Slepian functions on the sphere, generalized Gaussian quadrature rule

INSTITUTE OF PHYSICS PUBLISHING Inverse Problems (4 877 89 INVERSE PROBLEMS PII: S66-56(4769-7 Slepian functions on the sphere, generalized Gaussian quadrature rule L Miranian Department of Mathematics, University of California, Berkeley CA, 947, USA E-mail: luiza@math.berkeley.edu Received October 3, in final form March 4 Published April 4 Online at stacks.iop.org/ip//877 (DOI:.88/66-56//3/4 Abstract Denote by K the operator of time band time limiting on the surface of the sphere and consider the problem of computing singular vectors of K. This problem can be reduced to a simpler task of computing eigenfunctions of a differential operator, if a differential operator, which commutes with K and has a simple spectrum, can be exhibited. In Grünbaum et al (98 SIAM J. Appl. Math. 4 94 55 such a second-order differential operator commuting with K on the appropriate subspaces was constructed. In this paper, this algebraic property of commutativity is used to produce an efficient numerical scheme for computing a convenient basis for the space of singular vectors of K. The basis forms an extended Chebyshev system, and a generalized Gaussian quadrature rule for such a basis is presented.. Introduction The fundamental problem of recovering a time-limited function from the knowledge of its Fourier transform on a certain band of frequencies is a central chapter in signal processing. This problem plays an important role in many aspects of image processing since it underlies the question of how to make optimal use of the available information that is always limited and corrupted by noise. The remarkable series of papers [ 8], by Slepian, Landau and Pollak in connection with the issue of time band-limited signals has had a tremendous influence on many areas of engineering, science and mathematics. This work puts some of the pioneering work of Shannon [9] on firmer ground. Their starting points were fairly applied aspects of communication theory, optics, lasers, etc, but it became apparent that the ideas were applicable to many other situations. The work presented in this paper deals with the case when the real line is replaced by the surface of the sphere. Here, the mathematical and computational issue is to get good approximations to the appropriate Slepian functions. In this instance Slepian functions refer to a basis for the space of eigenfunctions of the operator K which is obtained by the 66-56/4/3877+6$3. 4 IOP Publishing Ltd Printed in the UK 877

878 L Miranian successive application of the operations of time, band and time limiting. In the classical case of the real line, the computation of Slepian functions is done using the fact that the classical differential operator, resulting from the Laplacian by separation of variables in prolate spheroidal coordinates, happens to commute with the time band-limiting integral operator. This commuting differential operator has a simple spectrum, hence its eigenfunctions form a basis for the space of eigenfunctions of the integral operator. In the case of the surface of the sphere, it was shown in [], that for a polar cap as well as for two symmetrically placed caps (one at each pole, a certain second-order differential operator D = d [ ( x (b x d ] L(L +x m (b x dx dx x defined on the interval [b, ] commutes with K on the spaces of functions whose dependence on φ is of the form e imφ. The operator D has a simple spectrum, hence its eigenfunctions, which are also eigenfunctions of K, can serve as a basis for the space of eigenfunctions of K. The same applies to the complement, in the sphere, of one or two polar caps. If the region in question has less symmetry, then one can always consider the integral operator, but the search for a commuting local operator has proved elusive. This algebraic property of existence of a commutative differential operator holds the key to a good algorithm. Since the object to be produced is a basis for the time band time-limited functions on a certain region of the sphere, one needs to expand easily in this basis, which requires an efficient quadrature rule for evaluating inner products. It is well known that for a system of functions that forms an extended Chebyshev system, a generalized Gaussian quadrature rule always exists. In [] a method of obtaining such a quadrature rule using the appropriate continuation scheme and well-chosen starting points for Newton s method is described. The operator K happens to be a finite rank Fredholm operator, which implies that it has only a finite number of non-zero eigenvalues. There are many ways to compute the null space of the operator K, but the algebraic property discussed above makes it possible to replace the computation of the eigenfunctions of K by the computation of the eigenfunctions of D. This not only simplifies the task from the numerical point of view, but also produces orthogonal functions which form an extended Chebyshev system, assuring the existence of an efficient quadrature rule. In order to numerically compute the eigenfunctions (of an appropriate self-adjoint extension of D we expand them in the basis of the shifted Legendre polynomials, and reduce the problem to the computation of generalized eigenvalues and eigenvectors of certain sparse matrices. We then use the method that has been advocated recently in []. The object produced is a good basis for the space of the eigenfunctions of K in the case when the regions of interest are either a polar cap or a spherical belt bounded by two parallels. Among the applications envisaged is the problem discussed in [] involving gravity field missions. Due to launch conditions or engineering reasons, the sampling is done not on the whole surface of the earth but on one of the type of regions mentioned above. The computation of the so-called Slepian functions, in this case, has been attempted in the geodesy community without taking recourse to the mathematical and computational advantages that are derived from exploiting the results in [] and the very recent note []. For other geodesy applications, see [3, 4]. In the first of these papers the region where the function is known is the surface of the oceans. This paper is organized as follows. In section. the problem of computing eigenfunctions of the integral operator K is discussed. In section. some properties of shifted Legendre polynomials are recalled. Sections.3 and.4 describe a method for computing Slepian

Slepian functions on the sphere, generalized Gaussian quadrature rule 879 functions using the differential operator. In section.5 the generalized Gaussian quadrature rule for the Slepian functions on the spherical cap is presented and concluding remarks are in section 3.. Numerical computation of the eigenproblem.. Direct computation of eigenfunctions of the integral operator K In this section, an attempt to compute eigenvalues/eigenvectors of the integral operator directly is described, and a more efficient alternative is suggested. As discussed in [], denote by A the polar cap θ arccos(b, φ π. Then the operator K is the finite convolution integral operator. Denote then (Kf (u = u = (sin θ cos φ,sin θ sin φ,cos θ, L P l ( u, u f (u du = A l= A ( L l= m= l where Y lm (u is the usual spherical harmonic with l Y lm (uy lm (u f(u du, and or π Y lm (x, φy lm (x, φ dx dφ =, l +(l m! Y lm (u = Y lm (cos θ,φ = 4π (l + m! P l m (cos θe imφ, l +(l m! Y lm (x, φ = 4π (l + m! P l m (x e imφ, x = cos θ. In the formulae above Pl m (x denotes the associated Legendre polynomial. The operator K is a singular Fredholm operator with rank (L +, hence it has only (L + non-zero eigenvalues. In the following proposition, some properties of the eigenvalues and eigenvectors of K are summarized. Proposition. Consider the finite (L + rank symmetric Fredholm operator as defined above and let H m be the subspaces of functions on the polar cap whose φ dependence is given by e imφ. Then (i There are L +linearly independent orthogonal eigenfunctions of K that belong to the space H ; consequently K has only L +distinct non-zero eigenvalues that correspond to eigenfunctions in H. (ii The L(L +/ non-zero eigenvalues of K have multiplicity : in both subspaces H m and H m K has a simple spectrum, i.e. L m +non-zero distinct eigenvalues, where m =,...,L. (iii L m +eigenfunctions of K belong to H m for all m =,...,L and are orthogonal.

88 L Miranian Proof. Let us fix L and m, and see what the integral operator looks like on H m, i.e. take g(u = f(cos θe i mφ = f(xe i mφ and apply the operator K to it: ( L l Kf(u= Y lm (uy lm (u g(u du A l= m= l π ( L l = Y lm (x, φy lm (x,φ f(x e i mφ dx dφ b l= m= l π ( L l l +(l m! = 4π (l + m! P l m (x e imφ Pl m (x e imφ f(x e i mφ dx dφ = b b L l= m= l l= m= l l l +(l m! 4π (l + m! P l m L = e i mφ l +(l m! b π l= m To simplify the notation denote (l + m! P l m L K(m,x,x l +(l m! = π l= m as the kernel of K in subspace H m. The following observations can be made: π (xpl m (x f (x (e imφ e iφ ( m+ m dφ dx (xp m l (x f (x dx. (l + m! P l m (xp m l (x (i The associated Legendre polynomials Pl m are linearly independent and K(m,x,x = K(m,x,xdefines a singular symmetric Fredholm operator which has only L m + distinct non-zero eigenvalues for all m =,...,L. (ii If m =, then K has only L + distinct non-zero eigenvalues in H, and because of the symmetry of the kernel K(,x,x the corresponding eigenvectors are orthogonal. (iii K(m,x,x = K( m, x, x. The symmetry of the kernel K(m,x,x implies that the eigenvectors of K in H m corresponding to different eigenvalues are orthogonal. (iv If m, then in both H m and H m the operator K has the same kernel, hence L m + eigenvalues of K will be duplicated for every m =,...,L. Attempts to compute the eigenfunctions of the integral operator KF n (u = µ n F n (u directly have not been fruitful. In the experiments discussed below the integral operator with L = 3 was discretized using the Gaussian quadrature rule with N x = -point grid in x = cos(θ variable and N φ = -point grid in φ variable. Let T be the matrix obtained after discretization of the integral operator K. The disadvantages of this method are: (i The size of the matrix T is N x N φ = 44; it depends quadratically on the grid size, which makes computing its eigenvalues/eigenvectors an intensive task. In the alternative approach a grid of 4 points in the x variable is used, whereas in the direct approach it is computationally infeasible because of the reason just mentioned. (ii Only eigenfunctions corresponding to non-zero eigenvalues could be computed directly, hence a procedure that would produce an orthogonal basis for the null space of K is needed. Below is the summary of some numerical experiments. Only the eigenfunctions of T that correspond to non-zero eigenvalues are meaningful, so only these are considered in the text below.

Slepian functions on the sphere, generalized Gaussian quadrature rule 88 4 4 3 3 3...3.4.5.6.7.8.9 3...3.4.5.6.7.8.9 3 3.5.5.5.5.5.5.5.5.5...3.4.5.6.7.8.9.5...3.4.5.6.7.8.9.5.5.5.5.5.5.5.5.5...3.4.5.6.7.8.9.5...3.4.5.6.7.8.9.5.5.5.5...3.4.5.6.7.8.9...3.4.5.6.7.8.9 Figure. Results of the direct discretization (four left-hand figures and eigenfunctions obtained by using the differential operator D (four right-hand figures for m =,,, 3. Functions presented are numbered according to the number of roots, i.e. G 3,G,G, G ; L = 3. (i Exactly four eigenfunctions of T are in subspace H ; three eigenfunctions in H ± ; two eigenfunctions in H ± and one eigenfunction in H ±3. (ii Eigenfunctions F j (x e ±imφ with j =,...,L m +oft in each H ±m are very close to these produced by the alternative procedure below (see figure.

88 L Miranian (iii Eigenfunctions of T corresponding to the eigenvalue cannot be computed by direct discretization. There is an efficient alternative to the procedure described above. It produces an orthonormal basis for the space of eigenfunctions of K. Consider the following second-order differential operator on the interval [b, ]: D = d dx [ ( x (b x d dx ] L(L +x m (b x x. A look at [] will show that this is the appropriate operator D that commutes with the K built there when acting on H m. The operator D defines a Sturm Liouville problem, hence it has a simple spectrum and orthogonal eigenfunctions. Because it commutes with K we can say that the eigenfunctions of D are also eigenfunctions of K when acting on H m. Since we search for an orthonormal basis in the space of eigenfunctions of K, the eigenfunctions of D provide us with such a basis. In order to compute numerically the eigenfunctions (of an appropriate self-adjoint extension of D we expand them in the basis of the shifted Legendre polynomials, and reduce the problem to (generalized eigenproblem for some well-structured matrices... Shifted Legendre polynomials In this section, an overview of some facts about shifted Legendre polynomials is presented. These facts will be used to design a procedure for computing eigenvalues/eigenvectors of the differential operator D. Define shifted Legendre polynomials to be the solutions of the following second-order differential equation: (b x( xs n +(x b S n n(n +S n =. Denote b := (+b/,b := ( b/. The following properties of S n will be useful later: (i recursion relation S n = b S n + b (n + n + S n+ + b n n + S n ; (ii derivative ( x(b xs n = b n(n + n + (S n+ S n ; (iii normalized shifted Legendre polynomials S k S k (k +/b..3. Computation of the eigenproblem DF n = λ n F n : case m = In this section, the problem of computing the eigenfunctions of the differential operator D with m = is reduced to the problem of computing eigenvectors of a certain symmetric tridiagonal matrix. Consider the following eigenproblem: ( d dx [ ( x (b x d dx ] L(L +x F n = λ n F n. (

Slepian functions on the sphere, generalized Gaussian quadrature rule 883 Let ( a n,an,an,... be the coefficients of the expansion of F n (x in the basis of shifted Legendre polynomials F n = ak n S k. k= After substituting F n (x into (, using properties (i, (ii and the linear independence of S k one obtains a recursion relation C k a k + B k+ a k+ + A k a k λ n a k =, ( where A k = k(k +(+b L(L +b B k = k k + b [(k (k + L(L +] C k = k + k + b [k(k + L(L +]. After rewriting ( for the normalized polynomials using property (iii, the recursion relation can be written in the matrix form as Mā n = λ n ā n, (3 where ā n = ( ā n, ān,... T are the coefficients of the expansion in the basis of the normalized shifted Legendre polynomials; M k,k = k(k +(+b L(L +b, b (k + M k,k+ = [k(k + L(L +], (k +3(k + M k+,k = b (k + (k +3(k + [k(k + L(L +], k =,,... and the remainder of the entries of the matrix being zero. Using ā n obtained from (3, we express F n = k= ān S k k. The numerical evidence strongly suggests that the coefficients āk n decay very fast. In practice, this allows us to compute F n = N k= ān S k k,for certain large values of N..4. Computation of the eigenproblem DG n = µ n G n : case m> In this section, the problem of computing eigenfunctions of the differential operator D with m>isreduced to a generalized matrix eigenproblem. Consider the following eigenproblem: [( x (b xg n ] L(L +xg n m (b x G x n = µ n G n. (4 Introduce the following simplifying notation: (i Matrix S = ( S, S, S 3,..., where S k are normalized shifted Legendre polynomials. (ii Matrix A with columns A k = ( ā k, āk, āk,... T, where ā k j are the coefficients of the expansion of F k (x in the basis of normalized shifted Legendre polynomials. (iii F k = S j āj k = SAk. j=

884 L Miranian (iv Also, xf k = = xs j aj k = j= j= ā k j = SQA k, j= ( b S j + a k j ( b S j + b (j + j + S j+ + b j j + S j b (j + (j +(j +3 S j+ + where the symmetric tridiagonal matrix Q has entries Q k,k = b, Q k,k+ = Q k+,k = b (k + (k +3(k +, with =,,,... Similarly x F k = SQ A k. (v Denote c n = ( c n,cn,cn,... T. (vi Matrix = diag(λ,λ,...; A = MA. (vii Denote I to be the identity matrix. b j S j (j +(j Since the functions F k form an orthonormal basis for L [b, ], one could expand G n in the basis of F k, i.e. G n = ck n F k(x = ck n SAk = SAc n = γk n S k (x, (5 k= k= where γ n = (γ n,γn,...t = Ac n. After substituting (5 into(4 and using the fact that F k satisfies ( one obtains c n k (λ k F k m (b x F x k µ n F k =, or ck n (λ kf k ( x m (b xf k µ n F k ( x =. (6 k= Using the notation above, (6 can be written as ck n (λ ks(i Q A k m S(b QA k µ n S(I Q A k k= = S ck n (λ k(i Q A k m (b QA k µ n (I Q A k =. k= Since the functions S k are linearly independent, one obtains = ck n (λ k(i Q A k m (b QA k µ n (I Q A k k= k= = (I Q A c n m (b QAc n µ n (I Q Ac n = (I Q MAc n m (b QAc n µ n (I Q Ac n = ((I Q M m (b Q µ n (I Q Ac n,

Slepian functions on the sphere, generalized Gaussian quadrature rule 885 which implies the generalized eigenproblem (I Q (M µ n Iγ n = m (b Qγ n. (7 Although the coefficients γ n can be computed from the elegant matrix eigenproblem (7, numerical experiments have shown that the coefficients of the expansion G n (x = k= γ k n S k (x decay slowly. However, the scheme above can be significantly improved, as we explain now. Taking into account boundary conditions, write G n = ( x m/ g n (x, for a certain function g n (x. After rewriting (4 in terms of g n (x we arrive at the following eigenproblem: [( x (b xg n ] +mx(x bg n + ((m +m L Lx mb(m +g n = µ n g n. (8 Now, the scheme described above can be applied to the function g n (x = k= cn k F k for some coefficients ck n using the differential equation (8. In this case the derivation of the generalized matrix eigenproblem is similar to that performed at the beginning of this section. After elaborate calculations one arrives at (I Q(M µ n Iα n = (mb(m +(I Q +mq 3 m(m +(Q Q α n, (9 where b Q 3 (k, k = (k +3(k, b Q 3 (k +,k= Q 3 (k, k + = (k +3(k, b (k + Q 3 (k, k + = (k +3 (k +(k +5, b (k + Q 3 (k +,k= (k +3 (k +(k +5, E(k, k = b k(k +, Q 3 = Q 3 E for k =,,, 3,...; matrices Q and M were defined before, and Ac n = α n. Observe that g n (x = ck n F k(x = ck n SAk = S ck n Ak = SAc n = Sα n, k= k= which means that while c n are coefficients of the expansion of g n (x in the basis of F k (x s, ᾱ n s are coefficients of the expansion of g n (x in the basis of normalized shifted Legendre polynomials S k s. From (9 we can compute g n = k= αn S k k, and obtain G n (x = ( x m/ g n (x. Experiments suggest that coefficients of the expansions of g n = k= αn S k k decay very rapidly. Moreover, it is not hard to see that if G n (x = ( x m/ g n (x = ( x m/ k= αn S k k is an eigenfunction of K that corresponds to a non-zero eigenvalue, then ᾱp n = for all p>l m. In order to observe this recall that on the subspace H m kernel of K is K(x,x = ( x m/ ( x m/ Z L m (x, x, where Z L m (x, x is a symmetric polynomial in x and x of degree L m. Then, α n S p (xg n (x ( p = dx b ( x m/ = S p (x K(m,x,x G b ( x m/ n (x dx dx ξ n b = ( S ξ n G n (x p (x ( x m/ K(m,x,x dx dx b b k=

886 L Miranian ( K(m,x,x ξ n b b ( x S m/ p (x dx dx = ( ( x ξ n m/ Z L m (x, x S p (x dx dx = b b for all p>l m, since the Legendre polynomial S p (x of degree p is orthogonal to x k for all k<p. Because of the very rapid decay of the coefficients αk n, eigenfunctions G n can be computed efficiently. It is very important to sort eigenvalues (along with corresponding eigenvectors of (9 in the ascending order before computing the sum g n = N k= ān S k k for some appropriate finite N. In figure one can see the decay of the coefficients of the expansion in the case m =,L = and b = for the eigenfunctions G 5,G, G 3,G 8 ; m = 4,L = 7 and b = /for the eigenfunction G 5,G, G 3,G 8. In figure 3 eigenfunctions G k of D are presented, for k =, 5,,, 35, 5,L = 5, m =,b = /..5. Construction of generalized Gaussian quadratures for the eigenfunctions of D In this section, an algorithm for constructing the generalized Gaussian quadrature rule for the eigenfunctions of D is described and results of some numerical experiments are presented. Functions G n (x form a complete orthonormal basis for the space L [b, ], and a quadrature rule for computing integrals of the form b f(xg n(x dx efficiently for various functions f(xis needed. In this instance, an efficient quadrature rule refers to a quadrature such that the error decreases at least exponentially as a function of the number of nodes used in the integration. The eigenfunctions G n form an extended Chebyshev system, and according to the principal result of [6], there exists a unique n-point generalized Gaussian quadrature rule with weight W :[b, ] R +. Nodes (x,...,x n and weights (w,...,w n of the quadrature satisfy a system of nonlinear equations n w i G j (x i = G j (xw(x dx; j =,...,n. ( i= b Newton s method for this system of equations is always quadratically convergent, since the fact that eigenfunctions form an extended Chebyshev system implies the Jacobian of the system being always nonsingular ([5, lemma.6]. In order to provide a good starting point for Newton s method we use the continuation scheme suggested in [5]. A numerical algorithm for obtaining weights and nodes of n-point generalized Gaussian quadrature is described below. (i As a starting point of the algorithm take the n roots of G n, denote them as r = (r,...,r n. (ii Use a continuation scheme, i.e. let the weight functions ω: [, ] [b, ] R + be defined by the formula n ω(α,x = αw(x + ( α δ(x r j, j= where δ denotes the Dirac delta function. Observe that when α =, the weight function is equal to the desired weight function W(x, and when α = the Gaussian weights and nodes are w i =,x i = r i for i =,...,n.

Slepian functions on the sphere, generalized Gaussian quadrature rule 887 5 5 5 5 5 5 5 5 3 35 4 5 5 5 5 3 35 4 5 4 6 8 5 4 6 8 5 5 5 5 3 35 4 5 5 5 3 35 4 5 5 5 5 5 5 5 5 5 3 35 4 5 5 5 3 35 4 4 6 8 4 6 8 4 6 8 4 6 5 5 5 3 35 4 8 5 5 5 3 35 4 Figure. Top four plots: L =,m =,b = ; bottom four plots: L = 7,m = 4,b = /; graphs correspond to absolute values of the first 35 coefficients of the expansions for eigenfunctions G 5,G,G 3,G 8 versus their index. (iii Damped Newton s method (at every iteration of Newton s method search for an appropriate step size along the direction prescribed by regular Newton s method is used to solve the system on every step of the continuation scheme.

888 L Miranian 6. 5.4 4.6 3.8..4.6.8.5.55.6.65.7.75.8.85.9.95 7 3.5.55.6.65.7.75.8.85.9.95 6 5 4 5 3 3.5.55.6.65.7.75.8.85.9.95 5.5.55.6.65.7.75.8.85.9.95 4 5 8 6 4 5 5 4 6.5.55.6.65.7.75.8.85.9.95.5.55.6.65.7.75.8.85.9.95 Figure 3. Eigenfunctions G k of D for k =, 5,,, 35, 5; b = /,L= 5,m=. In the numerical experiments conducted using MATLAB, the quadrature rule ( for various weight functions was obtained. In particular, quadratures with W(x =, x b were constructed and the resulting weights and nodes are presented in tables and. To see the accuracy of the quadratures, the test function f(x= sin 5 (x cos(7x( x, for instance, is integrated with relative and absolute errors of 7.8 and.4 6 with a 5-point quadrature, where W(x =. The test function f(x= x (x b 3/ ( x 5/ is integrated with relative and absolute errors of 5.65 and.4 with a -point quadrature, where W(x = x b.

Slepian functions on the sphere, generalized Gaussian quadrature rule 889 Table. Nodes and weights of a n = 5-, -, -point generalized Gaussian quadrature; L = 4; m = ; b = ; W(x =. N Nodes x i Weights w i 5 3.6 43 65 969 785 7.837 57 65 65 48.59 475 56 56 9.767 635 4 98 835 3.76 79 47 998 3.56 589 66 68 83 6.48 57 48 864 47.658 94 3 35 3 8.78 8 38 38 97.93 3 576 68 38 8.64 975 547 63 66 3.7 769 35 4 833 4.57 94 4 38 89 5.3 75 65 5 8.8 7 38 97 56 7.975 363 584 47 85.4 5 383 7 495.63 9 59 876 557 3. 379 4 7 77.86 97 636 98 55 4.59 563 7 38 97.435 765 5 777 34 6.55 7 94 96 98.47 7 93 36 89 7.483 55 46 653.358 98 67 78 45 8.78 6 7 55 3.83 34 96 7 349 9.6 9 6 676 84 6.587 759 9 56 947.33 389 565 53 658 3 5.936 75 445 69 434 3.8 75 444 5 983.38 563 354 93 6.994 46 439 8 85.69 9 36 4 83 5.555 435 9 5 35.95 99 56 39 8.895 76 63 636 96 3.76 3 368 537 99.3 3 95 75 97 4.48 5 676 477 47.785 9 84 999 33 5.9 93 3 886 898.34 488 8 98 68 5.889 56 75 3 475.96 759 639 57 856 6.5 739 863 5 73 3.637 955 855 843 94 7.3 445 659 594 4.36 43 335 53 53 7.4 77 345 9 7 5.7 48 43 587 686 7.669 557 687 33 547 5.889 757 75 35 7.733 7 747 3 6.656 984 698 634 46 7.585 359 3 84 3 7.398 53 6 783 987 7.4 896 745 966 8.89 84 7 86 4 6.58 79 947 4 9 8.76 46 686 75 344 5.7 45 693 98 534 9.4 733 69 36 755 4.68 74 4 95 86 9.63 698 984 9 738 3.33 967 754 66 58 9.886 6 4 97 869.899 54 577 56 On average, the algorithm described above performs only one step of the continuation scheme. On every step of the continuation scheme it does about seven steps of Newton s iteration with around three step-size adjustments per each iteration. The functions G n e imφ form a complete, orthonormal basis for functions on the spherical cap, hence we need to compute double integrals in colatitudinal and longitudinal variables. The procedure described above produces the nodes and weights for the colatitudinal variable. Integration with respect to the longitudinal variable can be done by a Gaussian quadrature as well, so the final quadrature rule is π n φ ( n f(x,φdx dφ = f(x k,φ i wk x, b i= w φ i k=

89 L Miranian Table. Nodes and weights of a n = 5-, -, -point generalized Gaussian quadrature; L = 3; m = ; b = ; W(x = x b. N Nodes x i Weights w i 5 5.7 88 969 74 534.34 55 38 88 86.3 63 5 9 88 8.884 898 67 98 555 4.356 4 4 5 46.74 654 53 9 344 7. 764 5 37 7.37 749 99 88 69 9.6 789 94 435 7.544 365 4 946 95.47 6 86 4 878 3.56 66 33 34 735 3 5.868 587 47 593 844.47 3 5 7 768.34 3 4 34 694 3.4 85 68 88 99.3 59 684 79 8 5.44 49 7 47 6 3.545 7 566 78 7 7.96 34 3 75 67 4.946 547 898 79 779. 669 6 9 473 6.4 444 375 669 494.64 567 46 85 98 7.83 84 46 38 84.49 57 587 384 354 8.96 668 473 79 68 9.335 975 93 94 83 9.79 9 8 57 93 5.58 77 75 89 774 3.959 64 57 989 759 3 4.979 655 538 89 7 4.58 745 856 87.99 69 85 64 446 3 3.559 4 977 6 9 4.47 865 45 33 35 3 6.3 587 478 637 355 7.97 664 68 99 845 3 9.86 54 37 98 763.3 39 88 45 87.46 36 87 947 48.759 39 754 49 365.99 943 64 3.36 55 4 95.49 785 459 883 93 3.3 45 3 783 3.8 6 75 648 356 3.77 573 73 4 895 3.88 74 965 396 798 4.4 387 4 8 9 4.55 689 74 93 45 5.58 684 6 339 364 5.3 99 69 4 78 5.6 58 8 77 5 6.85 48 47 94 9 6.3 67 8 364 54 6.847 43 949 754 737 6.7 3 94 364 78 7.577 49 536 99 6.44 3 96 3 8.5 96 475 975 498 5.786 968 99 983 98 8.843 6 938 4 78 5. 48 486 594 96 9.33 6 653 96 3 4.55 7 6 39 53 9.697 8 394 998 446.93 37 8 6 3 9.93 585 38 35 836.56 35 454 9 496 where nodes and weights x k,wk x are products of the scheme presented before, and nodes and weights φ i,w φ i are those that correspond to the quadrature rule for longitudinal variable with n φ being the desired number of nodes. Figures 4 and 5 show the nodes of the 4-point and -point quadratures on the sphere, where L =,m =,b = /and L = 5,m =,b = / correspondingly. The nodes tend to concentrate more at the bottom of the spherical cap rather than around the pole. A consequence of this fact is that the quadrature scheme described in this section does not produce the familiar north pole oversampling problem.

Slepian functions on the sphere, generalized Gaussian quadrature rule 89.5.5.5.5.5.5 Figure 4. Nodes for 4-point quadrature, b = /,L=,m=..5.5.5.5.5.5 Figure 5. Nodes for -point quadrature, b = /,L= 5,m=. 3. Conclusions An efficient numerical scheme for evaluating a basis for the set of singular vectors of a time band time-limiting operator for certain regions on the surface has been presented. The basis functions form an orthonormal set, as well as an extended Chebyshev system, which guarantees the existence of a generalized Gaussian quadrature rule. An algorithm for computing weights and nodes for such a quadrature rule is presented. The nodes produced tend to concentrate not at the pole of the sphere, but at the bottom of the spherical cap, not exhibiting a common problematic effect called north pole oversampling.

89 L Miranian Acknowledgments The author is very grateful to Professor F A Grünbaum for many invaluable discussions. The author would also like to thank Professor W Kahan for a discussion on the Gaussian quadrature rule and for providing very useful references on the subject. References [] Grünbaum F A, Longhi L and Perlstadt M 98 Differential operators commuting with finite convolution integral operators: some non-abelian examples SIAM J. Appl. Math. 4 94 55 [] Slepian D and Pollak H O 96 Prolate spheroidal wave functions, Fourier analysis and uncertainty, I Bell Syst. Tech. J. 4 43 64 [3] Landau H J and Pollak H O 96 Prolate spheroidal wave functions, Fourier analysis and uncertainty, II Bell Syst. Tech. J. 4 65 84 [4] Landau H J and Pollak H O 96 Prolate spheroidal wave functions, Fourier analysis and uncertainty, III Bell Syst. Tech. J. 4 95 336 [5] Slepian D 964 Prolate spheroidal wave functions, Fourier analysis and uncertainty, IV Bell Syst. Tech. J. 43 39 58 [6] Slepian D 983 Some comments on Fourier analysis, uncertainty and modeling SIAM Rev. 5 379 93 [7] Slepian D 976 On bandwidth Proc. IEEE 63 9 3 [8] Slepian D 978 Prolate spheroidal wave functions, Fourier analysis and uncertainty, V Bell Syst. Tech. J. 57 37 43 [9] Shannon C E 948 A mathematical theory of communication Bell Syst. Tech. J. 7 379 43 Shannon C E 948 Bell Syst. Tech. J. 7 63 56 [] Xiao H, Rokhlin V and Yarvin N Prolate spheroidal wave functions, quadrature, and interpolation Inverse Problems 7 85 38 [] Albertella A, Sanso F and Sneeuw N 999 Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere J. Geodesy 73 436 47 [] Grünbaum F A and Miranian L The magic of the prolate spheroidal functions in various setups Proc. SPIE 4478 5 6 [3] Hwang Ch 99 Spectral analysis using orthonormal functions with a case study on the sea surface topography Geophys. J. Int. 5 48 6 [4] Albertella A and Sneeuw N The analysis of radiometric data with Slepian functions Phys. Chem. Earth A 5 667 7 [5] Cheng H, Rokhlin V and Yarvin N 999 Nonlinear optimization, quadrature, and interpolation SIAM. J. Optim. 9 9 3 [6] Karlin S and Studden W 966 Tchebyshev Systems with Applications in Analysis and Statistics (New York: Wiley-Interscience