Sobolev estimates for constructive uniform-grid FFT interpolatory ...

Sobolev estimates for constructive uniform-grid FFT interpolatory approximations of spherical functions V. Dom´ınguez∗

M. Ganesh†

arXiv:1407.5152v1 [math.NA] 19 Jul 2014

Abstract The fast Fourier transform (FFT) based matrix-free ansatz interpolatory approximations of periodic functions are fundamental for efficient realization in several applications. In this work we design, analyze, and implement similar constructive interpolatory approximations of spherical functions, using samples of the unknown functions at the poles and at the unikπ form spherical-polar grid locations ( jπ N , N ), for j = 1, . . . , N − 1, k = 0, . . . , 2N − 1. The spherical matrix-free interpolation operator range space consists of a selective subspace of two dimensional trigonometric polynomials which are rich enough to contain all spherical polynomials of degree less than N . Using the O(N 2 ) data, the spherical interpolatory approximation is efficiently constructed by applying the FFT techniques (in both azimuthal and latitudinal variables) with only O(N 2 log N ) complexity. We describe the construction details using the FFT operators and provide complete convergence analysis of the interpolatory approximation in the Sobolev space framework that are well suited for quantification of various computer models. We prove that the rate of spectrally accurate convergence of the interpolatory approximations in Sobolev norms (of order zero and one) are similar (up to a log term) to that of the best approximation in the finite dimensional ansatz space. Efficient interpolatory quadratures on the sphere are important for several applications including radiation transport and wave propagation computer models. We use our matrix-free interpolatory approximations to construct robust FFT-based quadrature rules for a wide class of non-, mildly-, and stronglyoscillatory integrands on the sphere. We provide numerical experiments to demonstrate fast evaluation of the algorithm and various theoretical results presented in the article.

Key words:

Interpolation, Spherical functions, Sobolev norms, Cubature, Spherical integrals

AMS subject classification: 42A15, 65D32, 33C55

∗

Departamento Ingenier´ıa Matem´ atica e Inform´ atica, Universidad P´ ublica de Navarra, Campus de Tudela, 31500, Tudela, Spain. ([email protected]) † Department of Applied Mathematics & Statistics, Colorado School of Mines, Golden, CO 80401. ([email protected])

1

1

Introduction

Approximation of functions defined on the sphere is important for realization and understanding of various processes described in the spherical-polar coordinate system. In particular, approximation of an unknown spherical function with the requirement of exactly reproducing the function at certain locations on the sphere, namely the interpolatory approximation, plays an important role in designing efficient discrete computer models of various continuous systems. A key tool for several large scale simulations is the FFT-based representation (in polar coordinates) of the polynomial interpolatory approximation of a function defined on the circle (and in general any periodic function). The polar-coordinate/periodic case analytical interpolatory representation facilitates fast construction of the approximation without the need to solve any matrix system of interpolation constraint equations. A general approach in approximation theory is to seek a solution (that satisfy certain modeling constraints) in the space of polynomials. Within the space of spherical polynomials, it is an open problem to construct such a powerful matrix-free representation of interpolatory approximations of spherical functions. Indeed, if a standard constraint that the spherical interpolation operator (with truncated Fourier series anstaz) should exactly reproduce polynomials of degree, say N ≥ 3, is imposed, then it is impossible to construct a matrix-free polynomial interpolatory approximation [23]. This important two decade old work of Sloan [23] resulted in addressing several theoretical and practical questions, including efficient design of points on the sphere, see [4, 5, 16, 21, 24] and extensive references therein. It is still an open problem to prove the numerically observed O(N ) Lebesgue constant growth of some of the very efficient matrix-dependent spherical polynomial interpolator operators [24]. As discussed in detail in [4, 16, 24], the quality of spherical interpolatory approximation (determined by the Lebesgue constant of the interpolation operator) is important. Further, mathematically establishing error estimates of the approximation is crucial for quantifying the validity of various computer models that use such approximations. For applications, in addition to providing a fast procedure for evaluating interpolatory spherical approximations, it is important to prove associated error estimates in the L2 and Sobolev (energy) norms. This is because robust error estimates in various approximate computer models are usually established in such norms. Our main focus in this article is on such practical (matrix-free and FFT-based) considerations and associated robust mathematical analysis. To this end, we do not require that the interpolatory spherical approximations need to be in the space of spherical polynomials. In [7, 9, 14] a finite dimensional space χN (containing the space of spherical polynomials of degree less than N ) was introduced. Well-posedness of the χN -space based interpolation problem was established in [7], using equally spaced 2N azimuthal angles in [0, 2π) and arbitrary N + 1 elevation latitudinal angles in [0, π] so that the total number of interpolation points on the sphere (including the north and south poles) is equal to the dimension of χN . For the special choice of the N + 1 non-uniform latitudinal angles in [0, π] that are based on Gauss-Lobatto points, as shown in [14], the χN -space interpolation problem is matrix-free and Sobolev error estimates for this spherical approximation was proved in [7]. If the N + 1 latitudinal angles in [0, π] are equally spaced, then the χN -space uniquely solvable interpolation problem was also shown to be matrix-free in [14] and the growth of the Lebesgue constant of this interpolation operator is only O(log2 N ). This Lebesgue constant growth (and hence error estimates in the uniform norm) was proved in [9, 14]. As demonstrated by Sloan and Womersley in [24] for benchmark smooth and non-smooth functions, this non-polynomial interpolatory spherical approximation, with proven optimal Lebesgue constant, perform better than several matrix-dependent polynomial interpolatory approximations. 2

Practical construction and analysis of the matrix-free interpolation operator in this article is completely different from that in [14]. The main aim of this article is on the efficient construction of uniform-grid interpolatory spherical approximation using only the FFT operators and to provide robust mathematical analysis for quantifying the interpolatory approximations in the Sobolev norm. This article is the final of the three part constructive approximation theory and implementation work by Ganesh et al. [14, 7] on the non-polynomial range space S2 interpolation framework introduced in [9] (for a 3D potential theory computer model). Our new FFT-based construction and Soboleve space analysis presented in this article have potential applications in various large scale computer models that require approximation of spherical functions. In particular, in our future work, we shall focus on two important classes of specific applications of the interpolatory approximations developed in this article: (i) Deterministic and stochastic three dimensional wave propagation models, for evaluation of statistical quantities and uncertainty quantification in multiple particle configurations [10, 11, 12, 13]; and (ii) Advanced radiation transport (RT) computer models [17, 18]. The acoustic and electromagnetic wave propagation Galerkin computer models developed by Ganesh and Hawkins [10, 11, 12, 13] depend on local spherical-polar coordinate system based approximations (of surface currents and integrals). These coordinate systems are imposed either on various patches on the surface of a single scatterer [11] or on individual particles in multiple particle configurations [10, 12, 13]. The future advanced algorithms and analysis for the three dimensional wave propagation models will be based on the interpolatory collocation version of deterministic and stochastic algorithms in [10, 11, 12, 13]. These algorithms will also require efficient interpolatory cubature rules for medium to highly oscillatory integrals on the sphere. The linear RT equation (RTE) poses a significant computational challenge, even for the next generation of super computers, because of the high-dimensional phase space on which it is posed. In general, the solution of the RTE (an integro-differential equation) is a function of seven independent variables: one temporal variable, three spatial variables, one energy variable and two angular spherical-polar coordinate variables (describing the direction of radiation motion). The integral part of the RTE is an integral on the sphere and, in practice, integrands with limited smoothness properties on the sphere (similar to functions in χN ). The industrial standard approach hitherto is to apply cubature on the sphere with certain symmetry properties, such as that in [3]. However, a recent derivation [2] demonstrates that interpolatory approximation based cubature on the sphere are efficient. As discussed in the conclusion section in [2], solutions to the RTE in practice are poorly behaved in the angular variables. Hence interpolatory approximations and associated interpolatory cubature on the sphere based on approximations in the non-polynomial space χN will facilitate developing future advance RT computer models. In addition to the requirement of interpolatory approximations in both the above classes of applications, an important common tool required in these future application work is an efficient FFT evaluation based interpolatory cubature on the sphere with non-, mildly-, and stronglyoscillatory integrands and quantify the error in such cubature rules for integrands with limited smoothness properties. After developing (i) an FFT-based interpolatory approximation in Section 2; (ii) introducing a functional framework in Section 3; and (iii) proving the quality of our spherical interpolatory approximations in Section 4, we develop an efficient FFT-based interpolatory cubature on the sphere with error estimates in Section 5. Numerical results in Section 6 (and Appendix A) demonstrate various constructive and theoretical results developed and proved in this article and efficiency over a recent matrix-free interpolatory construction [7]. We conclude this article in Appendix B with proofs of some technical results stated in Section 3. This will also be of independent interest for analysis/applications on rotationally invariant manifolds [15]. 3

2

An interpolatory approximation of spherical functions

b ∈ S2 , using the standard convention: Let S2 be the unit sphere in R3 parameterized, for x θ, φ ∈ R.

b = p(θ, φ) := (sin θ cos φ, sin θ sin φ, cos θ), x

(1)

For any continuous functional F ◦ defined on the sphere, we denote F := F ◦ ◦ p and observe that F (θ, φ) = F (θ + 2π, φ) = F (θ, φ + 2π),

F (θ, φ) = F (−θ, φ + π),

∀θ, φ ∈ R.

(2)

Conversely, for any continuous functional F on R2 satisfying (2), there exists a unique associated function F ◦ on the sphere. Motivated by this observation, we define the space  C := F : R2 → C : F is continuous and satisfies (2) (3) which in view of (2) can be identified with C(S2 ), the space of complex valued continuous functionals on the unit sphere. In this work we will introduce a trigonometric interpolant for functions F ◦ on C(S2 ) using the following details: For N ∈ N with N ≥ 2, consider the equally spaced grid points θj =

jπ , N

φk =

kπ , N

j, k ∈ Z.

We recall that, for any φ ∈ R, the north and south poles are respectively p(θ0 , φ) = n and p(θN , φ) = s. Using (1), the parametrized uniform grid GN = {(θj , φk ) : j = 0, . . . , N, k = 0, . . . , 2N − 1}

(4)

in [0, π] × [0, 2π) corresponds to a grid of 2N (N − 1) + 2 distinct points on the sphere. We also note that, similar to (2), we have p(θj , φk ) = p(θj , φk+2N ) = p(θj+2N , φk ),

p(θj , φk ) = p(θj+N , φk−N ).

(5)

It is convenient to introduce the space of even and odd functions DeN DoN

:= span hcos jθ : j = 0, . . . , N i = span hcosj θ : j = 0, . . . , N i

(6)

j

:= span hsin jθ : j = 1, . . . , N + 1i = span hsin θ cos θ : j = 0, . . . , N i,

(7)

and then, as in [7, 9, 14], we consider a 2N 2 − 2N + 2 dimensional subspace of C, defined as  X X χN := p0 (θ) + sin2 θ pm (θ) exp(imφ) + pm (θ) exp(imφ) : −N 1 and Cs,t independent of f . Moreover, pm N f (0) = f (0),

and

pm N f (π) = f (π).

Proof. For m 6= 0 we can choose pm N to be the truncated partial sum Tm Nf

:=

X

m fbm (n)Qm n ∈ spanhQn : n ≤ N − 1i ⊂

0≤n≤N −1



DeN −1 , DoN −2 ,

if m is even, if m is odd,

with the choice of Qm n = 0 for n < |m| in (29) so that the sum above is void for n < |m|. The s implies definition of the norms of Wm 2 kTm s = N f − f kWm

X
n + 1 2s |fbm (n)|2 ≤ N + 1 2s−2t kf k2 t . Wm 2 2

(51)

n≥N m We observe that (50) holds actually for any t ≥ s and also that for m 6= 0, pm N f (0) = pN f (π) = 0 1 and f ∈ Wm vanishes at {0, π}. For m = 0, we cannot ensure that T0N f ({0, π}) = f ({0, π}) which prompts us to consider a different projection. In this case, we choose p0N to be the interpolant

p0N ∈ DeN ,

such that f (ηj ) = f (ηj ), 14

j = 0, . . . , N

where {cos ηj }j are the Gauss-Lobatto quadrature points, which includes the endpoints, that is, η0 = 0, and ηN = π. In [7, Appendix A, Proposition 6] we proved that for all t > 1 there exists Ct > 0 such that kp0N f − f kW00 ≤ Ct N −t kf kW0t , ∀f ∈ W0t . (52) Using (49) and (51), we first obtain kp0N f − f kW0s

≤ kp0N (f − T0N f )kW0s + kT0N f − f kW0s
s−t ≤ CN s kp0N (f − T0N f )kW00 + N + 21 kf kW0t
s−t   kf kW0t ≤ C (N + tf rac12)s kp0N f − f kW00 + kT0N f − f kW00 + N + 21 i h

s s−t kf kW0t . ≤ C N + 12 kp0N f − f kW00 + 2 N + 12

and hence the desired result (50) follows by applying (52)



Lemma 4.3 ([7, Proposition 4]) For f ∈ Z01 , there exists C > 0 such that  1/2 1/2  kf kH 0 ≤ C kf kL2 + kf kL2 kf 0 kL2 sin

#

DeN ,

For the next result, we introduce sN , seN ∈ of the sin(·) function on [0, π]. That is, DeN

sN

∈

seN

∈ DeN

such that such that

sin

sin

the orthogonal and interpolating approximations

sN (θj ) = sin θj , j = 0, . . . , N, Z π (e sN (θ) − sin θ)pN (θ) dθ = 0,

(53a) ∀pN ∈ DeN .

(53b)

0

Lemma 4.4 For all N ≥ 1 2 2 , ksN − seN kL∞ (0,π) ≤ . πN πN Proof. We prove the result for even N . The odd N case follows similarly. Straightforward calculations show that for θ ∈ [0, π] ke sN − sin(·)kL∞ (0,π) ≤

sin θ =

∞ 2 4 X cos 2jθ − . π π 4j 2 − 1 j=1

On the other hand, we have the aliasing effect

qeN cos((2j + 2`N ) · ) = cos(2j · ) = qeN cos((−2j + 2`N ) · ) ,

0 ≤ j ≤ N,

∀ ` ∈ Z.

These properties imply that N/2

seN

=

sN (θ) =

2 4 X cos 2jθ − π π 4j 2 − 1 j=1   ∞ X 2 2 1− π (2`N )2 − 1 `=1

N/2−1 

 ∞  X 4 X 1 1 1 − + + cos 2jθ π 4j 2 − 1 (2j + 2`N )2 − 1 (−2j + 2`N )2 − 1 j=1 `=1   ∞ X 4 1 1 − + cos N θ. π N2 − 1 (N + 2`N )2 − 1 `=1

15

Then, for any θ ∈ [0, π], 4 |e sN (θ) − sin θ| ≤ π

∞ X j=N/2+1

1 2 2 = ≤ . −1 π(N + 1) πN

(54)

4j 2

On the other hand, |e sN (θ) − sN (θ)| ≤

 ∞ 1 4 X π (`N )2 − 1 `=2

+

N/2−1 ∞  X X j=1

=

=

=

4 π 4 π 2 π

X ∞ `=1

`=1

1 1 + (2j + 2`N )2 − 1 ((N − 2j) + (2` − 1)N )2 − 1



 N/2−1 ∞ X X 1 1 + (N + `N )2 − 1 (2j + `N )2 − 1 j=1

N/2 ∞ X X j=1 `=1 ∞ X j=N +1

`=1

N ∞ 1 2 XX 1 ≤ 2 (2j + `N ) − 1 π (j + `N )2 − 1 j=1 `=1

1 2 = . j2 − 1 πN 

In order to prove the next result, it is convenient to consider the quadrature rules L1N g

:=

N π X00 g N

kπ N




L2N g

,

N π X (j−1/2)π g( N ). := N j=1

k=0

Since L1N

 cos(m·) =

π, if m = 2`N , 0, otherwise,

L2N

 cos(m·) =

(−1)` π, if m = 2`N , 0, otherwise,

we easily deduce the equalities L1N gN = L2N gN =

π

Z

gN (θ) dθ,

∀gN ∈ De2N −1 ,

0

and therefore L1N (|gN |2 ) = L2N (|gN |2 ) = kgN k2H 0 , #

∀gN ∈ DeN −1 ∪ DoN −1 .

(55)

Moreover, L1N (|gN |2 ) ≤ 2kgN k2H 0 , #

L2N (|gN |2 ) ≤ kgN k2H 0 , #

∀gN ∈ DeN .

(56)

Besides, for g sufficiently smooth L2N g − L1N g = −4

∞ Z X `=0

π

g(θ) cos(2N (1 + 2`)θ) dθ.

0

16

(57)

That is, the difference in the quadrature rules is four times the sum of the Fourier coefficients in the cosine series of order 2(1 + 2`)N . The well known estimate for the error of the composite rectangular rule Z π Z π 1 LN g − ≤ πN −1 |g 0 (θ)| dθ (58) g(θ) dθ 0

0

will be used repeatedly in this section. Further it is useful to note the relation
L12N = 12 L1N + L2N .

(59)

1 , with m 6= 0, then f is continuous with In the proofs below we use the fact that f ∈ Wm f (0) = f (π) = 0. 1 with m 6= 0. Then Proposition 4.5 Suppose that Hypothesis 1 holds. Let f ∈ Wm   −1 kqm kf 0 kL2 + N −1/2 kqm N f kL2 ≤ C kf kL2 + N N f kH 0 sin

sin

sin

(60)

#

with C independent of f and m. Proof. Since 2 kqm N f kL2sin

π

Z

2 |qm N f (θ)| sN (θ) dθ

=

Z +

0

π 2 |qm N f (θ)| (sin θ − sN (θ)) dθ,

(61)

0

using Lemma 4.4, 2 kqm N f kL2sin

Z ≤ 0

π

2 1 −1 m gN (θ) dθ + CN −1 kqm kqN f k2H 0 N f kH 0 = L2N gN + CN #

#

2 e where gN := |qm N f | sN ∈ D3N . Applying (59) we deduce the bound
2 1 −1 m 2 1 1 kqN f kH 0 . kqm N f kL2 ≤ LN gN + 2 LN gN − LN gN + CN #

sin

(62)

Since f (0) = f (π) = 0 and qm N f (θj ) = f (θj ), for all j = 0, . . . , N , using Lemma 4.4 we easily deduce the following bound: L1N gN

=

N −1 N −1 π X π X m 2 |f (θj )| sin θj + |qN f (θj )|2 (sN (θj ) − sin θj ) N N j=1

≤

1 N

N −1 X

j=1

2 −1 1 2 |f (θj )|2 sin θj + CN −1 L1N (|qm LN (|qm N f | ) =: E1 + CN N f | ).

(63)

j=1

Also, using (56), 2 m 2 L1N (|qm N f | ) ≤ 2kqN f kH 0 .

(64)

#

On the other hand, E1 in (63) can be bounded using the error of the rectangular rule (58):   Z π Z π 2 1 2 2 |f (θ)| sin θ dθ + LN (|f | sin()) − E1 = |f (θ)| sin θ dθ 0 0 Z π π 2 ≤ kf kL2 + |(|f (θ)|2 sin θ)0 | dθ sin N 0 Z Z π π π π 2 2 ≤ kf kL2 + |f (θ)| dθ + |2f (θ)f 0 (θ)| sin θ dθ sin N 0 N 0   Z π ≤ kf k2L2 + CN −1 kf k2L2 + kf kL2 kf 0 kL2 + |f (θ)f 0 (θ)| sin θ dθ sin

≤

kf k2L2 sin

sin

sin

+ CN

−2

kf 0 k2L2 . sin

sin

0

(65) 17

We stress that in (65) we have used Lemma 4.3 and, in the last step, the inequality Z π 0 kf kL2 kf kL2 + |f (θ)f 0 (θ)| sin θ ≤ N kf kL2 + N −1 kf 0 kL2 . sin

sin

sin

0

sin

2 e Finally, using the fact that |qm N f | sN ∈ D3N and applying (57) and Hypothesis 1, we deduce the bound Z π
1 2 2 |qm LN gN − L1N gN = −2 N f | (θ)sN (θ) cos(2N θ) dθ 2 Z0 π 2 |qm = −2 N f | (θ) sin θ cos(2N θ) dθ 0 Z π 2 −2 |qm N f | (θ)(sin θ − sN (θ)) cos(2N θ) dθ 0 Z π (2) (3) 2 −1 m |qm kqN f k2H 0 , (66) ≤ max{cH , cH } N f | (θ) sin θ dθ + CN #

0

(2)

(3)

where we have used again Lemma 4.4. Collecting (63)-(66) in (62) (with cH := max{cH , cH }) we conclude   2 2 −2 2 (1 − cH )kqm kf 0 k2L2 + N −1 kqm N f kL2 ≤ C kf kL2 + N N f kH 0 . sin

sin

sin

#

Hence the desired result (60) follows.



Corollary 4.6 Suppose that Hypothesis 1 holds. For all r > 1 there exists Cr > 0 so that for r [with f (0) = f (π) = 0 for m = 0] all f ∈ Wm   −1 −r r , kqm kf k kf k (67) 0 ≤ C kf kW 0 + N 1 + N W W N f kWm m m m with Cr independent of N , m and f . Proof. In light of Proposition 4.5, we just have to bound −1/2 m N −1/2 kqm kqN f − f kH 0 + N −1/2 kf kH 0 . N f kH 0 ≤ N #

#

(68)

#

For the second term we can apply Lemma 4.3 and the inequality 2ab ≤ N 1/2 a + N −1/2 b, to show that   1/2 1/2
N −1/2 kf kH 0 ≤ CN −1/2 kf kL2 + kf kL2 kf 0 kL2 ≤ C kf kL2 + N −1 kf 0 kL2 . (69) sin

#

sin

sin

sin

sin

On the other hand, (48) implies that for all r > 1 there exists Cr so that −r N −1/2 kqm kf kH r−1/2 ≤ Cr0 N −r kf kWmr , N f − f k H 0 ≤ Cr N #

(70)

#

where we have applied in the last step Proposition 3.2. We note that Cr is again independent of m. Applying (69) and (70) in (68) we deduce the bound   −1 N −1/2 kqm kf kWm1 + N −r kf kWmr N f kH 0 ≤ Cr kf kH 0 + N #

#

and hence the desired results (67) follows.

 18

1 , it is convenient to introduce the notation To prove stability estimates in Wm π

Z

kf k2L2 r sin

|f (θ)|2 sinr θ dθ,

:= 0

(in particular with r = 1, −1, −2 or −3) and use the equivalence of norms described in Theorem 3.1 involving the two terms |m|kqm N f kL2

and

sin−1

0 k(qm N f ) kL2 . sin

(71)

The second term is easily controlled by using inverse inequality and the results developed so far. Lemma 4.7 Suppose that Hypothesis 1 holds. There exists C > 0 independent of m, f and N such that   0 −1 (72) k(qm kf kWm0 + N −1 kf kWm1 + N −r kf kWmr . N f ) kL2 ≤ CN sin

Proof. Lemma 4.1 and the equivalent norms presented in Theorem 3.1 allow us to conclude, as a byproduct, the inverse inequality 0 m k(qm N f ) kL2 ≤ CN kqN f kL2 . sin

sin

Hence applying Corollary 4.6 will lead to the derivation of the bound (72).

(73) 

The analysis of the first term in (71) is rather more delicate. Thus, before entering in the analysis we need to prove some technical results. Lemma 4.8 There exists C > 0 so that for all rN ∈ DeN −2 with N ≥ 2, |rN (0)|2 + |rN (π)|2 ≤ CN 2 log N L1N (|rN |2 sin(·)).

(74)

Proof. For each j = 1, . . . , N − 1 define j+1 LN sin2 θj j := (−1)

sin N θ ∈ DeN −2 . N sin θ(cos θ − cos θj )

It is a simple exercise to verify that LN j (θi )

 =

1, 0,

i = j, otherwise.

e Thus, {LN j } is the Lagrange basis for the interpolation problem on DN −2 with grid points −1 {θj }N j=1 . Thus

rN =

N −1 X

rN (θj )LN j .

j=1

Since j+1 2 cos2 (θj /2), LN j (0) = (−1)

19

we obtain |rN (0)|

2

≤ ≤

 NX −1

2 2|rN (θj )|

j=1  4N 2

π

2N ≤ π

 NX −1

−1

sin

 θj

j=1

 N −1 π X 2 |rN (θj )| sin θj N

 1 1 L (|rN |2 sin(·)), j N

X 1≤j≤N/2

(75)

j=1

(76)

where we have used the inequality sin θN −j = sin θj ≥

2θj 2j = , π N

∀j = 0, . . . , bN/2c.

Hence, for N ≥ 2, the desired result (74) for rN (0) follows from the inequality X 1≤j≤N/2

1 ≤ 2 log N. j

The bound for rN (π) in (74) can be established analogously.



Next we establish bounds for qm N , by investigating separately the cases for even m (i.e., e operator qN ) and for odd m (i.e., operator qoN ). Proposition 4.9 Suppose that Hypothesis 1 holds. There exists C > 0 such that for any integer 2 , m 6= 0 and f ∈ W2m |2m| kqeN f kL2

sin−1

  −1 ≤ C kf kW2m kf kW2m . 1 + N 2

(77)

Proof. We assume throughout this proof that f is a real valued function. We consider the function g2N (θ) := |(qeN f (θ))|2 / sin2 (θ)s2N (θ) ∈ De4N −4 . Using the definition of s˜2N in (53) and Lemma 4.4, Z π e Z π Z π e qN f 2 qN f 2 e 2 kqN f kL2 = g2N (θ) dθ + s2N (θ) − s2N (θ)) dθ se2N (θ) dθ = (e −1 sin θ sin θ sin 0 0 0 1 ≤ L12N g2N + kqe f k2 2 πN N Lsin−2 1 = L1N g2N + 12 (L2N g2N − L1N g2N ) + kqe f k2 2 . (78) πN N Lsin−2 Proceeding similarly as in (66), using Hypothesis 1 (for qeN f (θ)/sin θ ∈ DoN −2 ) and again Lemma 4.4, we obtain Z π 2 1 1 gN (θ) cos(2N θ) dθ 2 (LN g2N − LN g2N ) = −2 0 Z π e Z π e qN f (θ) 2 qN f (θ) 2 s2N (θ) − sin θ) cos(2N θ) dθ = −2 sin θ sin θ cos 2N θ dθ − 2 sin θ (e 0 0 Z π e qN f (θ) 2 2 (2) e 2 ≤ cH sin θ sin θ dθ + πN kqN f kL2sin−2 0 (79) 20

Using (79) in (78) and the identity 2
L1N |qeN f / sin(·) = kqeN f k2L2

sin−2

,

we easily derive the bound kqeN f k2L2

≤

sin−1

1

h (2)

1 − cH

L1N g2N +

2
i 3 1 LN |qeN f / sin(·) . πN

(80)

The first term in the above bound can be estimated as follows. Using the definition of s2N , (58) and the Cauchy-Schwarz inequality (combined with the inequality 2ab ≤ N a2 + N −1 b2 ), we obtain L1N g2N

N −1 N −1 N −1 π X |f (θj )|2 π X |qeN f (θj )|2 π X |qeN f (θj )|2 = s (θ ) = 2N j N N sin θj N sin θj sin2 θj j=1 j=1 j=1   Z π Z π Z π dθ π 0 2 dθ 2 dθ 2|f (θ)f (θ)| ≤ |f (θ)| |f (θ)| + + sin θ N 0 sin θ sin2 θ 0 0  
3π π kf k2L2 + kf k2L2 + 2kf 0 k2L2 . ≤ 1+ 2 −1 −3 2 2N sin sin sin−1

=

(81)

On the other hand using the fact that qeN f ∈ De2N , we obtain (qeN f / sin)(0) = (qeN f / sin)(π) = 0. Thus it is an even function which also satisfies (qeN f )(0) = f (0) = 0 = f (π) = (qeN f )(π). We have L1N



2

qeN f / sin(·)

N −1 N −1 π X qeN f (θj ) 2 π X f (θj ) 2 = sin θj = N sin θj N j=1 j=1   Z π Z π Z π π dθ |f (θ)|2 0 2 dθ dθ + + 2f (θ)f (θ) ≤ 2 |f (θ)| N sin2 θ sin3 θ sin2 θ 0 0 0 h
 i ≤ N π + 12 kf k2L2 + N −2 (3π + 21 )kf k2L2 + πkf 0 k2L2 . sin−1

sin−3

sin−1

In other words, h 2  N −1 L1N |qeN f / sin(·) ≤ C kf k2L2

sin−1

 + N −2 kf 0 k2L2

sin−1

+ kf k2L2

sin−3

i .

(82)

Plugging (81) and (82) in (80), and taking into account the definitions of the equivalent norms k · kZm  1 and k · kZ 2 (see Theorem 3.1), we obtain the desired result (77). m For the next result, we recall that the equivalence norms relation in (37) is valid only for |m| ≥ 2. Proposition 4.10 Suppose that Hypothesis 1 holds. There exists C > 0 such that for any 2 f ∈ W2m+1 with m 6= −1, 0, |2m + 1| kqoN f kL2

sin−1

≤ (C + C 0

p   log N ) kf kW2m+1 + N −1 kf kW2m+1 . 1 2 21

(83)

Proof. Following the same steps as in the proof of Proposition 4.9, using Hypothesis 1 again with j = 1 (because qoN f (θ)/ sin(θ) ∈ DeN −2 ), we obtain 2 i 3 1 o 1 h 1 L g + L q f / sin(·) . (84) kqoN f k2L2 ≤ 2N N N N (1) πN sin−1 1−c H

First term can be treated as in (81) to get h L1N g2N ≤ C kf k2L2 + N −2 kf k2L2 sin−1

sin−3

+ kf 0 k2L2

sin−1


i .

(85)

The main difference compared with the even 2m case dealt in the previous Proposition arises in the second term, since we now have   N −1  2 π X |f (θj )|2 π  qoN f  2  qoN f  2 1 o LN qN f / sin(·) = (0) + (π) + 2N sin sin N sin2 θj j=1

=: S1 + S2 . The first term S1 did not appear in the proof of Proposition 4.9, since q
oN f ∈ DoN −2 , or, equiva
lently, qoN f / sin ∈ DeN −2 . Thus, we can expect (qoN f / sin (0), (qoN f / sin (π) 6= 0. Clearly, the second term can be bounded as in (82):  i h + kf k2L2 . (86) N −1 S2 ≤ C kf k2L2 + N −2 kf 0 k2L2 sin−1

sin−1

sin−3

For S1 we apply Lemma 4.8 and we follow arguments similar to the derivation of (81) to obtain N

−1

S1 ≤ C log N

L1N

 N −1 π X |f (θj )|2 = C log N N sin θj j=1 i + N −2 kf 0 k2L2 + N −2 kf k2L2 . 

|qoN f |2 / sin(·)




h ≤ C 0 log N kf k2L2

sin−1

sin−1

(87)

sin−3

Thus we have proved the inequality kqoN f k2L2

sin−1

h ≤ (C1 + C2 log N ) kf k2L2

sin−1

+ N −2 kf 0 k2L2

sin−1

+ N −2 kf k2L2

The desired result (83) now follows from Theorem 3.1.

sin−3

i . 

s norms. Now we are ready to establish convergence estimates for qm N similar to (48) in k · kWm

2, Theorem 4.11 Suppose that Hypothesis 1 holds. There exists C > 0 so that for any f ∈ Wm   kqm ≤ C kf kWm0 + N −1 kf kWm1 + N −2 kf kWm2 , (88a) 0 N f kWm h i −1 kq2m ≤ C N kf kW2m kf kW2m , (88b) 1 0 + kf kW 1 + N 2 N f kW2m 2m h i p . (88c) + kf kW2m+1 + N −1 kf kW2m+1 kq2m+1 f kW2m+1 ≤ C(1 + log N ) N kf kW2m+1 1 0 1 2 N

Moreover, for all r ≥ 2 there exists Cr > 0 so that for all m kqm 0 N f − f kWm kq2m N f

− f kW2m 1

kq2m+1 f − f kW2m+1 1 N

≤ CN −r kf kWmr ,

(89a)

1−r

(89b)

≤ CN

r , kf kW2m p r . ≤ C(1 + log N )N 1−r kf kW2m+1

22

(89c)

2 , then f vanishes at 0, π. Then, for m 6= 0, Proof. We recall that for m 6= 0, if f ∈ Wm Corollary 4.6 and Proposition 4.9 yield the bounds (88a) and (88b) for m 6= 0. For m = 0, and under the additional assumption that f (0) = f (π) = 0, the bound (88a) was also established in Corollary 4.6. The estimate (88b) follows from combining (88a) and the inverse inequalities stated in Lemma 4.1. For m 6= 0, (88b) follows from Lemma 4.7 and Proposition 4.9. For m = 0, −1, (88c) can be deduced similarly, via the inverse inequalities applied to estimate (88a). If m 6= 0, −1, (88c) is a consequence of Lemma 4.7 and Proposition 4.10. With pm N being the projection introduced in Proposition 4.2, we observe that m m m kqm N f − f kL2 ≤ kqN (f − pN f )kL2 + kpN f − f kL2 . sin

sin

sin

m Further, f − pm N f (0) = f − pN f (π) = 0 even for m = 0. Thus Corollary 4.6 (or (88) in the cases proven up to now) can be applied to derive the bound   −1 −2 kqm ≤ Cr kf − pm kf − pm kf − pm 1 + N 2 , N f − f kL2 N f kL2 + N N f kWm N f kWm







sin

sin

where C is independent of m and f . Now (89a) follows from Proposition 4.2. To prove (89b), we proceed as before, using Lemma 4.7 and Proposition 4.9, to obtain the inequality i h −1 2m 2m 2m + N kf − p f k + kf − p f k ≤ C N kf − p f k kq2m f − f k 2 1 0 1 Wm . N N N N W2m W2m W2m Hence Proposition 4.2 yields the estimate (89b). The procedure for proving estimate (89c) is completely analogous. We note that (88a)-(88b) for m = 0 in the general case (i.e., for functions not vanishing at {0, π}) can be now deduced from (89a)-(89b).  Next we are ready conclude Section 4 by proving the main spectrally accurate convergence result of this article, namely, Theorem 2.4.

4.3

Proof of Theorem 2.4

The proof starts from (46) where we have derived kQN F − F k2Hs ≤ E1 + E2 + E3 with E1 :=

X

2 kqm s N Fm F − Fm F kWm

(90)

m 2 kqm s N (ρN F − Fm F )kWm

(91)

−N +1≤m≤N

E2 :=

X −N +1≤m≤N

E3

=

 X m≥N +1

+

X  kFm F k2Wms

(92)

m≤−N

For the sake of simplicity we can restrict ourselves to consider only F ∈ span {Ynm } which makes the sums above to be finite. The general case can be deduced by a density argument. Moreover, we can take s ∈ {0, 1} since the result for intermediate values of s follows from the theory of interpolation of Sobolev spaces [19]. 23

The third term can be estimated with the help of (30) and (32), as follows: For t ≥ s, X X
2s−2t N 2s−2t kFm F k2Wmt ≤ N 2(s−t) kF k2Ht . (93) E3 ≤ |m| + 21 kFm F k2Wmt ≤ |m|≥N

|m|≥N

For E1 , we japply (89) to obtain p E1 ≤ C(1 + log N )s N 2s−2t

X

kFm F k2Wmt ≤ C(1 +

p log N )s N 2s−2t kF k2Ht .

(94)

−N +1≤m≤N

Regarding E2 , we apply, sequentially, the definition of ρm N in (45), inequality (38) and estimates (89) to deduce X X 2 E2 ≤ kqm s N Fm+2`N F kWm ≤

−N +1≤m≤N `6=0 X X
2 5 s s kqm N Fm+2`N F kWm+2`N 4 −N +1≤m≤N `6=0

≤


5 s 4

X

2 kqm s N Fm F kWm

|m|≥N

i X h p ≤ C(1 + log N )N 2s kFm F k2Wm0 + N −2 kFm F k2Wm1 + N −4 kFm F k2Wm2 |m|≥N

p ≤ C(1 + log N )N 2s−2t kF kWmt .

(95)

In the last step we have proceeded as in (93). Gathering bounds (94)-(95), we obtain the spectrally accurate convergence estimate (23) in Theorem 2.4.

5

A FFT-based interpolatory cubature on the sphere

As described in the introduction, interpolatory cubature rules on the sphere are important in several applications, including the radiative transfer and wave propagation models. Using the FFT-based spherical interpolatory operator, for a (wavenumber) parameter κ, we develop a cubature rule to approximate the following (non–, mild–, and highly–oscillatory) integral on the sphere: Z π Z 2π ZZ Iκ F := F (θ, φ) exp(iκ cos θ) sin θ dφ dθ = F ◦ (x) exp(iκx · [0, 0, 1])dS(x).(96) 0

S2

0

In the integral above the parameter κ is a real number. Therefore, (96) includes standard integrals as well as a class of highly–oscillatory integrals for large values of κ. In wave propagation applications, the [0, 0, 1] corresponds to the direction of the incident wave. The rotationally invariant property of the sphere facilitates fixing such an incident direction. The above integral occurs, for example, in developing efficient computer models to simulate scattered wave (and its far-field) from an acoustically/electromagnetically small, medium, and large closed obstacles [10, 11, 12, 13] with compact simply connected surface, leading to surface integral reformulations of the model on the sphere. The integral for the κ = 0 case occurs in potential theory and radiative transport models. For the FFT-based efficient cubature approximation of the integral, we first consider a Filontype product integration interpolatory approximation Z π Z 2π
IN,κ F := QN F (θ, φ) exp(iκ cos θ) sin θ dφ dθ. (97) 0

0

24

Using the representation X

(QN F )(θ, φ) = p0 (θ) +

sin θ pm (θ) exp(imφ) +

−N/2 4 |Iκ (F ) − Iκ,N (F )| ≤ Cr κ−` N η(`)−r kF kHr ,

(103)

where Cr independent of N and   0, 3/2, η(`) :=  4,

6

if ` = 0, if ` = 1, if ` = 2.

Numerical experiments

In this section we demonstrate the main interpolatory spectrally accurate approximation result (23) and the high-order cubature approximation result (103) for functions with various order of smoothness. We also demonstrate that the full FFT-based interpolatory approximation developed in this article using the uniform-grid and the QN operator is faster, even for small to medium sized data locations, than another efficient similarly accurate interpolation operator gl Qgl N . We developed the operator QN in [7], using Gauss-Lobatto points in latitudinal angle, that facilitates the use of the standard FFT only in the azimuthal variable. For calculation of the Ht norms, for t = 0, 1, used in (23), we apply the following integral based formulas: Z πZ 2π kF k2H0 := |F (θ, φ)|2 sin θ dφ dθ (104) 0 0 2 2 Z π Z 2π Z π Z 2π ∂F ∂F 1 1 2 2 kF kH1 := kF kH0 (θ, φ) dφdθ + (θ, φ) sin θ dφ dθ. (105) 4 ∂φ sin θ ∂θ 0 0 0 0 Except for some trivial cases, the above norms cannot be evaluated exactly. We computed the above norms for tabulated results in this section using over 150, 000 quadrature points on the sphere, taking into account that some of the functions considered in this section have only limited smoothness properties and hence require fine grids to compute with sufficiently high accuracy.

Experiment #1 (Approximation of smooth and limited smooth functions) For the first set of experiments we consider interpolatory approximation of test functions: F1◦ (x, y, z) :=

1 , 4+x+y+z

Fj◦ (x, y, z) := (1−x2 )5/2−j yz,

for j = 2, 3

and

Recalling (1)-(2), the corresponding equivalent functions are F1 (θ, φ) :=

1 4 + sin θ cos φ + sin θ sin φ + cos θ

Fj (θ, φ) := (1 − sin2 θ cos2 φ)5/2−j sin θ sin φ cos θ, 26

j = 2, 3.

(x, y, z) ∈ S2 .

Clearly F1◦ is a smooth function, and hence our theoretical result (23) suggests superalgebraic convergence QN F1◦ to F1◦ in both the H0 and H1 norms. Computational results in Table 1 validate the theoretical result and demonstrate the power of obtaining machine precision accurate approximation of the smooth function with N = 32. N 008 016 032

kQN F1 − F1 kH0 4.86E-06 2.02E-11 5.78E-15

EoC 17.9 11.8

kQN F1 − F1 kH1 4.40E-05 3.37E-10 7.82E-15

EoC 17.0 15.4

Table 1: Approximation of F1 by QN F1 : Errors and estimate order of convergence (EoC) The functions F2◦ and F3◦ have only limited regularity. It can be shown that, for any ε > 0, F2◦ ∈ H4−ε and F3 ∈ H2−ε . Indeed, using an atlas with local charts around the singularities [points (±1, 0, 0) ∈ S2 ] one can easily see that the Sobolev regularity of Fj◦ are the same as the functions F˜j◦ (y, z) := (y 2 + z 2 )5/2−j y. Hence according to our theoretical result (23), the estimated order of convergence (EoC) in approximating F2 by QN F2 in the H0 , H1 norms are respectively almost 4 and 3 and that for F3 by QN F3 are respectively almost 2 and 1. Computational results in Table 2 validate the theoretical result (23). N 008 016 032 064 128

kQN F2 − F2 kH0 1.43E-03 8.14E-05 5.01E-06 3.12E-07 1.87E-08

EoC

N 008 016 032 064 128

kQN F3 − F3 kH0 2.76E-02 6.92E-03 1.73E-03 4.33E-04 1.08E-04

EoC

4.14 4.02 4.01 4.06

2.00 2.00 2.00 2.00

kQN F2 − F2 kH1 1.41E-02 1.55E-03 1.90E-04 2.36E-05 3.07E-06

EoC

kQN F3 − F3 kH1 3.21E-01 1.58E-01 7.93E-02 3.97E-02 1.99E-02

EoC

3.18 3.03 3.00 2.95

1.02 1.00 1.00 0.99

Table 2: Approximation of F2 and F3 by QN F2 and QN F3 : Errors and EoC

Experiment #2 (Accuracy and fast evaluation comparison with a recent work) For this experiment we compare the performance, in terms of error and computation time, of the FFT-based interpolant developed in this article with the interpolant considered in [7] (and first proposed, not analyzed, in [14]). This interpolant shares the same discrete space, χN , and the nodes in the azimuthal angle {φj }. The difference is on the nodes in the latitudinal angle N which were chosen in [7] to be the non-uniform grid points {θi = arccos ηi }N i=0 where {ηi }i=0 are the Gauss-Lobatto points of the quadrature rule for approximating integrals in [−1, 1]. In other words, η0 = −1, ηN = 1 and ηi for i = 1, . . . , N − 1 are the roots of PN0 (x) where PN is the Lagrange polynomial of degree N . We recall that this non-uniform Gauss-Lobatto points based interpolant is denoted as Qgl N. 27

In [7] we proved that for F ∈ Ht and s = 0, 1, s−t s kQgl kF kHt , N F − F kH ≤ CN

which is, up to the (log N )s/2 term, identical to the error estimate in (23) that we proved for the FFT-based operator QN . For our comparison testing purpose, we have chosen the function " 2  2  2 #3/2 1 1 1 √ − sin θ cos φ + √ − sin θ sin φ + √ − cos θ F4 (θ, φ) = ∈ H4−ε 3 3 3 √ √ √ b ∈ S2 with x∗ = [1/ 3, 1/ 3, 1/ 3]. which corresponds to the function F4◦ (b x) = |b x − x∗ |3 , x The error in H0 and H1 norms and associated EoC are depicted in Table 3 for QN and Qgl N. Similar to our established theoretical results, we observe from Table 3 that although Qgl performs N slightly better, the difference is not significant and the estimated orders of convergence are roughly the same. N 008 016 032 064 128

kQN F4 − F4 kH0 1.48e-03 1.00e-04 6.16e-06 3.66e-07 2.63e-08

EoC

N 008 016 032 064 128

kQN F4 − F4 kH1 1.43e-02 1.88e-03 2.27e-04 2.69e-05 3.82e-06

EoC

3.89 4.02 4.07 3.80

2.93 3.05 3.08 2.82

kQgl N F4 − F4 kH0 1.41e-03 9.03e-05 5.68e-06 3.61e-07 2.47e-08

EoC

kQgl N F4 − F4 kH1 1.50e-02 1.90e-03 2.20e-04 2.83e-05 4.26e-06

EoC

3.97 3.99 3.98 3.87

2.99 3.11 2.96 2.73

Table 3: Approximation of F4 by QN F4 and Qgl F4 : H0 case (top) and H1 case (bottom). The computational effort required for both interplants are however important. In Figure 1, we show the low computational cost of the spherical interpolant constructed in this article compared to even the efficient matrix-free interpolant developed in [7]. The difference in the performance and computational complexity can be easily explained just by examining both interpolants. For construction of the QN based approximation, we proceed as follows (see Proposition 4.5): first, we apply N − 1 FFT transforms of 2N elements and next N + 1 DCT/DST (discrete sine/cosine transform) . Thus the overall computational complexity for the QN operator approximation is O(N 2 log N ) operations. For construction of the Qgl N based approximation, the first step is similar, with N − 1 FFT transforms of 2N elements, but the second step (in the non-uniform grid latitudinal angle) is different: two different (polynomial) interpolation prob2m−1 lems of N + 1 (p2m ) have to be solved N times which amounts about O(N 3 ) N ) and N − 1 (pN operations. Moreover, the interpolant QN developed in this article can be used in a natural way with nested grids which can be easily exploited to construct error estimates almost for free, or, if the data node points are doubled, previous function evaluations can be reused to construct the associated updated interpolatory approximation.

28

Figure 1: Performance of the operator QN (*-values) compared to that of Qgl N (o–values) for various data parameter N = 2j , [top: j = 7, 8, 9, 10 and bottom: j = 11, 12, 13, 14] applied to the data obtained using the function F4 . The ordinate values are the CPUTIME (in minutes) obtained using a single core of a Intel Xeon-5570 2.93GHz processor. The QN operator facilitates application of the standard FFT in both the azimuthal and latitudinal angles while the Qgl N with non-uniform Gauss-Lobatto based points in the latitudinal angles provides the standard FFT evaluation only in the azimuthal variable. 29

Experiment #3 (Performance of the FFT-based cubature) Using the smooth integrand function F2 and for the limited smooth integrand F4 , we have tested the convergence of the cubature rule developed in Section 5 to approximate the integral (96), for κ = 0, 10, 102 , . . . , 106 , corresponding to the total non-, mildly-, and highly-oscillatory integrands. The results for all these cases are given in Tables 4 and 5. Each row of Tables 4 and 5 corresponds to a fixed N . We clearly observe the κ−2 decay of the error, for fixed N . Reading along the columns corresponds to varying values of N . For the smooth function F1 , based on the column results in Table 4, we see the superalgebraic convergence in N , as proved in Section 1. For a fixed κ, for the case of the integrand function F4 with limited regularity, we observe from Table 5 the convergence in general is better than the estimated theoretical result, suggesting that our estimated error result and analysis could be improved in some cases. N \κ 004 008 016

0 5.36E-05 1.17E-08 4.44E-16

1 1.03E-04 1.28E-08 1.28E-14

10 5.01E-05 9.14E-07 2.15E-13

100 6.77E-07 4.28E-09 9.89E-14

103 4.55E-09 5.04E-11 7.01E-16

104 6.93E-11 4.35E-13 2.23E-18

105 7.22E-13 4.24E-15 1.79E-19

106 6.83E-15 4.38E-17 6.07E-20

Table 4: |Iκ (F1 ) − Iκ,N (F1 )| for various parameters κ and N N \κ 004 008 016 032 064 128 256

0 1.04E-04 7.92E-05 4.14E-06 2.22E-08 2.51E-09 1.54E-10 3.33E-12

1 2.67E-03 8.62E-05 4.20E-06 2.33E-08 2.51E-09 1.54E-10 3.31E-12

10 1.72E-03 5.75E-04 9.47E-06 8.44E-08 2.73E-09 1.56E-10 3.34E-12

100 2.40E-05 3.15E-06 1.31E-07 2.26E-08 1.83E-08 4.20E-10 4.93E-12

103 1.64E-07 2.95E-08 1.02E-09 3.67E-11 4.98E-12 2.21E-12 4.26E-13

104 2.48E-09 3.08E-10 2.01E-11 1.00E-12 3.78E-14 5.97E-15 7.06E-16

105 2.58E-11 3.10E-12 2.11E-13 1.07E-14 4.02E-16 1.03E-16 1.11E-17

106 2.44E-13 3.08E-14 1.99E-15 1.01E-16 3.93E-18 9.62E-19 2.58E-19

Table 5: |Iκ (F4 ) − Iκ,N (F4 )| for various parameters κ and N

A

Discussion on and verification of Hypothesis 1

In this section, we present details required to computationally verify the inequalities (21) in Hypothesis 1 and demonstrate that the hypothesis holds for almost all practical values of N . To this end, we first rewrite Hypothesis 1 in a computationally convenient form. Using the cosine change of variables x = cos θ and the Chebyshev polynomial T2N , we rewrite inequality (21) as Z

1

−2

2

2 α

|pN −2 (x)| (1 − x ) T2N (x) dx ≤ −1

cαH

Z

1

|pN −2 (x)|2 (1 − x2 )α dx,

∀pN −2 ∈ PN −2 , (106)

−1

for α = 0, 1, 2 where cαH < 1. Next we consider a polynomial of degree N − 2 represented using the Chebyshev basis: pN −2 =

N −2 X j=0

30

βj+1 Tj .

Then (106) is equivalent to the coefficient based inequality −2

N −1 X

βi βj bij (α, N ) ≤ cαH

i,j=1

N −1 X

βi βj aij (α)

(107)

i,j=1

where Z

1

aij (α) := −1 Z 1

bij (α, N ) :=

Ti−1 (x)Tj−1 (x)(1 − x2 )α dx, Ti−1 (x)Tj−1 (x)T2N (x)(1 − x2 )α dx.

−1

Observe that these quantities are easily computable using the identities
1 Ti+j + T|i−j| Ti Tj = 2 T2N = 2TN2 − 1 1 1 1 1 1 (1 − x2 ) = − T2 (x) + , (1 − x2 )2 = T22 (x) − T2 (x) + 2 2 4 2 4  Z 1 2 − i2 −1 , for even i, Ti (x) dx = 0, otherwise. −1

(108a) (108b) (108c) (108d)

Thus, for numerical verification of the Hypothesis 1, we used the follow algorithm: R1 • Construct an auxiliary matrix C sufficiently large with Ci+1,j+1 = −1 Ti Tj using (108a) and (108d) . −1 N −1 • From C, construct A(α, N ) := (aij (α))N i,j=1 and B(α, N ) := (bij (α, N ))i,j=1 by applying (108a)–(108d). Hence

aij (0) = cij  1 1 aij (1) = − ci+2,j + c|i−2|+1,j + cij 4 2  1 aij (2) = ci+2,j+2 + c|i−2|+1,j+2 + ci+2,|j−2|+1 + c|i−2|+1,|j−2|+1 16  1 1 − ci+2,j + c|i−2|+1,j + cij 4 4  1 bij (α, N ) = ai+N,j+N (α) + a|i−N |+1,j (α) + ai,|j−N |+1 (α) + a|i−N |+1,|j−N |+1 (α) − ai,j (α). 2 • Compute the minimum of the generalized Rayleigh quotient for A and B cH (N ; α) := −2 min

b∈RN −1

b> B(α, N )b b> A(α, N )b

(109)

Both A(α, N ) and B(α, N ) are symmetric. Moreover, A(α, N ) is positive definite. Thus, we can compute the Cholesky factorization A = R> R so that (109) is equivalent to compute the smallest algebraic eigenvalue of the matrix R−> B(α, N )R−1 . We have implemented the above algorithm by precomputing C that facilities acceleration of the algorithm for several values of N . The graphs of cH (·; α) for 2 ≤ N ≤ 214 are depicted in Figure 2 demonstrating the validity of the hypothesis for most practically useful values of N .

31

CH(N;0)

1 0.5 0 4

24

45

66 N

86

107

128

4

24

45

66 N

86

107

128

4

24

45

66 N

86

107

128

129

2838

5547

8256 N

10965

13674

16384

2838

5547

8256 N

10965

13674

16384

2838

5547

8256 N

10965

13674

16384

CH(N;1)

−3

1 0 −1 −2 −3

x 10

CH(N;2)

1 0.5 0

CH(N;0)

1 0.5 0

CH(N;1)

1 0.5 0 129

CH(N;2)

1 0.5 0 129

Figure 2: Plots of computed values of cH (N ; α) defined in (109) for α = 0, 1, 2, respectively in the first three subplots, for 2 ≤ N ≤ 27 ; and in the last three subplots for 27 < N ≤ 214 . These plots, demonstrating cH (N ; α) < 1, numerically validate (for almost all practical cases) (j) that in (21), we have cH < 1, for j = 1, 2, 3. 32

B

Proofs results in Section 3

In this section we provide proofs of Theorem 3.1 and Proposition 3.2.

B.1

Proof of Theorem 3.1

Proof. First we recall that the space  H1 = F : R2 → C : F ∈ H0 , ∇S2 F ∈ H0 × H0 is equipped with the norm kF k2H1

:= =

1 kF k2H0 + k∇S2 F k2H0 ×H0 4 Z Z 1 π 2π |F (θ, φ)|2 sin θ dθ 4 0 0 2 Z π Z 2π Z πZ ∂F 1 + ∂φ (θ, φ) sin θ dθ dφ + 0

0

0

0

2π

2 ∂F sin θ dθ dφ. (θ, φ) ∂θ

(110)

Since, |∇S2 (f ⊗ em )(θ, φ)|2 =

i 1 h m2 2 0 2 , |f (θ)| + |f (θ)| 2π sin2 θ

we obtain for all m ∈ Z, kf k2Wm1

=

1 4

Z

π 2

2

Z

|f (θ)| sin θ dθ + m

0

0

π

dθ |f (θ)| + sin θ 2

Z

π

|f 0 (θ)|2 sin θ dθ.

(111)

0

For |m| ≥ 1, (36) follows from (111). Next we prove that if |m| ≥ 2, Z π Z Z π 1 9m2 π dθ dθ kf k2Wm2 = |f (θ)|2 sin θ dθ + |f (θ)|2 + (m4 − 4m2 ) |f (θ)|2 3 16 0 4 0 sin θ sin θ Z π Z π Z π 0 1 dθ + |f 0 (θ)|2 sin θ dθ + (1 + 2m2 ) |f 0 (θ)|2 + |f 00 (θ)|2 sin θ dθ. 4 0 sin θ 0 0 (112) It is convenient to recall that 1 1 kF k2H2 := k∆2S F k2H0 + k∇2S F k2H0 ×H0 + kF k2H0 . 2 16 Without loss of generality, we assume f to be a real valued function. The proof of (112) requires more calculations and application of integration by parts several times to take care of some cross products appearing in the integral form of the norm. Without loss of generality, for 2 a fixed m ∈ Z, we can assume f ∈ span {Qm n : n ≥ |m|} because this subspace is dense in Wm . Observe that f (0) = f 0 (0) = f (π) = f 0 (π) = 0. (113) Since 1 1 k∇S2 f ⊗ em k2H0 ×H0 + kf ⊗ em k2H0 4 16

=

33

Z Z m2 π 2 dθ 1 π 0 f (θ) + |f (θ)|2 sin θ dθ 4 0 sin θ 4 0 Z π 1 + f 2 (θ) sin θ dθ, 16 0

it is sufficient to analyze the term containing the Laplace-Beltrami operator: Z π Z 2π |∆S2 (f ⊗ em )(θ, φ)|2 sin θ dθ dφ k∆2S (f ⊗ em )k2H0 = 0 0 Z π 2
0 2 m 1 0 − = sin2 θ f (θ) + sin θ sin θf (θ) sin θ dθ 0 Z π Z π
0 2 dθ 1 2 4 0 dθ f (θ) = m + sin θf (θ) sin3 θ 0 0 sin θ Z π
0 dθ 2 f (θ) sin θf 0 (θ) =: m4 I1 + I2 − 2m2 I3 . −2m sin2 θ 0 Using (113) and integration by parts, Z πh 2
0 i cos θ 0 I2 = |f (θ)|2 + sin θ|f 00 (θ)|2 + cos θ |f 0 (θ)|2 dθ sin θ 0 Z π Z π Z π cos2 θ 0 00 2 2 |f 0 (θ)|2 sin θ dθ = sin θ|f (θ)| dθ + |f (θ)| dθ + sin θ 0 0 0 Z π Z π 0 2 dθ = |f (θ)| + sin θ|f 00 (θ)|2 dθ. sin θ 0 0 Proceeding similarly, we derive Z π 0 1 I3 = − f (θ) f 0 (θ) sin θ dθ 2 sin θ Z π Z0 π
0 cos θ 0 2 dθ + f 2 (θ) = − |f (θ)| dθ sin θ sin2 θ 0 0 Z π Z π 1
2 cos2 θ dθ + + f 2 (θ) dθ = − |f 0 (θ)|2 3 sin θ sin θ sin θ 0 0 Z π Z π Z π dθ dθ dθ |f 0 (θ)|2 = − − f 2 (θ) +2 f 2 (θ) 3 . sin θ sin θ sin θ 0 0 0

(114)

(115)

(116)

Inserting (115)-(116) in (114), we obtain (112). The inequalities 1 9m2 + + (m4 − 4m2 ) ≤ m4 < 3m4 , 16 4

1 + (1 + 2m2 ) ≤ 3m2 , 4

∀|m| ≥ 2

with (112) imply the first inequality of (37). For |m| ≥ 3 the second inequality of (37) is simply a consequence of the inequalities m4 − 4m2 > m4 /2 >

m4 , 6

1 + 2m2 > m2 ≥

m2 . 6

The case |m| = 2, has to be analyzed separately since one of the crucial terms, the third term in (112), vanishes: Using (112), we obtain Z π Z π Z π 0 2 dθ 2 2 dθ kf kWm2 ≥ 9 |f (θ)| +9 |f (θ)| + |f 00 (θ)|2 sin θ dθ. (117) sin θ sin θ 0 0 0  As before it suffices to consider f to be real valued and that f ∈ span Q2n : n ≥ 2 . Note that Z π Z π Z π dθ dθ cos2 θ 2 2 f (θ) 3 = f (θ) + f 2 (θ) 3 dθ. (118) sin θ sin θ sin θ 0 0 0 34

Applying integration by parts to the second term and using (113) we obtain Z

π

f 2 (θ)

0

cos2 θ dθ = sin3 θ

Z

π

0


 cos θ  f (θ)f 0 (θ) log(tan(θ/2)) + dθ. sin2 θ

(119)

Notice that for θ ∈ (0, π/2], sin θ| log tan(θ/2)| ≤ 2 tan(θ/2)| log tan(θ/2)| ≤ 2e−1 ≤ 1.

(120)

By symmetry, we can extend this bound for any θ ∈ (0, π). With the help of (120) and the inequality 2ab ≤ a2 + b2 , from (119) we obtain Z 0

π

cos2 θ f (θ) 3 dθ ≤ sin θ 2

≤

Z π f (θ)f 0 (θ) dθ + f (θ)f 0 (θ) dθ sin θ sin2 θ 0 Z π  Z π0 Z dθ 1 π 2 dθ 1 2 0 2 dθ f (θ) |f (θ)| + f (θ) 3 . +2 2 0 sin θ sin θ 2 0 sin θ 0

Z

π

(121)

Inserting (121) in (118) we easily derive Z Z π Z Z π dθ dθ 1 π 2 dθ 3 π 2 dθ f (θ) + |f 0 (θ)|2 + f (θ) 3 f 2 (θ) 3 ≤ 2 sin θ sin θ 2 sin θ sin θ 0 0 0 0 and therefore Z 0

π

dθ f (θ) 3 sin θ 2

Z ≤ 3 0

π

dθ f (θ) +2 sin θ 2

Z

π

|f 0 (θ)|2

0

dθ . sin θ

(122)

From (117) and (122), we obtain Z π Z π Z π dθ 0 2 dθ 2 2 |f (θ)| |f 00 (θ)|2 sin θ dθ 6kf kWm2 ≥ 54 f (θ) + 54 +6 sin θ sin θ 0 0 0  Z π  Z π dθ dθ ≥ 16 3 f 2 (θ) +2 |f 0 (θ)|2 sin θ sin θ Z π0 Z π0 dθ +4 |f 0 (θ)|2 + |f 00 (θ)|2 sin θ dθ sin θ 0 0 Z π Z π Z π 0 2 dθ 2 dθ ≥ 16 |f (θ)| |f (θ)| + |f 00 (θ)|2 sin θ dθ. 3 +4 sin θ sin θ 0 0 0 Hence the inequalities in (37) hold.

B.2



Proof of Proposition 3.2

Proof. Denote by Γ the maximum circle in S2 , parametrized by q(θ) := (sin θ, 0, cos θ).

(123)

Given f ◦ : Γ → C we denote f = f ◦ ◦ q : R → C. The norm in the Sobolev space H r (Γ) can be then defined with the help of q and (41): kf ◦ kH r (Γ) := kf kH#r . 35

The second ingredient we will use in this proof is the trace operator γΓ which can be shown to be continuous from Hr+1/2 (S2 ) onto H r (Γ) for all r > 0 (see [6, 19] for a proof of this result in Rn ; the proof can be easily extended by using local charts of the unit sphere and the equivalent definitions of the Sobolev spaces involved). r , consider the mapping Given f ∈ Wm √ Pm f := 2π(γΓ F ◦ ) ◦ q, F ◦ := (f ⊗ em ) ◦ p−1 . Observe that F ◦ ∈ Hr (S2 ) and that actually f = Pm f , that is Pm is simply the identity operator. Moreover, √ √ kPm f kH#r ≤ 2πkγΓ kHr+1/2 (S2 )→H r (Γ) kF ◦ kHr+1/2 (S2 ) = 2πkγΓ kHr+1/2 (S2 )→H r (Γ) kf kW r+1/2 , m

where kγΓ kHr+1/2 (S2 )→H r (Γ) is the continuity constant of γΓ as a linear operator from Hr+1/2 (S2 ) onto H r (Γ). Hence we obtain (43). 0 ∼ L2 , the first equation in (44) is clear whereas the second equation in (44) for Since Wm = sin m = 0 follows directly from (111) and (42). Finally, if m 6= 0 using f (0) = 0 we observe that Z 0

π/2

dθ |f (θ)| sin θ 2

Z

π/2

= 0

Z

2 Z  Z π/2 √  Z θ 1 θ 0 θ 0 2 f (ξ) dξ dθ ≤ |f (ξ)| dξ dθ sin θ 0 sin θ 0 0

π/2

≤ C

|f 0 (ξ)|2 dξ.

0

Proceeding similarly, but using now that f (π) = 0, we can bound the integral in (π/2, π) and hence conclude that Z π Z π dθ |f (θ)|2 ≤C |f 0 (ξ)|2 dξ. sin θ 0 0 Equation (36) now yields that √ kf kWm1 ≤

5 kf kZm 1 ≤ C(1 + |m|)kf kH 1 . # 2 

Acknowledgment The research of the first author was supported, in part, by grant DMS-1216889 from the National Science Foundation. Support of the Colorado Golden Energy Computing Organization (GECO) is gratefully acknowledged.

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