Non-linearity versus linearity in ridge approximation - CiteSeerX

Non-linearity versus linearity in ridge approximation K.I. Oskolkov

Abstract

The goal is to compare free (non-linear), equispaced ridge and algebraic polynomial approx2 2 2 imations RfrN [f ]; Req N [f ]; EN [f ] of2 individual functions f (x) in the norm of L (IB ), IB { the unit disc jxj  1 on the plane IR . By de nition

RfrN [f ] := inf kf ? Rk; Req kf ? Rk; EN [f ] := P 2P min kf ? P k: N [f ] := Rmin 2W R2WNfr

eq

2

N ?1

N

fr the set of all N -term linear combinations of planar wave functions R(x) = Here, PN WW(Nx
denotes eq j ) of arbitrary pro les Wj (x); x 2 IR1 and directions fj gN 1 j 1 ; WN { the subset k l 2 fr of WN corresponding to N equispaced directions, and PN ?1 := Span fx1 x2 gk+l 0 such that if
f (x) = 0; jxj < 1; f 2 L2 (IB2 ), then

  RfrN [f ]  c" ReqN [f ] exp(?N ") + ReqN ? " [f ] : 2 3

fr On the other hand, RfrN [f ]  1c Req N [f ]. Thus, for f = fharm; RN [f ] is \almost square times eq better" than RN [f ]. However, these ultra-convergence rates are achieved at the expense of collapse of wave vectors. On the contrary, non-linearity in Rfr does not bring any essential gain in q approximation eq orders, say, for all radial functions. If f (x) = f (jxj), then E2N [f ]  RN [f ]  23 E2N (f ) and q RfrN [f ]  sup">0 3(1+" ") Req (1+")N [f ] : These problems are elaborated via Fourier { Chebyshev analysis in IB2 and arising duality between ridge approximation and optimization of quadrature formulas, in the sense of Kolmogorov { Nikol'skii [1], on classes of trigonometric polynomials. 2

1

1

Introduction

We consider here a special case of the general problem of ridge approximation. First of all, we restrict ourselves to the case of (complex valued) of two real variables f (x) = f (x1; x2), supported q 2 functions 2 2 in the unit disc IB := fx : jxj := x1 + x2  1g on the real Euclidean plane IR2. Further, we assume that f (x) 2 L2(IB2), and focus on the approximation problem exclusively in the norm of the Hilbert space L2(IB2), Z Z  kf (x); L2(IB2)k := IB jf (x)j2 (dx) ; where (dx) := dx dx denotes the normalized Lebesgue measure on IB2. Let us introduce some other notations. x
y will denote the usual inner product of vectors x; y 2 IR2; S 1 - the unit circle jxj = 1;  = (#) := hcos #; sin #i ; # 2 [0; 2) { polar parametrization of S 1. Further, for N = 1; 2; : : :, we will apply the vector notations #~ := f#j gN1 2 IRN for N element sets of directional angles; j := hcos #j ; sin #j i; ~ = ~(#~ ) := fj gN1 . Let us consider the following sets W (#~ ); WNeq; WNfr of ridge functions { N -terms linear combinations of planar waves: ) ( N [ X ~ W (#~ ); W (#) := R(x) = Wj (x
j ) ; #~ 2 IRN : WNfr := 1 #~2IRN ) ( N X j eq WN := R(x) = Wj (x
j ) : #j = N ; j = 0; : : : ; N ? 1 : 1 2

2

1

2

1

In the above de nitions, Wj (x); x 2 IR1 are single-variate functions (pro les of waves). Clearly, in the de nition of W (#~ ) we can con ne the components #j of #~ to the interval [0; ), and consider only non-degenerate #~, i. e. the case when #j are pairwise non-congruent mod. Thus, W (#~ ); WNfr ; WNeq consist of N -term linear combinations of planar waves of arbitrary pro les; W (#~ ) corresponds to a xed set of directional angles #~ 2 IRN ; WNfr { the widest collection of all functions of such type, WNeq { the particular case of WN#~ with N equispaced wave vectors. Our goal is to study, for a xed function f (x) 2 L2 (IB2), the extremal problems R(#~ ); Rfr; Req associated with the following quantities: R[f; #~] := inf ~ kf ? R; L2(IB2)k; #~ 2 IRN ; RfrN [f ] := inf kf ? R; L2(IB2)k; R2WNfr

R2W (#)

ReqN [f ] := Rmin kf ? R; L2(IB2)k: 2W eq

N

2

Obviously, RfrN [f ] = inf #~2IRN R[f; #~]  Req N [f ]: Of particular interest is to clarify structural (geometric, dierential, etc.) conditions on the given f (x) when freedom in selection of wave vectors fj gN1 in the problem Rfr brings an essential advantage in approximation rates over those associated with Req. Quantitatively, this advantage is expressed by order relations RfrN [f ] = o (Req N [f ]) ; N ! 1. A partial answer is given below in theorem 3 (see also corollary 1) regarding two important types of functions: radials and harmonics. A very essential dierence between the problems Rfr and Req cosists in non-linearity of Rfr. The latter is associated with a complete freedom in the choice of wave vectors fj gN1 , that are allowed to be selected optimally for a given function f (x). On the contrary, for each xed #~ 2 IRN the problem R[f; #~] is linear and the solution is provided by the orthogonal projection in L2(IB2) onto the corresponding subspace L2(IB2)#~ , cf. also theorem 4 below. Further, non-existence and non-uniqueness of the element of best ridge approximation are quite typical for the problem RfrN , if N  2. This can be seen from the following.   j ?1 1) If j  1; W (x); jxj  1 { a smooth single variate function and f (x; #) := @#@ W (x
), then fr for each xed # one has Rj [f ] = 0. This follows from consideration of the angular derivative as limit of divided dierences. This simple observation directs to a natural completion of W (#~ ) and WNfr by the following sets of collapsed ridge functions: 8 9 0 Nj 1 ! >  ? 1 > < = X X @ X W N (#~ ) := >:R(x) = @ @# Wj; (x
)A : Nj = N > ; #~ = f#j g ; #=#j j j  =1 (obviously, the class W N (#~ ) is non-trivial only if dim #~  N ). Respectively, the extremal problem Rfr can be \sliced" as follows: RN [f; #~ ] := inf ~ kf (x) ? R(x)k; RM;N [f ] := ~ infM RN [f; #~]; RfrN [f ] = 1min R [f ]: M N M;N R2W N (#)

#2IR

A particular case is approximation by completely collapsed ridge functions:



!j?1 N X

@ 2 2

f (x) ? RfrN [f ]  Rcol W j (x
 ); L (IB )

; N [f; #] := fW inf N

j (x)gj j =1 @# =1

(1)

which is in a sense the direct counterpart of equispaced ridge approximation. 2) Denote, respectively, PN1 := Span fxk gkN and PN2 := Span fxk1 xl2gk+lN the subspaces of algebraic polynomials of degree N in one and two real variables. If the components #j of #~ 2 IRN 3

are pairwise non-congruent mod, then (cf. e. g. [2] and theorem 1 below) every polynomial P (x) 2 PN2 ?1 can be represented as a linear combination of planar wave polynomials of degree N ? 1 N X (2) P (x) = Pj (x
j ); Pj (x) 2 PN1 ?1; or PN2 ?1  W (#~ ); R[P; #~] = 0: j =1

Thus, the element of best ridge approximation is not unique for all algebraic polynomials. Further, the classical quantities { best algebraic polynomial approximations EN [f ] := min kf ? P; L2(IB2)k P 2PN2 ?1

majorize ridge approximations for every non-degenerate #~ 2 IRN R[f; #~]  EN [f ]: (3) The solution of the problem of ridge approximation of the given function f (x) depends upon the Chebyshev orthogonal momenta an(f; #) generated by Fourier analysis in IB2 : Z x ; n = 0; 1; : : : : (4) an(f; #) := f (x)un(x
) (dx); un(x) := sin (np+ 1) arccos IB 1 ? x2 By means of the latter, the problem RN (f ) is split into an in nite series of Kolmogorov { Nikol'skii type problems, cf. [1], concerning optimal quadrature formulas for recovery of linear functionals Z 2 Fn (f )[T ] := an(f; #)T (#) (d#); n = 0; 1; : : : : 2

0

In the case of a general function f (x) 2 L2(IB2), momenta an(f; #) are trigonometric polynomials of n-th order, satisfying an(f; # + )  (?1)n an(f; #). Let us denote Tn the whole subspace of trigonometric polynomials possessing this property, i. e. Tn := Span feim#gjmjn(2); here and below we use the notation jmj  n(2) for the set of integers m with jmj  n and m  n (mod 2). Further,   denote (d#) := 2d# the normalized Lebesgue measure on S 1, kT; L22 k := R02 jT (#)j2 (d#) , and let o n IB(Tn) := T 2 Tn : kT; L22 k  1 ; IB(Pn1) := fP (z) 2 Pn1 : kP (ei#); L22 k  1g: A quadrature formula (#~; w~ )[T ] with the nodes #~ := f#j gN1 2 IRN and weights w~ := fwj gN1 2 C N is a point-values functional N X (#~; w~ )[T ] := wj T (#j ); 1 2

j =1

4

and the following quantities are typical for Kolmogorov { Nikol'skii setting, cf. [1] of the problem on optimization of quadrature formulas for recovery of linear functionals: Z 2 ~ ~ a(f; #)T (#) (d#) ? (#; w~ )[T ] ; #~ 2 IRN ; Qn[a; #] := w~inf sup 2C N T 2IB(Tn ) 0 opt ~ (5) Qeqn;N [a] := Qn[a; #~ ]; #j = j N ; Qn;N [a] := #~2infIRN Qn[a; #]: In the above, a = a(#) is a xed trigonometric polynomial, a 2 Tn . Theorem 1 Let f (x) 2 L2(IB2); #~ 2 IRN where the coordinates #j are pairwise non-congruent mod. Then v v u uX 1 1   X u 2 t (n + 1) Qn[an(f ); #~]2 : (6) R[f; #~] = t (n + 1) Qn[an(f ); #~ ] ; RfrN [f ] = ~ infN u #2IR

n=N

In particular,

n=N

v u 1 2 fr t X (n + 1) Qopt [ a ( f )] RN [f ]  u n;N n

(7)

PN2 ?1  WN#~ ; R[f; #~ ]  EN [f ]:

(8)

n=N

and

Like in free ridge approximation, in Kolmogorov { Nikol'skii problem we should admit collapsed quadrature formulas, involving linear combinations of point values of linear dierential operators of total degree  N ? 1. A particular case is represented by completely collapsed quadrature formulas col(P; #)[T ] := P d#d T (#); P 2 PN1 ?1. A proper version of the quantities Qn[a; #~] answering this case is Z 2 col col Qn;N [a; #] := P 2Pinf sup  0 a(f; ')T (') (d') ?  (P; #)[T ] ; N? 1

1

T 2IB(Tn )

and according to (6) free ridge approximations can be estimated from above as follows: v u 1  col 2 X u t RfrN [f ]  Rcol [ f ] = inf ( n + 1) Q [ a ( f ) ; # ] n N n;N # n=N

(9)

Momenta an(f; #), cf. (4), are especially simple for radial or harmonic functions in IB2, i. e. when f (x) = f (jxj) or, respectively,
f (x) = 0; jxj < 1 (in the sequel, we will apply the self 5

explanatory notations f = frad; f = fharm in these cases). For f = frad the n-th momentum is a constant, an(f; #) = n (and all while for f = fharm it is a monomial of the highest frequency, an(f; #) = nein# + n e?in# (cf. lemma 1 below) It is clear from theorem 1 what special cases of Kolmogorov { Nikol'skii problems should be solved. In the case of f = frad, we need to solve the problem concerning Qopt n;N [1] { optimal recovery R 2   is non-trivial only for even n; n  N ). For of the averages 0 T (#) (d#) of TR 2 Tn (the   problem 2  in# ? in# (d#), i. e. linear combinations of the f = fharm, we need the recovery of 0 T (#) e + e R senior Fourier coecients T^(n) := 02 T (#)ein# (d#). opt in# ] are of a quite analogous nature Seemingly, the problems concerning Qopt n;N [1] and Qn;N [e simply because all coecients of polynomials in IB(Tn ) possess \equal rights". Moreover, the problem concerning recovery of T^(n) can be also reformulated, cf. (50) below, as that of complex polynomials P (z) 2 IB(Pn1) in the center z = 0 of the disc jzj  1 via their values P (zj ) on the circumference S 1 = fz : jzj = 1g: N X in# ] = Qopt [1; IB(P 1)] := (10) inf N sup P (0) ? wj P (zj ) : Qopt n n;N [e n;N N fzj g 2S ; fwj g P 2IB(Pn ) j =1 1

1

1

1

Thus, the following conjectures are plausible: 1) optimal nodal points #~ = f#j gN1 should be \uniformly distributed"; 2) if the nodal de ciency is essential, i. e. the ratio Nn is small, then it is impossible to recover Fourier coecients of all polynomials in IB(Tn ) with a small error: neither of the quantities opt in# ] can be small. Qopt n;N [1]; Qn;N [e in# ] (see Theorem 2 However, these conjectures fail to be true in the part concerning Qopt n;N [e below). Recovery of the senior Fourier coecient (or the value P (0) of algebraic polynomials P (z) 2 IB(Pn1), cf. (10)) with a small global error on the class IB(Tn) is possible even if the sampling number N is much smaller than n, and it is rather the ratio pNn that governs this eect.

Theorem 2 Let n; N be positive integers, n even, n  2N . Then

s s  1 1 ? 2N   Qopt [1]  2 1 ? 2N  : n;N 2 n+1 n+2 Furthermore, the following relations hold true for n  N  5  p ? pN  N opt ?  in# col  in# n e  Qn;N [e ]  Qn;N [e ; 0]  min 1; 2ne n : 2

2

6

(11) (12)

The meaning of the next statement is the following. 1) Solutions of the problem Req (equispaced ridge approximation) for f = frad and f = fharm are qualitatively and quantitatively the same, and in essence coincide with E . 2) For f = frad, freedom in the choice of directions, in particular, the eect of collapse is not associated with any essential gain in approximation orders of Rfr compared with Req or E . 3) On the contrary, for f = fharm free ridge approximation Rfr is \almost square times better" than Req, due to the eect of collapse of wave vectors.

Theorem 3 The following relations hold true

v s u 1 (E2qN [f ])2 X u 2 E [f ]  Req[f ]  E [f ] ; f = f ; eq ; RN [f ] = t2 2N rad N 2 3 2N q=1 4q ? 1 s 1 E [f ]  Req[f ]  E [f ] ; f = f ; (13) N harm N 3 N +1 s s M ? N M ? N Req [f ] ; f = f ; (14) E RfrN [f ]  Msup rad 2M [f ]  sup 2M 2M M M N N p ? pN  ? 8 fr col M e EN [f ]  RN [f ]  RN [f ]  Mmin 2Me EN [f ] + EM +1[f ] ; N  5; f = fharm : (15) N 2

Corollary 1 If f = frad then R

fr [f ] N

s

s " " eq  sup [f ] : E R 2(1+")N [f ]  sup ">0 2(1 + ") ">0 3(1 + ") (1+")N

If f = fharm then

(i) 8" > 0 9c" > 0 : RfrN [f ]  c" (ReqN [f ] exp(?N " ) + ReqN ? " [f ]); (ii) 9 > 0 : EN ? [f ] = o (EN [f ]) =) RfrN [f ] = o (ReqN [f ]) ; ? fr ?2+" ); N ! 1: (iii) 9 > 0 : Req N [f ] = O(N ) =) 8" > 0 : RN [f ] = O(N 2 3

2

7

2

Proofs

2.1 Chebyshev { Fourier analysis in L2(IB2)

For a given non-negative integer n, let Dn (#) denote the Dirichlet kernel for the subspace Tn : X im# sin (n + 1)# Dn (#) := e = sin # : jmjn(2) Obviously,

Z 2 (16) T (#) = [T  Dn ](#) := 0 T (')Dn(' ? #) (d#); T 2 Tn : Further, let as above un(x) denote the n-th Chebyshev polynomial of the second kind, i. e. un(x) = x ; jxj  1: Dn (arccos x) = sin(np+1)1?arccos x Chebyshev { Fourier analysis in L2(IB2) consists in the orthogonal expansion ! Z 1 L (IB ) 2 X (17) (n + 1)an(f; #)un(x
) (d#); f (x) = 2

2

2

0

n=0

cf. [2] { [4]. For each xed  2 S 1, the corresponding planar wave Chebyshev polynomial un (x
) is orthogonal in L2(IB2) to all polynomials of degree  n ? 1: Z un (x
)P (x) (dx) = 0; 8P (x) 2 Pn2?1: (18) IB2

As mentioned above, the ortogonal momenta, or Fourier { Chebyshev coecients an(f; #), are trigonometric polynomials an(f ) 2 Tn. The Parceval identity is given by Z 2 1 1

2 1 X



2 2

2 X 2

f; L (IB ) = (n + 1) an(f ); L2 = 2 (n + 1) 0 jan(f; #)j2 d#: (19) n=0 n=0 Furthermore, if fan(#)g1 n=0 is a sequence of trigonometric polynomials satisfying the conditions 1

2

X  (n + 1)

an; L22

< 1; a n 2 Tn ; n=0

then (Plancherel's theorem) there exists a function f (x) 2 L2(IB2), unique up to a set of Lebesgue measure 0, such that an(f; #)  an (#); n = 0; 1; : : : : 8

Orthogonal projection Proj N [f ](x) in L2(IB2) of a function f (x) 2 L2(IB2) onto the subspace of algebraic polynomials of degree N ? 1 is given by the partial sum of rst N terms of the expansion (17), ! Z 2 NX ?1 (n + 1)an(f; #)un(x
) (d#); N = 1; 2; : : : ; Proj N [f ](x) = and in particular

0

n=0

v uX 1 t (n + 1) kan (f ); L22 k2 : EN [f ] = u

(20)

n=N

2.2 Momenta of radial, harmonic and planar wave functions

Lemma 1 1) If f (x) = g(jxj2), where g(x) 2 L2(0; 1) and g(x) L =(0;1) P1=0 g l (x) is the Fourier { 2

Legendre expansion of g(x), then

a2+1(f ) = 0; a2 (f ) = p g ;  = 0; 1; : : : : (21) 2 + 1   n f^(?n)e?in' + f^(n)ein' ; 0  r < 1 is the standard representation 2) If f (r') = f^(0) + P1 r n=1 of a harmonic function f = fharm 2 L2(IB2 ) in the polar coordinates x = r', then ?in# ^ in# ^ (22) a0(f ) = f^(0); an(f; #) = f (?n)e n ++1 f (n)e ; n = 1; 2; : : : : p 3) Let !(x) := 2 1 ? x2; jxj  1; W (x) 2 L2! (?1; 1) and W (x) L! (=?1;1) P1n=0 W nun (x) { the Fourier { Chebyshev expansion of W , ' 2 S 1 { a xed unit vector. Then W (x
') 2 L2 (IB2 ) and   (n + 1)(# ? ') ; n = 1; 2; : : : : an (W (x
'); #) = nW+n1 Dn (# ? ') = W(nnsin (23) + 1) sin (# ? ') 2

Proof. For the proof of (21), see e.g. [5]. Let us also note the following relations between Legendre

and Chebyshev polynomials of the second kind: Z 2 Z 2 2 u2 (x
) (d#) = pl (jxj ) ;  = 0; 1; : : : : u2+1(x
) (d#) = 0; 2 + 1 0 0 9

For the proof of claim 2), let us note that the terms pm(x) := rm eim' are harmonic algebraic R  2 polynomials, pm (x) 2 Pm. By (18), IB un(x
)pm (x) (dx) = 0 for all m < n. The same relations are valid for m > n. Indeed, for each xed = un (r cos (# ? ')) is a trigonometric R 2jxj = r  0; un(x
)im' polynomial in ', of degree n, so that 0 un(r cos (# ? '))e d' = 0, if m > n. Thus, for the proof of (22) we need to consider only the case m = n. Let Tn(x) := cos (n arccos x); jxj P1 denote the n-th Chebyshev polynomial of the rst kind. We have u0(x) = T0(x); un(x) = 2 0mn(2) Tm(x) for n  1, and Tn(x) = 2n?1 xn + q(x), where q(x) 2 Pn1?2 . Thus un(x) = 2n?1 xn + q1(x); q1(x) 2 Pn1?2 and for xed n  1; r > 0 and #, un(r cos (# ?R ')) = 2n (r cos (# ? '))n + t (ei') = 2rn cos n(# ? ') + t1 (ei') where t; t1 2 Tn?2. Consequently, 02 un(r cos (# ? '))ein' (d') = rn ein# and  Z1 Z Z 1 Z 2 n  in'  in#  r2n+1 dr; r e un(r cos (# ? ')) (d') dr = 2e pn (x)un(x
) (dx) = 2 r 2

0

IB2

0

0

whence the relations (22) follow. Let us also note that for f = fharm 2 L 1 jf^(n)j2 X kf ; L2 (IB2)k2 = : n=?1 jnj + 1 Further, for a xed vector x; un(x
) is a trigonometric polynomial in #, and un (x
) 2 Tn. Thus, by (16) Z 2 W n W n u (x
'); D n (# ? ')un (x
 ) (d#) = n+1 n 0 n+1 and (23) follow from (17). 2 (IB2 ) 1

2.3 Proof of theorem 1

Now let us consider a ridge function R(x) = PNj=1 Wj (x
j ) 2 L2(IB2). Then the momenta of R(x) are linear combinations of shifted Dirichlet kernels: N X an(R; #) = n +1 1 W j;n Dn (# ? #j ); (24) j =1 where W j;n denotes the n-th Fourier { Chebyshev coecient of the function Wj (x) 2 L2! (?1; 1), cf. (23). Let us upply vector notations: N N X ~ nj2 := X jW j;nj2; U~
V~ = Uj Vj ; W~ n := fW j;ngNj=1 2 C N ; jW j =1

1

j =1

Note that the condition (x) 2 L2(IB2 ) does not guarantee the existence of the boundary values ( ). f

f

10

let also Dn (#~ ) denote the N N symmetric matrix fDn (#j ?#k)gNj;k=1 , and Dn0 (#~) := Dn(#~ )?Dn (0)I = Dn (#~ ) ? (n + 1)I , where I is the N -th identity matrix; z - the conjugate of a complex number z. Lemma 2 Let n be a positive integer, N { a natural number, #~ = f#j gN1 2 IRN , where #j are pairwise distinct mod. Further, let a(#) be a xed polynomial of the class Tn ; ~a(#~ ) := fa(#j )gN1 . Then:



N X

a(#) ? wj Dn (# ? #j ); L2 (IB2)

: 1) Qn[a; #~] = w~min (25) 2C N

j =1 2) Denote w~ = fwj gN1 the vector of optimal weights, or, which is the same, the minimizer of the extremal problem on the right of (25). Then w~ satis es the following system of N linear equations: N X j =1

wj Dn (#k ? #j ) = a(#k ); k = 1; : : : ; N; or Dn (#~ )w~ = ~a(#~ ):

3) rank Dn (#~ ) = dimSpan fDn (# ? #j )gNj=1 = min(N; n + 1): q 4) Qn[a; #~] = 0; n  N ? 1; Qn[a; #~] = ka; L22 k2 ? w~
~a(#~); n  N; where w~ is the vector of optimal weights. 5) sup kDn0 (#~ )kl 7!l = C (#~) < 1; 2

n

2

i. e., the l2 7! l2-norm of the matrix Dn0 (#~ ) is uniformly bounded in n.

Proof. First of all,

Z



N N X X 2



w T ( # ) a ( # ) T ( # )  ( d# ) ? j j :

a(#) ? wj Dn (# ? #j )

= T 2sup IB(T  ) 0 j =1

j =1

n

This relation is a corollary of (16): 1 Z 2 0 Z 2 N N X X @a(#) ? wj Dn (# ? #j )A T (#) (d#): a(#)T (#) (d#) ? wj T (#j ) = 0

j =1

0

j =1

and the resonance case of Cauchy inequality. Thus, (25) follows from the de nition of Qn[a; #~ ]. 11

(26) (27) (28) (29)

Next, (26) follows from (25), because Z 2 Z 2 a(#)Dn(# ? #k ) (d#) = a(#k ): Dn (# ? #j )Dn (# ? #k ) (d#) = Dn (#j ? #k ); 0

0

The system (26) is consistent for every polynomial a(#) 2 Tn . Since dim Tn = n + 1, by Lagrange interpolation over #~, from here we conclude that rank Dn (#~) = dimfDn (#~ )w~ : w~ 2 C N g = dimf~a(#~ ) : a(#) 2 Tn g = min(N; n + 1): Further, dimSpan fDn (# ? #j )gNj=1 = rank Dn (#~ ), because Dn (#~ ) is the Gramm matrix of the system fDn (# ? #j )gNj=1 , which completes the proof of (27). Since Span fDn (# ? #j )gNj=1 = Tn for n  N ? 1, the equalities Qn(a; #~) = 0; n  N ? 1 follow from (27). Remark. Linear independence of the system fDn(# ? #j )gNj=1 for n  N ? 1 and (27) is not a new result, cf. e.g. [2]. The equality

2

N  i X

2

a(#) ? wj Dn (# ? #j ); L2(IB2)

= ka; L22k2 ? w~
~a(#~ ) Qn [a; #~) = w~min N 2C

j =1 is a corollary of (26). Finally, the entries of the matrix Dn0 (#~ ) are majorized by maxj6=k j csc (#j ? #k )j, which implies (29). Theorem 1 follows from the Parceval identity (19), claim 3) of Lemma 1, (23) and the de nition P N ~ of Qn[a; #]. Indeed, by (24) the n-th momentum of a ridge function R(x) = j=1 Wj (x
j ) is a linear combination of shifted Dirichlet kernels: N  X an(R; #) = wj;nDn (# ? #j ); wj;n := nW+j;n1 ; j = 1; : : : ; N: (30) j =1 Therefore, the selection of optimal pro les Wj (x) for a given f (x) 2 L2(IB2) and a non-degenerate set of directional angles #~ = f#j gNj=1 2 IRN is performed in the following three steps. Step 1. Find the point values of the Chebyshev momenta Z an(f; #j ) = f (x)un(x
j ) (dx); j = 1; : : : ; N; n = 0; 1; : : : : IB2

12

Step 2. For each xed n = 0; 1; : : :, solve the system of N linear equations N X Dn (#j ? #k )wj;n = an(f; #k ); k = 1; : : : ; N: j =1

with respect to the unknown w~ n = fwj;ngNj=1 . Note, that all these systems are cosistent. However, the solution is not unique, if n  N ? 2 (\low frequences"), and is unique for all n  N ? 1 (\high frequences"). Step 3. Let 1 X Wj (x) := (n + 1)wj;nun (x); j = 1; : : : ; N: n=0

Due to non-uniqueness on Step 2, optimal pro les Wj (x) are always non-unique, if N  2. However, it is easy to see that these pro les are \unique up to low frequences" { in the orthogonal complement ? P 2 := L2 (IB2 ) P 2 of the subspace of algebraic polynomials P 2 within L2 (IB2 ). Thus, N ?1 N ?1 N ?1 the set of optimal pro les fWj (x)gN1 = fWj (f; #~ ; x; #~ )gN1 , i. e. the minimizer in the problem RN (f ? Proj N [f ]; #~), is uniquelly de ned. ~ (x) = fWj (x)gN1 for a set of N univariate functions In the next statement we apply the notation W and let v v u u N 1 u X u 2 t X jW~ nj2 ; M = 1; 2; : : : ; EM [W~ ] := t (EM [Wj (x
j )]) = u j =1

n=M

Theorem 4 Assume that the components #j of #~ 2 IRN are pairwise distinct mod , and let R(x) = P N W (x
 ). Then j j =1 j   1  ~ (31) EM [W ] = 1 + O#~ M EM [R]; M ! 1; EM [W~ ]  C (#~ )EM [R]; M  N ? 1; where the constant in O#~ and C (#~) depend only upon #~; Further, the operator



N X

N W~ N : f (x) 7! W~ N (f ) = fWj (f; x)gj=1 := arg min

f (x) ? Wj (x
j ); L2(IB2)

j =1

p

is well-de ned, linear and bounded from ? PN2 ?1 into L~ !;N , where !(x) = 2 1 ? x2 and

9 8 N = <

X ~ L~!;N

= kWj (x); L2! (?1; 1)k < 1 : L~!;N := :W~ (x) = fWj (x)gNj=1 :

W; ; j =1 13

Proof. According to (20) and (30),

1 1 D (#~ ) X X n ~ n
W ~ n (EM [R])2 = (n + 1)kan(R); L22 k2 = W n=M n=M n + 1 1 0 1 D0 (#~ ) 1 0 (#~ )  ~ 2 X  X D ~ ~ ~ n
W ~ n ; ~ n W n 2    A @ = jW nj ? n + 1 W n
W n = EM [W ] ? n+1 n=M

n=M

(32)

and making use of (29), we further have 1 0 1 ~  ~ 2 X Dn (#~ ) ~ ~  C (#~) X ~ 2 = C (#) E [W  j W j  W
W n n n M + 1 M +1 M ] ; n=M n + 1 n=M ~ ]  C (#~)EM [R] for and the asymptotic formula in (31) follows. This also implies the estimate EM [W all suciently large M  M0(#~ ). To prove that the same estimate is true for all remaining M , i. e. N ? 1  M < M0(#~), we note that, according to (27), for n  M  N ? 1, all matrices Dn (#~ ) are strictly positive de nite, and thus (cf. (32)) X X Dn (#~ ) ~ ~  W n
W n: jW~ nj2  C 0(#~ ) n + 1 ~ ~ M n 0 then Qopt N [1; IB(Tn )] are bounded below, i. e. QN [1; IB(Tn )]  c" > 0. Using the latter result, it was established in [5] that 8" > 0; 9c" > 0 : RfrN (frad)  c"E2(1+")N (frad). (11) and (14) represent more explicit versions of these results. Preliminary upper estimates of the righthand side of (48) based on Chebyshev polynomials TM (x) appeared in discussions with my colleagues at USC P. Petrushev, B. Popov and O. Trifonov. A subsequent improvement of the type (49) was later communicated to the author by I.I. Sharapudinov. Recently, using properties of discrete Chebyshev polynomials, Sharapudinov [23] proved that the condition pNn ! 1 is necessary and sucient for v u n X u col 1 2 t QN [1; IB(Pn] = min (1 ? P (0)) + P 2(m) ! 0:

P 2PN1

p

m=1

(In view of this result, it seems likely that the factor 2n on the right of (49) can be substituted by a constant.) That the condition pNn ! 1 is necessary, follows also from the lower estimate (47) of errors of quadrature formulas with free nodes.

2.7 Acknowledgements

The author is grateful to his colleagues at USC R. DeVore, R. Howard, P. Petrushev, B. Popov, V.N. Temlyakov, O. Trifonov for many useful discussions and valuable comments. V.N. Temlyakov indicated on the above mentioned results in [20]. I.I. Sharapudinov communicated with the author concerning discrete Chebyshev polynomials. The idea of using incomplete formulas of rectangles in upper estimates of Qopt n;N [1; IB(Tn )] belongs to E.A. Rakhmanov. The lower estimate (47) of 1 fr QN [1; IB(Pn] was established in a close collaboration with B.S. Kashin during his recent visit to USC (April { May 1998). The author also had a number of very useful e-mail communications with B. Bojanov and V. Totik concerning optimal quadrature formulas. An especially warm gratitude is directed to S.M. Nikol'skii, who visited USC in October { November 1997. The author enjoyed several personal discussions with him of the results of the present paper. The work was supported by the NSF Grant DMS-9706883.

26

References [1] S.M. Nikol'skii. Quadrature formulas, 2nd ed., \Nauka", Moscow, 1974 (in Russian). (Translation into English of 1st ed.: Hindustan, Delhi, 1964.) [2] M.E. Davison snd F.A. Grunbaum. Tomographic reconstruction with arbitrary directions. Comm. Pure and Appl. Math., 34(1981), pp. 77 { 120. [3] B. Logan, L. Schepp. Optimal reconstruction of a function from its projections. Duke Mathematical Journal, 42(1975), pp. 645 { 659. [4] A.G. Ramm, A.I. Katsevich. The Radon transform and local tomography. 1996, CRC Press, Inc., 485 pp. ISBN 0-8493-9492-9 [5] K.I. Oskolkov. Ridge approximation, Chebyshev { Fourier analysis and optimal quadrature formulas. Proc. Steklov Math. Inst., 219 (1997), pp. 269 { 285 (in Russian). [6] D.L. Donoho and I.M. Johnstone. Projection-based approximation and a duality with kernel methods. The Annals of Statistics, 17 (1989), No. 1, pp. 58 { 106. [7] H.L. Montgomery. Ten lectures on the interface between analytic number theory and harmonic analysis. expository lectures from the NSF-CBMS Regional Conference held at Kansas State University, Manhattan, Kansas, May 22-25, 1990. (Regional conference series in mathematics, ISSN 0160-7642; no.84), AMS, 220 pp. [8] M. Lachance, E.B. Sa and R.S. Varga. Inequalities for polynomials with a prescribed zero. Mathematische Zeitschrift, 168 (1979), pp. 105-116. [9] H. Groemer. Geometric applications of Fourier series and spherical harmonics. (Encyclopedia of Mathematics and its applications, 61). Cambridge University Press, 1996, 329 pp. [10] R.A. DeVore, K.I. Oskolkov and P.P. Petrushev. Approximation by feed-forward neural networks. Annals of Numerical Mathematics, 4 (1997), pp. 261 { 287. [11] P.P. Petrushev. Approximation by ridge functions and neural networks. Research report 1997:18, Industrial mathematics initiative. USC, 41 pp. To appear in: SIAM Journal on Mathematical Analysis (1998). 27

[12] V.E. Majorov. On best approximation by ridge functions. Preprint, Department of Mathematics, Technion, Haifa, Israel, January 14, 1998. 27 pp. [13] V.E. Majorov. On best approximation by ridge functions. Preprint, Department of Mathematics, Technion, Haifa, Israel, 1997. 16 pp. [14] V.N. Temlyakov. On approximation by ridge functions. Preprint, Department of Mathematics, University of South Carolina, 1996. 12 pp. [15] N.P. Korneichuk. Splines in approximation theory, \Nauka", 1984, 352 pp. (in Russian). [16] V.P. Motornyi. On the best quadrature formula Pnk=1 pk f (xk ) for certain classes of periodic dierentiable functions. { Izvestiia AN SSSR, Ser. matem., 38, no. 3 (1974), pp. 583 { 614 (in Russian). [17] A.A. Zhensykbaev. Monosplines of minimal norm and optimal quadrature formulas. { Uspekhi Matem. Nauk, 36, no. 4 (1981), pp. 107 { 159 (in Russian). [18] K.I. Oskolkov. On the optimality of the quadrature formula with equidistant nodes on classes of periodic functions. Soviet Math. Dokl., 20(1979), No. 6, pp. 1211 { 1214. [19] K.I. Oskolkov. On optimal quadrature formulae on certain classes of periodic functions. Appl. Math. Optim., 8(1982), pp. 245 { 263. [20] V.N. Temlyakov. Approximation of periodic functions. ISBN 1-56072-131-6, Nova Science Publishers, Inc. (Ch. 1, Theorem 1.3, and Ch. 2, Theorem 1.1). [21] B.S. Kashin. On certain properties of trigonometric polynomials with the uniform norm. Trudy MIAN, 145 (1980), pp. 111 { 116; English transl. in: Proceedings of Steklov Math. Inst., issue 1 (1981)). [22] B.S. Kashin. On certain properties of the space of trigonometric polynomials in connection with uniform convergence. Soobsch. AN Gruz. SSR, 93(1979), pp. 281 { 284 (in Russian). [23] I.I. Sharapudinov. On a new application of Chebyshev polynomials orthogonal on equispaced nets. Manuscript (1998), 6pp. Submitted to Mathematical Notes (in Russian).

28