i;j )) j

)

j O(i;j )f (i;j )() ? O(i;j )sN(i;j )() j 2?(i;j )(; k(i;j )) kf (i;j )khmi;j ? (i;j ) (i;j ) (i;j ) 4Fm;; fAn g j ? k j kf khmi;j ? (i;j ) (i;j ) (i;j ) 4Fm;; fAn g (N ) kf khmi;j ? :

)

(O(i;j )Kh_ mi;j ? (; ) ? O(i;ji;j) Kh_ mi;j ? (k(i;j ); ); f_ (i;j ))hmi;j ? : (

(

)

Summarizing our results we nd

O(i;j )f_ (i;j )() ? O(i;ji;j) f_ (i;j )(k(i;j ) ) = (

23

)

satis es the inequality

sup j O(i;j )f () ? O(i;j )sN () j 2

with

4Fm;;fAn g (N ) kf kh?m ; i; j = 1; 2; 3;

?

Fm;;fAn g = i;jmax F (i;j ) : =1;::;3 m;;fAn g

Now we want to show that f ? sN does not only tend to zero in its O(i;j ){components, if N ! 0 for N ! 1, but also in the components and, in addition, in the Euclidean norm. For that purpose we use the decomposition Theorem 2.4 and we are able to obtain

Corollary 3.4 The uniquely determined solution sN 2 h of the interpolation problem (3.10) satis es sup j f () ? sN () j 2

4(13 + 2C)Fm;;fAn g (N ) kf kh?m ; where C > 0 denotes a real number satisfying

C j

Z

o(i;j)G(( + 1); ; )d!() j : (3.11)


Proof: From Theorem 2.4 we know that for 2

f () ? sN () = o(1 ;1)(O(1;1)(f () ? sN ())) X 1 (i;j ) (i;j ) + 2 o (O (f () ? sN ())) (i;j )2f(2;2);(3;3)g

X

?

(i;j )2f(1;2);(1;3);(2;1);(3;1)g Z o(i;j) G( ; ; )(O(i;j)(f () ? sN ()))d!()

Z X 1 ( i;j ) 2 (i;j )2f(2;3);(3;2)g o G( ( + 1 ); ; ) (O(i;j )(f () ? sN ()))d!():

+

In what follows we want to give an estimate for each component. We have

j o(1;1)(O(1;1)(f () ? s

N ())) j

jj O(1;1)(f () ? s

j N ()) j j O(1;1)(f () ? sN ()) j;

and for i = 2; 3 j 21 o(i;i)(O(i;i)(f () ? sN ())) j 21 2 j O(i;i)(f () ? sN ()) j j O(i;i)(f () ? sN ()) j :

R

R

Since o(i;j ) : : :d!() = o(i;j ) : : :d!(), we nd for (i; j) 2 f(1; 2); (1; 3); (2;1); (3;1)g

Z

j o(i;j ) G(; ; )(O(i;j )(f () ? sN ()))d!() j

sup j O(i;j )(f () ? sN ()) j 2

sup

Z

2

Using

Z

j o(i;j )G( ; ; ) j d!():

j o(i;j )G(; ; )d!() j

= =

= =

Z

j r G(; ; ) j d!()

1 Z j r ln(1 ? ) j d!() 4 Z ? ( ) 1 4 j p1 ? j d!() 1 Z j 1 ? ( )2 j d!() 4 p 1 ? 1 Z 1 p1 + t dt 2; 2 ?1 1 ? t

24

we get for (i; j) 2 f(1; 2); (1; 3); (2;1);(3; 1)g

j o(i;j )

Z

G( ; ; )(O(i;j )(f () ? sN ()))d!() j

2 sup j O(i;j )(f () ? sN ()) j : 2

The same process for (i; j) 2 f(2; 3); (3; 2)g leads us to

Z

j o(i;j ) G( ( + 1 ); ; )

( i;j ) (O (f () ? sN ()))d!() j sup j O(i;j )(f () ? sN ()) j 2

Z

sup j 2

o(i;j)G(( + 1); ; )d!() j

C sup j O(i;j )(f () ? sN ()) j; 2

for all 2 and (i; j) 2 f(2; 3); (3; 2)g, where C > 0 satis es (3.11). Summarizing our results we get by virtue of Corollary 3.3 j f () ? sN () j (1 + 4 + 8 + 2C)Fm;;fAn g (N ) kf kh?m for every 2 . This proves Corollary 3.4. Corollary 3.4 means that we are able to approximate any f 2 h together with its O(i;j )-components uniformly on in a constructive way using spline interpolation provided that the XN {width N tends to zero as N ! 1. As is well{known, the set of all nite linear combinations of tensor spherical harmonics is dense in the space c(2) ( ). Hence, h is a dense subset of c(2) ( ), too. An extended version of Helly's theorem due to Yamabe (1959) shows us that to any g 2 c(2) ( ) and any (admissible) system XT , there exists an element f 2 h in an "-neighbourhood of g with O(i;ji;j) f (k(i;j )) = O(i;ji;j) g(k(i;j )) (

)

k

(

)

k

for k = oi;j ; : : :; T ? 1; i; j = 1; 2; 3. Thus we nally obtain Theorem 3.5 Let XT be a harm0;:::;m {admissible system. Suppose, that fXN g, N ! 1 is a sequence of harm0;:::;m {admissible systems such that XT XN for all N and N ! 0 as N ! 1. Then, to any g 2 c(2) ( ) and to every " > 0, there exist an integer N = N(") and a spherical spline sN 2 h such that O(i;ji;j) sN (i(i;j ) ) = O(i;ji;j) g(i(i;j )) (

k

)

(

k

)

for k = oi;j ; : : :; T ? 1; i; j = 1; 2; 3, and

sup j O(i;j )sN () ? O(i;j )g() j "; 2


?(i ? ) ( 12 H())

sup j sN () ? g() j ": 2

?r ( 12 H())

Theorem 3.5 gives the theoretical justi cation of the use of tensor spherical spline functions as basis system in problems of uniform approximation on the unit sphere.

?(i ? ) ( 21 H())

= o(2 ;3)( 21 H()) ?2r ( 21 H()) ?(i ? ) ( 12 H());

Appendix A | The Decomposition of the Hesse Matrix In order to clarify the decomposition of spherical tensor elds as described in Theorem 2.4, we discuss the decomposition of the Hesse matrix of a twice continuously dierentiable function. The Hesse matrix plays an important rôle in physical geodesy, in particular for the determination of the earth's gravity eld via satellite gradiometry (cf. e.g. Rummel (1986)). In this context, the Hesse matrix denotes the gravitational tensor. We assume that x 2 R3 has the representation x = r, 2 , in polar coordinates. Then the Hesse matrix h of a twice continuously dierentiable function H is de ned on by h() = @x@ @x@ H(x)jr=1 i;j=1;2;3 i j = rx rxH(x)jr=1 : The separation process of the dierential operators in radial and angular parts yields rx rx = ( @r@ + 1r r ) ( @r@ + r1 r ) @ )2 ? 1 r = ( @r r2 @ + 1 r @ + r1 r @r r @r + r12 r r ; such that

h() = (( @r@ )2H(x) ) + r

?H() + @r@ H(x) r=1

r=1

@ H(x)j ) +r ( @r r=1 +r r H(): Furthermore,

r r H() = 2r r ( 12 H()) = 2r r ( 12 H()) +r ( 21 H())

25

and

r ( @r@ (H(x))jr=1 ) = r ( @r@ H(x)jr=1 ) @ H(x)j ): +(i ? )( @r r=1

Summarizing our results we nally arrive at the following decomposition of the Hesse matrix:

h() =

! @r H(x) r=1

@ 2

o(1;1)

@ H(x) +o(1 ;2) ?H() + @r r=1 @ H(x) +o(2 ;1) ?H() + @r r=1 1 @ H(x) +o(2 ;2) ? 2 H() + @r +o(2;3)

1

r=1

2 H() : Finally, we are interested in expanding the gravitational tensor (at "altidude\ r0 > R) v(r0 ) = rx rxV (x)jr=ro

into a series in terms of tensor spherical harmonics, where V is the uniquely determined solution of the classical Dirichlet problem of potential theory corresponding to continuous boundary values on the sphere around the origin with radius R and r0 is assumed to be strictly greater than R (cf. Rummel and van Gelderen (1992) using Zerilli (1970) nomenclature): V (x) =

1 2X n+1 X n=0 j =1

R n+1

Vn;j jxj

Yn;j ();

with x = r; r > R, and Vn;j =

Z

V (R)Yn;j ()d!():

It is clear that x V (x)jjxj=r = 0 is equivalent to tr (rx rxV (x))jjxj=r = 0: 0

0


Therefore an easy calculation shows that v(r0) = n+1 1 2X n+1 (1;1) X () Vn;j (n + 1)(n + 2) rR n+3 yn;j 0 n=0 j =1

?

n+1 1 2X X

n=1 j =1

n+1 1 2X X

p n+1 (1;2) Vn;j (n + 2) n rR n+3 yn;j () 0

p n+1 (2;1) Vn;j (n + 2) n rR n+3 yn;j () 0 n=1 j =1 ? p12

n+1 1 2X X

n=0 j =1 1 2X n+1 X

n+1 (2;2) Vn;j (n + 1)(n + 2) rR n+3 yn;j () 0

n+1 (2;3) p Vn;j n(n ? 1 ) rR n+3 yn;j + p1 () 2 n=2 j =1 0

holds for all 2 . As an example we mention the gravitational tensor v of a xed mass point y inside the sphere (i.e. y = jyj; jyj < R): V (x) = jx ?1 yj ; jxj > R: On the one hand, v(r0) allows a series expansion in terms of tensor spherical harmonics with coecients Vn;j given by n Vn;j = 2n4+ 1 R1 jRyj Yn;j (): Written out in terms of Legendre polynomials we nd v(r0) = V1 () + V2 () ( ( ? ( )) + ( ? ( )) ) + V3 ()itan + V4()( ? ( )) ( ? ( )); where via the addition theorem the functions Vi are expressible as series in terms of the Legendre polynomials Pn and their derivatives. We omit the explicit expansions. On the other hand simple calculations yield

v(r0) = rx rx jx ? yj?1 jxj=r

0

= jr 3? yj5 (r0 ? y) (r0 ? y) ? jr 1? yj3 i: 0 0

Appendix B | The Choice of the Norm In the described approach for the tensor spherical spline interpolation all conclusions were drawn under a prede ned Hilbert space structure. Now we x our attention

26

on this Hilbert space structure which is given by the summable sequence fAn g or, equivalently, by the choice of the norm. Of course it is desired to choose a norm which is adopted to the speci c problem and to the speci c data. However, for numerical reasons, there are some restrictions for this choice. The condition number of the resulting linear equation is in uenced by this norm, as well as the possibility of a representation of the reproducing kernel in closed form. For this reason, the choice of the norm must be a compromise between numerical restrictions and theoretical intentions. In this context, we carry over some ideas of positive de nite functions (Micchelli 1986, Madych and Nelson 1988) to the spherical case, to have an appropriate tool for the decision, whether a given kernel is applicable or not. To simplify our investigations, we rst restrict ourselves to the scalar case, i. e. we consider only scalar spherical functions and data. This theory is immediately applicable to tensor functions of type (1; 1). To transfer these results to tensors with tangential parts, some straightforward modi cations are necessary. Since some emphasis lies on the choice of the sequence fAng, we change our notations in an obvious way. We start with an extension of De nition 3.1.

De nition B.1 Let fAn g ; fBng 2 A. fAng is called fBn g{summable of order m, if An 6= 0 for n m + 1 and

1 2n + 1 B X n 4 A 2 < 1:

n=m+1

n

Furthermore, fAn g is called summable of order m, if An 6= 0 for n m + 1 and 1 2n + 1 1 X 4 A 2 < 1:

n=m+1

n

To stress the dependence on the sequence fAn g, we denote by H(fAn g; ) the completion of the space of all in nitely dierentiable functions satisfying n+1 1 2X X n=0 j =1

j An j2 (F; Yn;j )2L < 1 2

with respect to the norm corresponding to the inner product (F; G)H(fAn g; ) = 1 2X n+1 X

n=0 j =1

j An j2 (F; Yn;j )L (G; Yn;j )L : 2

2

If no confusion is likely to arise we write H instead of H(fAn g; ). We know (cf. Theorem 3.1) that, if fAn g is summable, each function of H = H(fAn g; ) corresponds to


a continuous function in C (0)( ) and that there exists a reproducing kernel in H(fAn g; ). It is given by KH (; ) = KH(fAn g; ) (; ) = 1 2X n+1 X

j An j?2 Yn;j ()Yn;j (); ; 2 :

n=0 j =1

The space Hm = Harm0;:::;m , spanned by the set of all spherical harmonics of order m, is a nite dimensional Hilbert space with inner product corresponding to the norm

kF kHm

1 0 m 2n+1 X X j An j2 (F; Yn;j )2 A =@

1 2

n=0 j =1

and the reproducing kernel KHm (; ) =

n+1 m 2X X

n=0 j =1

j An j?2 Yn;j ()Yn;j ():

Let us denote by H?m = H?m (fAn g; ) the orthogonal complement of H?m in H(fAn g; ). The linear space H?m (fAn g; ) is a Hilbert space with inner product (; )H?m corresponding to the norm

kF kH?m

1 0 1 2n+1 X X j An j2 (F; Yn;j )2L A = @ n=m+1 j =1

1 2

2

and the reproducing kernel KH?m (; ) =

1 2X n+1 X

n=m+1 j =1

j An j?2 Yn;j ()Yn;j ():

Hence, H is the orthogonal direct sum of Hm and H?m with the inner product (; )H = (; )Hm + (; )H?m and the reproducing kernel KH (; ) = KHm (; ) + KH?m (; ): Before we can draw our attention on dierent kernels, we want to give conditions for the kernels to be suitable for spline interpolation. For this we introduce the concept of positive de nite functions as mentioned before. De nition B.2 Let K : [?1; +1] ! R be continuous. K is said to be conditionally positive de nite of order m, if for any distinct (admissible) system XN = f1; : : :; N g on and scalars a1 ; : : :; aN satisfying 0 Y ( ) : : : Y ( ) 1 0 a 1 1 0;1 N B B@ 0;1... 1 C . .. A @ .. CA = 0 ; . aN Ym;2m+1 (1) : : : Ym;2m+1 (N ) (B.1)

27

the quadratic form N X N X k=1 l=1

ak al K(k l )

is nonnegative. K is said to be conditionally strictly positive de nite of order m if the quadratic form is positive. Conditionally positive de niteness of order ?1 is simply called conditionally positive de niteness. In this case, the condition (B.1) has to be omitted. As an immediate consequence of De nition B.2 we obtain the following lemma. Lemma B.1 If K 2 C (0)[?1; +1] is conditionally strictly positive de nite, then Km? de ned by Km? (t) =

K(t) ?

0 m 2n + 1 Z+1 X @

n=0

2

?1

1 K(u)Pn(u)duA Pn(t);

for t 2 [?1; +1] is conditionally strictly positive de nite of order m.

Often it is not easy to determine whether a given continuous function K is conditionally strictly positive de nite. A direct method is to guarantee one of the following equivalent conditions. Theorem B.1 Let K : [?1; +1] ! R be continuous. Then the following statements are equivalent: (i)

ZZ

Km? ( )()()d!()d!() > 0

for all 2 C (1) ( ) with (; Yn;j )L 6= 0 for n =

m + 1; m + 2; : : :; j = 1; : : :; 2n + 1: (ii)

ZZ

2

Km? ( )Yn;j ()Yn;j ()d!()d!() > 0

for all Yn;j with n = m + 1; m + 2; : : :; j = 1; : : :; 2n + 1. (iii) K is conditionally strictly positive de nite of order m. (iv) The sequence fAn g with

1?1=2 0 +1 Z (B.2) An = @ 2n 2+ 1 Km? (u)Pn(u)duA ?1

is summable of order m.


Proof: The statement (i) ) (ii) is clear. Therefore we turn to (ii) ) (iii). Assume that (ii) is true. Then

it is obvious that

N X

0 r2n

Z Z

k=1

ak Yn;j (k )

!2

K ? ( )Yn;j ()Yn;j ()d!()d!() m

for positive values r and coecients a1 ; : : :; aN satisfying (B.1). But this shows that 0

Z Z

Km? ( )r ()r ()d!()d!();

where r () =

N X l=1

al Qr (j ); 2

and Qr is the Poisson kernel de ned by 1 ? r2 1 Qr (t) = 4 (1 + r2 ? 2rt)3=2 : Letting r ! 1 we see that Km? is conditionally strictly positive de nite of order m. Next we prove (iii) ) (iv). Suppose that 0

N X N X k=1 l=1

ak al Km? (k l )

for all a1 ; : : :; aN satisfying (B.1). Then, due to Aronszjain (1950), there corresponds one and only one class of functions forming a real Hilbert space (denoted by) Hm? (fAng; ) and admitting Km? ( ); ; 2 as reproducing kernel. Obviously, the system fA?n 1Yn;j g, n = m + 1; m + 2; : : :; j = 1; : : :; 2n + 1, with coecients An given by (B.2) is complete in Hm? (fAn g; ), i.e. Km? ( ) =

1 2X n+1 X

n=m+1 j =1

j An j?2 Yn;j ()Yn;j () (B.3)

in the sense of uniform convergence for all ; 2 with ?1 +1. Moreover, for all ; 2 , jKm? ( )j (Km? ( ))1=2 (Km? ( ))1=2; so that sup j Km? ( ) j (;)2

2

=

1 2n + 1 1 X 2 Pn (1) n=m+1 4 An 1 2n + 1 1 X n=m+1

be summable of order m, the representation (B.3) is valid. Consequently,

Z Z

4 A2n :

But this means that fAng is summable of order m. Finally we prove (iv) ) (i). Since fAng is assumed to

28

Km? ( )()()d!()d!() = n+1 1 1 2X X 2 A2 (; Yn;j )L > 0

n=m+1 n=1

2

n

for all 2 C (1) ( ) with (; Yn;j )L 6= 0; n = m + 1; m + 2; : : :; j = 1; : : :; 2n + 1. This states Theorem B.1. 2

The last theorem gives conditions for the decision whether a given kernel is suitable for the spline interpolation method. As examples, we want to de ne some possible kernels and hence the corresponding Hilbert spaces which are de ned by the underlying sequences. We will show examples of kernels of dierent types: classical Green functions with respect to and its iterations kernels related to pseudodierential operators multiquadric kernels. locally supported kernels ("axisymmetric nite elements\) Let us construct the Hilbert space H = H(fAn g; ) by choosing the inner product (; )H(fAng; ) corresponding to the norm kkH(fAn g; ) , given by

kF k2H(fAng; ) = 1 j Z F()Y ()d!j2 0;1 4

+

n+1 1 2X X

n=1 j =1

(n )2

Z

2

j F()Yn;j ()d!j :

That means the sequence fAng is given by An =

( q1

n=0 n = 1; 2; : : :

4

?n Obviously, fAn g is f(n + 12 ) g{summable for all < 2. For the reproducing kernel in H(fAn g; ) we nd 1 2n + 1 1 X KH (; ) = 1 + 4 ( )2 Pn ( ): n=1

n

Apart from an additive constant the series on the right hand side coincides with Green's function on the unit sphere corresponding to the dierential operator ()2 . It can be seen (cf. Freeden and Gervens (1993)) that in this case the semi{norm which is minimized by the


spline interpolant is related to the linearized bending energy of a thin beam. As an example of kernels related to pseudodierential operators on , we consider fAn g = fr?n=2(n) g; 0 < r < 1, where is a pseudodierential operator on with symbol ((n) ) such that (i) n 7! (n) is a real rational function (ii) there exists a positive constant C 0 with 0 hm; (fr?n= n g; )(; ); > if (k; l) = (1; 1) > > D G ; > r ; ) (; ); > < if h(k;m l)(f2r?fn=(1;2);n g(2; 1); (1; 3);(3; 1)g 2G (; ; > ? n= hm (fr n g; ) ); > > if (k; l) 2 f(2; 2); (3; 3)g > > 2D (D ? )G m; (fr?n= n g; ) (; ); > : rif (k;r l) 21f(2;h3); (3; 2)g: But this means that all Ghmk;l (fr?n= n g; ) (; ) are known as elementary functions, if Ghm; (fr?n= n g; ) (; ) is available as elementary function, which is of basic interest in numerical computations. To de ne kernels of multiquadric type, suppose that H is in C (0) [0; 1) \ C (1) (0; 1) with p (?1)j H (j )(pt) 0; t > 0; j m + 1; H (m+1) ( t) 6= const: Then it can be readily seen from results due to Micchelli (1986) that p K : t 7! K(t) = H( 2 ? 2t); t 2 [?1; +1]; is a conditionally strictly positive function (of order m). Examples of functions H satisfying the aforementioned conditions include (cf. Dyn (1989)): H(t) = (?1)m+1 t ; 2m < < 2m + 2 H(t) = (?1)m+1 tm ln t H(t) = (?1)m+1 (t2 + c2) =2 ; 2m < < 2m + 2; c > 0 H(t) = (?1)m+1 (t2 + c2)m ln (t2 + c2)1=2 : These functions include the multiquadric for m = 0. For geophysical applications, kernels of multiquadric type have been used by Hardy (1983). Finally, we turn to the " nite element\ type of kernel functions mentioned before. Consider e.g. the auxiliary function Bh(k) de ned by ( 0 for ? 1 t h ( k ) Bh (t) = (t?h)k for h < t 1 (1?h)k (

)

2

( )

(1 1)

2

( )

(1 1)

(1 1)

2

2

( )

( )

(1 1)

(

2

)

(1 1)

2

( )

( )

2

( )

for k = 0; 1; : : :, h 2 [0; 1) (other types of Bh(k) can in Cui et al. (1992)). nbe found o Obviously, the system ( k ) ( k ) Bh (1 :); : : :; Bh (N :) , i 6= k for i 6= k, forms a linearly independent system. Now let L(hk)(Z i ; k) = L(hk)(i k ) = Bh(k) (i )Bh(k) (k )d!()


for i; k = 1; : : :; N. Hence, according to our construction, t 7! L(hk) (t), t 2 [?1; 1], is conditionally strictly positive. Note that, for every 2 , L(hk)( :) is an axisymmetric locally supported function showing as support suppL(hk) ( :) = f 2 j ? 1 + 2h2 1g: Thus band matrix solvers can be applied in spline interpolation (of order ?1, i.e. when no polynomial exactness is required) which is of particular importance for numerical purposes.

References Aronszjain N (1950) Theory of Reproducing Kernels.

Trans. Am. Math. Soc. 68: 337{404 Backus GE (1966) Potentials for Tangent Tensor Fields on Spheroids. Arch. Ration. Mech. Analysis 22: 210{ 252 Backus GE (1967) Converting Vector and Tensor Equations to Scalar Equations in Spherical Coordinates. Geophys. J.R. Astr. Soc. 13: 71-101 Cui J, Freeden W, Witte B (1992) Gleichmaige Approximation mittels spharischer Finite{Elemente und ihre Anwendung in der Geodasie. Z. f. Vermessungswesen 117, Heft 5: 266-278 Dahlen FA, Smith ML (1975) The In uence of Rotation on the Free Oscillations of the Earth. Philos. Trans. R. Soc. Lond. A, 279: 583{629 Davis PJ (1963) Interpolation and Approximation. Blaisdell Publishing Company, New York, Toronto, London Dyn N (1989) Interpolation and Approximation by Radial and Related Functions. In: Chui CK, Schumaker LL, Ward JD eds, Approximation Theory VI, Vol. 1, Academic Press: 211{234 Euler HJ, Groten E, Hausch W, Stock B (1985)

Precise Geoid Computation at Mountaineous Coastlines in View of Vertical Datum Determinations. Vero. Zentralinst. fur Physik der Erde, Akad. der Wiss. der DDR 81, Teil I: 109-139 Euler HJ, Groten E, Hausch W, Kling T (1986) New Results Obtained for Detailed Geoid Approximations. Bolletino di Geodesia e Scienze Ani 4 Freeden W (1978) Eine Klasse von Integralformeln der Mathematischen Geodasie. Vero. Geod. Inst. RWTH Aachen, Nr. 27 Freeden W (1981) On Spherical Spline Interpolation and Approximation. Math. Meth. in the Appl. Sciences 3: 551-575 Freeden W (1982) On the Permanence Property in Spherical Spline Interpolation. Report No. 341, Department of Geodetic Science, Ohio State University, Columbus Freeden W (1990) Spherical Spline Approximation and its Application in Physical Geodesy. In: Vogel A et al. eds, Geophysical Data Inversion Methods and Applications. Vieweg{Verlag, Braunschweig, pp 79{104

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Freeden W, Gervens T (1993) Vector Spherical

Spline Interpolation (Basic Theory and Computational Aspects). Math. Meth. Appl. Sci. 16: 151{183 Freeden W, Hermann P (1986) Uniform Approximation by Spherical Spline Interpolation. Math. Z. 193: 265-275 Freeden W, Reuter R (1990) An Ecient Algorithm for the Generation of Homogeneous Harmonic Polynomials. In: Mason JC, Cox MG eds, Scienti c Software Systems, Chapman and Hall Mathematics, London, New York: 166-180 Freeden W, Witte B (1982) A Combined (Spline{) Interpolation and Smoothing Method for the Determination of the External Gravitational Potential from Heterogeneous Data. Bull. Geod. 56: 53{62 Georgiadou P, Grafarend E (1986) Global Vorticity and the De nition of the Rotation of a Deformable Earth. Gerlands Beitr. Geophysik 95: 516{528 Gervens T (1989) Vektorkugelfunktionen mit Anwendungen in der Theorie der elastischen Verformungen fur die Kugel. Doctoral Thesis, RWTH Aachen Girard DA (1989) A Fast "Monte{Carlo Cross{Validation\ Procedure for Large Least Squares Problems with Noisy Data. Numer. Math. 56, No. 1: 1{23 Grafarend E (1986) Three{Dimensional Deformation Analysis: Global Vector Spherical Harmonic and Local Finite Element Representation. Tectonophysics 130: 337-359 Groten E, Moritz H (1964) On the Accuracy of Geoid Heights and De ections of the Vertical. Institute of Geodesy, Photogrammetry and Cartography, 38, Ohio State University, Columbus Gurtin ME (1972) Theory of Elasticity. Handbuch der Physik Vol. VI, 2nd ed., Springer, Berlin v. Gysen H, Merry LC (1990) Towards a Cross{ Validated Spherical Spline Geoid for the South{ Western Cape, South Africa. In: Mueller II ed, Sea Surface Topography and the Geoid, International Association of Geodesy Symposia 104, Springer, New York, Berlin, Heidelberg, London: 53{60 Hardy RL (1986) Multiquadric Equations of Topography and Other Irregular Surfaces. J. Geophysical Res. 76: 1905{1915 James RW (1976) New Tensor Spherical Harmonics for Application to the Partial Dierential Equations of Mathematical Physics. Philosophical Transactions Royal Society A 281: 197{221 Jones MN (1980) Tensor Spherical Harmonics and Their Application to the Navier Equation of Dynamic Elasticity. Gerlands Beitr. Geophysik 89: 135{148 Jung K (1961) Schwerkraftverfahren in der Angewandten Geophysik. Akademische Verlagsgesellschaft, Leipzig Kling T, Becker M, Euler HJ, Groten E (1987)

Studien zur detaillierten Geoidberechnung. Deutsche Geodatische Kommission bei der Bayerischen Akademie der Wissenschaften, Reihe B, Nr. 285


Madych WR, Nelson SA (1988) Multivariate Interpo-

lation and Conditionally Positive De nite Functions. Approx. Theory and its Appl. : 77{89 Magnus W, Oberhettinger F, Soni RP (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 52, Springer, Berlin, Heidelberg, New York McClung HB (1989) The Elastic Sphere under Nonsymmetric Loading. J. of Elasticity 21: 1{26 Meissl P (1971a) On the Linearization of the Geodetic{ Boundary{Value Problem. Report No. 151, Department of Geodetic Science, Ohio State University, Columbus Meissl P (1971b) A Study of Covariance Functions Related to the Earth's Disturbing Potential. Report No. 152, Department of Geodetic Science, Ohio State University, Columbus Micchelli CA (1986) Interpolation of Scattered Data: Distance Matrices and Conditionally Positive De nite Functions. Constr. Approx. 2: 11{22 Mochizuki E (1988) Spherical Harmonic Decomposition of an Elastic Tensor. Geophys. J. 93: 521{526 Muller C (1966) Spherical Harmonics. Lecture Notes in Mathematics 17, Springer, Berlin, Heidelberg, New York Phinney RA (1973) Representation of the Elastic{ Gravitational Excitation of a Spherical Earth Model by Generalized Spherical Harmonics. Geophys J. R. Astr. Soc 34: 451{487 Rummel R (1975) Downward Continuation of Gravity Information from Satellite to Satellite Tracking or Satellite Gradiometry in Local Areas. Report No. 221, Department of Geodetic Science, Ohio State University, Columbus Rummel R (1986) Satellite Gradiometry. In: Sunkel H ed, Lecture Notes in Earth Sciences 7, Mathematical and Numerical Techniques in Physical Geodesy, Springer, Berlin: 318{363 Rummel R, Colombo OL (1985) Gravity Field Determination from Satellite Gradiometry. Bull. Geod. 59: 233-246 Rummel R, van Gelderen M (1992) Spectral Analysis of the Full Gravity Tensor. Geoph. J. 111: 159{169 Sacerdote F, Sanso F (1989) Some Problems Related to Satellite Gradiometry. Bull. Geod. 63: 405{415 Schwarz KP, Krynski J (1977) Improvement of the Geoid in Local Areas by Satellite Gradiometry. Bull. Geod 51: 163{176 Thorne KS (1980) Multipole Expansions of Gravitational Radiation. Rev. Mod. Phys. 52, No. 2, Part I: 299{339 Utreras FI (1990) Recent Results on Multivariate Smoothing Splines. Int. Series Numer. Math. (ISNM) 94: 299{312

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Wahba G (1981) Spline Interpolation and Smoothing on

the Sphere. SIAM J. Scienti c and Statistical Computing 2: 1-15 Wahba G (1982) Vector Splines on the Sphere with Application to the Estimation of Vorticity and Divergence from Discrete Noisy Data. ISNM 61: 407-429 Wahba G (1984) Surface Fitting with Scattered Noisy Data on Euclidian d{Space and on the Sphere. Rocky Mountain J. Math. 14: 281{299 Yamabe H (1959) On an Extension of Helly's Theorem. Osaka Math. J.: 15-22 Zerilli FJ (1970) Tensor Harmonics in Canonical Form for Gravitational Radiation and Other Applications. J. Math. Physics 11: 2203{2208

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