slepian functions on the sphere, generalized gaussian quadrature rule

INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS

Inverse Problems 20 (2004) 877–892 PII: S0266-5611(04)70691-7

Slepian functions on the sphere, generalized Gaussianquadrature rule

L Miranian

Department of Mathematics, University of California, Berkeley CA, 94720, USA

E-mail: [email protected]

Received 20 October 2003, in final form 1 March 2004Published 2 April 2004Online at stacks.iop.org/IP/20/877 (DOI: 10.1088/0266-5611/20/3/014)

AbstractDenote by K the operator of ‘time–band–time’ limiting on the surface of thesphere and consider the problem of computing singular vectors of K. Thisproblem can be reduced to a simpler task of computing eigenfunctions of adifferential operator, if a differential operator, which commutes with K and hasa simple spectrum, can be exhibited. In Grunbaum et al (1982 SIAM J. Appl.Math. 42 941–55) such a second-order differential operator commuting withK on the appropriate subspaces was constructed. In this paper, this algebraicproperty of commutativity is used to produce an efficient numerical scheme forcomputing a convenient basis for the space of singular vectors of K. The basisforms an extended Chebyshev system, and a generalized Gaussian quadraturerule for such a basis is presented.

1. Introduction

The fundamental problem of recovering a time-limited function from the knowledge of itsFourier 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 underliesthe question of how to make optimal use of the available information that is always limitedand corrupted by noise. The remarkable series of papers [2–8], by Slepian, Landau and Pollakin connection with the issue of time–band-limited signals has had a tremendous influence onmany areas of engineering, science and mathematics. This work puts some of the pioneeringwork of Shannon [9] on firmer ground. Their starting points were fairly applied aspects ofcommunication theory, optics, lasers, etc, but it became apparent that the ideas were applicableto many other situations.

The work presented in this paper deals with the case when the real line is replaced bythe surface of the sphere. Here, the mathematical and computational issue is to get goodapproximations 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

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878 L Miranian

successive application of the operations of time, band and time limiting. In the classicalcase of the real line, the computation of Slepian functions is done using the fact that theclassical differential operator, resulting from the Laplacian by separation of variables in prolatespheroidal coordinates, happens to commute with the time–band-limiting integral operator.This commuting differential operator has a simple spectrum, hence its eigenfunctions form abasis for the space of eigenfunctions of the integral operator.

In the case of the surface of the sphere, it was shown in [1], that for a polar cap as wellas for two symmetrically placed caps (one at each pole), a certain second-order differentialoperator

D = d

dx

[(1 − x2)(b − x)

d

dx

]− L(L + 2)x − m2(b − x)

1 − x2

defined on the interval [b, 1] commutes with K on the spaces of functions whose dependenceon φ is of the form eimφ . 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 inquestion has less symmetry, then one can always consider the integral operator, but the searchfor a commuting local operator has proved elusive.

This algebraic property of existence of a commutative differential operator holds the keyto a good algorithm. Since the object to be produced is a basis for the time–band–time-limitedfunctions on a certain region of the sphere, one needs to expand easily in this basis, whichrequires an efficient quadrature rule for evaluating inner products. It is well known that fora system of functions that forms an extended Chebyshev system, a generalized Gaussianquadrature rule always exists. In [10] a method of obtaining such a quadrature rule usingthe appropriate continuation scheme and well-chosen starting points for Newton’s method isdescribed. The operator K happens to be a finite rank Fredholm operator, which implies thatit has only a finite number of non-zero eigenvalues. There are many ways to compute thenull space of the operator K, but the algebraic property discussed above makes it possible toreplace the computation of the eigenfunctions of K by the computation of the eigenfunctionsof D. This not only simplifies the task from the numerical point of view, but also producesorthogonal functions which form an extended Chebyshev system, assuring the existence of anefficient quadrature rule.

In order to numerically compute the eigenfunctions (of an appropriate self-adjointextension) of D we expand them in the basis of the shifted Legendre polynomials, andreduce the problem to the computation of generalized eigenvalues and eigenvectors of certainsparse matrices. We then use the method that has been advocated recently in [10]. The objectproduced is a good basis for the space of the eigenfunctions of K in the case when the regionsof interest are either a polar cap or a spherical belt bounded by two parallels.

Among the applications envisaged is the problem discussed in [11] involving gravity fieldmissions. Due to launch conditions or engineering reasons, the sampling is done not on thewhole surface of the earth but on one of the type of regions mentioned above. The computationof the so-called Slepian functions, in this case, has been attempted in the geodesy communitywithout taking recourse to the mathematical and computational advantages that are derivedfrom exploiting the results in [1] and the very recent note [12]. For other geodesy applications,see [13, 14]. In the first of these papers the region where the function is known is the surfaceof the oceans.

This paper is organized as follows. In section 2.1 the problem of computing eigenfunctionsof the integral operator K is discussed. In section 2.2 some properties of shifted Legendrepolynomials are recalled. Sections 2.3 and 2.4 describe a method for computing Slepian

Slepian functions on the sphere, generalized Gaussian quadrature rule 879

functions using the differential operator. In section 2.5 the generalized Gaussian quadraturerule for the Slepian functions on the spherical cap is presented and concluding remarks are insection 3.

2. Numerical computation of the eigenproblem

2.1. Direct computation of eigenfunctions of the integral operator K

In this section, an attempt to compute eigenvalues/eigenvectors of the integral operator directlyis described, and a more efficient alternative is suggested.

As discussed in [1], denote by A the ‘polar cap’ 0 � θ � arccos(b), 0 � φ � 2π . Thenthe operator K is the ‘finite convolution integral operator’. Denote

u = (sin θ cos φ, sin θ sin φ, cos θ),

then

(Kf )(u) =∫

A

L∑l=0

Pl(〈u, u′〉)f (u′) du′ =∫

A

(L∑

l=0

l∑m=−l

Ylm(u)Ylm(u′)

)f (u′) du′,

where Ylm(u) is the usual ‘spherical harmonic’ with∫ 2π

0

∫ 1

−1Ylm(x, φ)Ylm(x, φ) dx dφ = 1,

and

Ylm(u) = Ylm(cos θ, φ) =√

2l + 1

4π

(l − m)!

(l + m)!P m

l (cos θ) eimφ,

or

Ylm(x, φ) =√

2l + 1

4π

(l − m)!

(l + m)!P m

l (x) eimφ, x = cos θ.

In the formulae above P ml (x) denotes the associated Legendre polynomial.

The operator K is a singular Fredholm operator with rank (L + 1)2, hence it has only(L + 1)2 non-zero eigenvalues. In the following proposition, some properties of the eigenvaluesand eigenvectors of K are summarized.

Proposition 1. Consider the finite (L + 1)2 rank symmetric Fredholm operator as definedabove and let Hm be the subspaces of functions on the polar cap whose φ dependence is givenby eimφ . Then

(i) There are L + 1 linearly independent orthogonal eigenfunctions of K that belong to thespace H0; consequently K has only L + 1 distinct non-zero eigenvalues that correspondto eigenfunctions in H0.

(ii) The L(L + 1)/2 non-zero eigenvalues of K have multiplicity 2: in both subspaces Hm

and H−m K has a simple spectrum, i.e. L − m + 1 non-zero distinct eigenvalues, wherem = 1, . . . , L − 1.

(iii) L − |m| + 1 eigenfunctions of K belong to Hm for all |m| = 1, . . . , L − 1 and areorthogonal.

880 L Miranian

Proof. Let us fix L and m, and see what the integral operator looks like on Hm, i.e. takeg(u) = f (cos θ) eimφ = f (x) eimφ and apply the operator K to it:

Kf (u) =∫

A

(L∑

l=0

l∑m=−l

Ylm(u)Ylm(u′)

)g(u′) du′

=∫ 2π

0

∫ 1

b

(L∑

l=0

l∑m=−l

Ylm(x, φ)Ylm(x ′, φ′)

)f (x ′) eimφ′

dx ′ dφ′

=∫ 2π

0

∫ 1

b

(L∑

l=0

l∑m=−l

2l + 1

4π

(l − m)!

(l + m)!P m

l (x) eimφP ml (x ′) e−imφ′

)f (x ′) eimφ′

dx ′ dφ′

=∫ 1

b

L∑l=0

l∑m=−l

2l + 1

4π

(l − m)!

(l + m)!P m

l (x)P ml (x ′)f (x ′)

(eimφ

∫ 2π

0eiφ′(−m+m) dφ′

)dx ′

= eimφ

∫ 1

b

L∑l=|m|

2l + 1

2π

(l − |m|)!(l + |m|)! P m

l (x)P ml (x ′)f (x ′) dx ′.

To simplify the notation denote

K(m, x, x ′) =L∑

l=|m|

2l + 1

2π

(l − |m|)!(l + |m|)! P m

l (x)P ml (x ′)

as the kernel of K in subspace Hm.The following observations can be made:

(i) The associated Legendre polynomials P ml are linearly independent and K(m, x, x ′) =

K(m, x ′, x) defines a singular symmetric Fredholm operator which has only L − |m| + 1distinct non-zero eigenvalues for all |m| = 0, . . . , L − 1.

(ii) If m = 0, then K has only L + 1 distinct non-zero eigenvalues in H0, and because of thesymmetry of the kernel K(0, 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 theeigenvectors of K in Hm corresponding to different eigenvalues are orthogonal.

(iv) If m �= 0, then in both Hm and H−m the operator K has the same kernel, hence L−|m|+ 1eigenvalues of K will be duplicated for every |m| = 1, . . . , L − 1. �

Attempts to compute the eigenfunctions of the integral operator KFn(u) = µnFn(u) directlyhave not been fruitful. In the experiments discussed below the integral operator with L = 3was discretized using the Gaussian quadrature rule with Nx = 21-point grid in x = cos(θ)

variable and Nφ = 21-point grid in φ variable. Let T be the matrix obtained after discretizationof the integral operator K. The disadvantages of this method are:

(i) The size of the matrix T is Nx Nφ = 441; it depends quadratically on the grid size,which makes computing its eigenvalues/eigenvectors an intensive task. In the alternativeapproach a grid of 104 points in the x variable is used, whereas in the direct approach itis 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 isneeded.

Below is the summary of some numerical experiments. Only the eigenfunctions of T thatcorrespond to non-zero eigenvalues are meaningful, so only these are considered in the textbelow.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–3

–2

–1

0

1

2

3

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–3

–2

–1

0

1

2

3

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

Figure 1. Results of the direct discretization (four left-hand figures) and eigenfunctions obtained byusing the differential operator D (four right-hand figures) for m = 0, 1, 2, 3. Functions presentedare numbered according to the number of roots, i.e. G3, G2, G1, G0; L = 3.

(i) Exactly four eigenfunctions of T are in subspace H0; three eigenfunctions in H±1; twoeigenfunctions in H±2 and one eigenfunction in H±3.

(ii) Eigenfunctions Fj (x) e±imφ with j = 0, . . . , L − |m| + 1 of T in each H±m are very closeto these produced by the alternative procedure below (see figure 1).

882 L Miranian

(iii) Eigenfunctions of T corresponding to the eigenvalue 0 cannot be computed by directdiscretization.

There is an efficient alternative to the procedure described above. It produces an orthonormalbasis for the space of eigenfunctions of K. Consider the following second-order differentialoperator on the interval [b, 1]:

D = d

dx

[(1 − x2)(b − x)

d

dx

]− L(L + 2)x − m2(b − x)

1 − x2.

A look at [1] will show that this is the appropriate operator D that commutes with the K builtthere when acting on Hm. The operator D defines a Sturm–Liouville problem, hence it has asimple spectrum and orthogonal eigenfunctions. Because it commutes with K we can say thatthe eigenfunctions of D are also eigenfunctions of K when acting on Hm. Since we search foran orthonormal basis in the space of eigenfunctions of K, the eigenfunctions of D provide uswith such a basis.

In order to compute numerically the eigenfunctions (of an appropriate self-adjointextension) of D we expand them in the basis of the shifted Legendre polynomials, and reducethe problem to (generalized) eigenproblem for some well-structured matrices.

2.2. 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 thedifferential operator D.

Define shifted Legendre polynomials to be the solutions of the following second-orderdifferential equation:

(b − x)(1 − x)S ′′n + 2(x − b1)S

′n − n(n + 1)Sn = 0.

Denote b1 := (1 + b)/2, b2 := (1 − b)/2. The following properties of Sn will be useful later:

(i) recursion relation

Sn = b1Sn +b2(n + 1)

2n + 1Sn+1 +

b2n

2n + 1Sn−1;

(ii) derivative

(1 − x)(b − x)S ′n = b2

n(n + 1)

2n + 1(Sn+1 − Sn−1);

(iii) normalized shifted Legendre polynomials

Sk ≡ Sk

√(2k + 1)/2b2.

2.3. Computation of the eigenproblem DFn = λnFn: case m = 0

In this section, the problem of computing the eigenfunctions of the differential operator D withm = 0 is reduced to the problem of computing eigenvectors of a certain symmetric tridiagonalmatrix.

Consider the following eigenproblem:(d

dx

[(1 − x2)(b − x)

d

dx

]− L(L + 2)x

)Fn = λnFn. (1)


Let(an

0 , an1 , an

2 , . . .)

be the coefficients of the expansion of Fn(x) in the basis of shiftedLegendre polynomials

Fn =∞∑

k=0

ank Sk.

After substituting Fn(x) into (1), using properties (i), (ii) and the linear independence of Sk

one obtains a recursion relation

Ck−1ak−1 + Bk+1ak+1 + Akak − λnak = 0, (2)

where

Ak = k(k + 1)(1 + b1) − L(L + 2)b1

Bk = k

2k + 1b2[(k − 1)(k + 1) − L(L + 2)]

Ck = k + 1

2k + 1b2[k(k + 2) − L(L + 2)].

After rewriting (2) for the normalized polynomials using property (iii), the recursion relationcan be written in the matrix form as

Man = λnan, (3)

where an = (an

0 , an1 , . . .

)Tare the coefficients of the expansion in the basis of the normalized

shifted Legendre polynomials;

Mk,k = k(k + 1)(1 + b1) − L(L + 2)b1,

Mk,k+1 = b2(k + 1)√(2k + 3)(2k + 1)

[k(k + 2) − L(L + 2)],

Mk+1,k = b2(k + 1)√(2k + 3)(2k + 1)

[k(k + 2) − L(L + 2)],

k = 0, 1, . . . and the remainder of the entries of the matrix being zero. Using an obtainedfrom (3), we express Fn = ∑∞

k=0 ank Sk . The numerical evidence strongly suggests that the

coefficients ank decay very fast. In practice, this allows us to compute Fn = ∑N

k=0 ank Sk , for

certain large values of N .

2.4. Computation of the eigenproblem DGn = µnGn: case m > 0

In this section, the problem of computing eigenfunctions of the differential operator D withm > 0 is reduced to a generalized matrix eigenproblem.

Consider the following eigenproblem:

[(1 − x2)(b − x)G′n]′ − L(L + 2)xGn − m2(b − x)

1 − x2Gn = µnGn. (4)

Introduce the following simplifying notation:

(i) Matrix S = (S0, S1, S3, . . .), where Sk are normalized shifted Legendre polynomials.

(ii) Matrix A with columns Ak = (ak

0, ak1, a

k2, . . .

)T, where ak

j are the coefficients of theexpansion of Fk(x) in the basis of normalized shifted Legendre polynomials.

(iii) Fk =∞∑

j=0

Sj akj = SAk .

884 L Miranian

(iv) Also,

xFk =∞∑

j=0

xSjakj =

∞∑j=0

akj

(b1Sj +

b2(j + 1)

2j + 1Sj+1 +

b2j

2j + 1Sj−1

)

=∞∑

j=0

akj

(b1Sj +

b2(j + 1)√(2j + 1)(2j + 3)

Sj+1 +b2j√

(2j + 1)(2j − 1)Sj−1

)

= SQAk,

where the symmetric tridiagonal matrix Q has entries

Qk,k = b1,

Qk,k+1 = Qk+1,k = b2(k + 1)√(2k + 3)(2k + 1)

,

with = 0, 1, 2, . . . . Similarly x2Fk = SQ2Ak .

(v) Denote cn = (cn

0 , cn1 , c

n2 , . . .

)T.

(vi) Matrix � = diag(λ0, λ1, . . .); A� = MA.(vii) Denote I to be the identity matrix.

Since the functions Fk form an orthonormal basis for L2[b, 1], one could expand Gn in thebasis of Fk , i.e.

Gn =∞∑

k=0

cnkFk(x) =

∞∑k=0

cnkSAk = SAcn =

∞∑k=0

γ nk Sk(x), (5)

where γ n = (γ n0 , γ n

1 , . . .)T = Acn. After substituting (5) into (4) and using the fact that Fk

satisfies (1) one obtains∑cnk

(λkFk − m2(b − x)

1 − x2Fk − µnFk

)= 0,

or∞∑

k=0

cnk (λkFk(1 − x2) − m2(b − x)Fk − µnFk(1 − x2)) = 0. (6)

Using the notation above, (6) can be written as∞∑

k=0

cnk (λkS(I − Q2)Ak − m2S(b − Q)Ak − µnS(I − Q2)Ak)

= S

∞∑k=0

cnk (λk(I − Q2)Ak − m2(b − Q)Ak − µn(I − Q2)Ak) = 0.

Since the functions Sk are linearly independent, one obtains

0 =∞∑

k=0

cnk (λk(I − Q2)Ak − m2(b − Q)Ak − µn(I − Q2)Ak)

= (I − Q2)A�cn − m2(b − Q)Acn − µn(I − Q2)Acn

= (I − Q2)MAcn − m2(b − Q)Acn − µn(I − Q2)Acn

= ((I − Q2)M − m2(b − Q) − µn(I − Q2))Acn,


which implies the generalized eigenproblem

(I − Q2)(M − µnI)γ n = m2(b − Q)γ 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 Gn(x) =∑∞

k=0 γ nk Sk(x) decay slowly. However, the scheme above can be significantly improved,

as we explain now. Taking into account boundary conditions, write Gn = (1 − x2)m/2gn(x),for a certain function gn(x). After rewriting (4) in terms of gn(x) we arrive at the followingeigenproblem:

[(1 − x2)(b − x)g′n]′ + 2mx(x − b)g′

n + ((m2 + 2m − L2 − 2L)x − mb(m + 1))gn = µngn.

(8)

Now, the scheme described above can be applied to the function gn(x) = ∑∞k=0 cn

kFk forsome coefficients cn

k using the differential equation (8). In this case the derivation of thegeneralized matrix eigenproblem is similar to that performed at the beginning of this section.After elaborate calculations one arrives at

(I − Q)(M − µnI)αn = (mb(m + 1)(I − Q) + 2mQ3 − m(m + 2)(Q − Q2))αn, (9)

where

Q3(k, k) = −b2

(2k + 3)(2k − 1),

Q3(k + 1, k) = −Q3(k, k + 1) = b1

(2k + 3)(2k − 1),

Q3(k, k + 2) = −b2(k + 1)

(2k + 3)√

(2k + 1)(2k + 5),

Q3(k + 2, k) = b2(k + 2)

(2k + 3)√

(2k + 1)(2k + 5),

E(k, k) = b2k(k + 1), Q3 = Q3E

for k = 0, 1, 2, 3, . . . ; matrices Q and M were defined before, and Acn = αn. Observe that

gn(x) =∞∑

k=0

cnkFk(x) =

∞∑k=0

cnkSAk = S

∞∑k=0

cnkA

k = SAcn = Sαn,

which means that while cn are coefficients of the expansion of gn(x) in the basis ofFk(x)′ s, αn s are coefficients of the expansion of gn(x) in the basis of normalized shiftedLegendre polynomials S ′

ks. From (9) we can compute gn = ∑∞k=0 αn

k Sk , and obtainGn(x) = (1 − x2)m/2gn(x).

Experiments suggest that coefficients of the expansions of gn = ∑∞k=0 αn

k Sk decayvery rapidly. Moreover, it is not hard to see that if Gn(x) = (1 − x2)m/2gn(x) =(1 − x2)m/2 ∑∞

k=0 αnk Sk is an eigenfunction of K that corresponds to a non-zero eigenvalue,

then αnp = 0 for all p > L − m. In order to observe this recall that on the subspace Hm kernel

of K is K(x, x ′) = (1 − x2)m/2(1 − x ′2)m/2ZL−m(x, x ′), where ZL−m(x, x ′) is a symmetricpolynomial in x and x ′ of degree L − m. Then,

∣∣αnp

∣∣ =∣∣∣∣∫ 1

b

Sp(x)Gn(x)

(1 − x2)m/2dx

∣∣∣∣ =∣∣∣∣∫ 1

b

Sp(x)

(1 − x2)m/2

1

ξn

(∫ 1

b

K(m, x, x ′)Gn(x′) dx ′

)dx

∣∣∣∣= 1

|ξn|∣∣∣∣∫ 1

b

Gn(x′)

(∫ 1

b

Sp(x)

(1 − x2)m/2K(m, x, x ′) dx

)dx ′

∣∣∣∣

886 L Miranian

� 1

|ξn|

√∫ 1

b

(∫ 1

b

K(m, x, x ′)(1 − x2)m/2

Sp(x) dx

)2

dx ′

= 1

|ξn|

√∫ 1

b

(∫ 1

b

(1 − x ′2)m/2ZL−m(x, x ′)Sp(x) dx

)2

dx ′ = 0

for all p > L − m, since the Legendre polynomial Sp(x) of degree p is orthogonal to xk forall k < p.

Because of the very rapid decay of the coefficients αnk , eigenfunctions Gn 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 gn = ∑N

k=0 ank Sk for some appropriate

finite N .In figure 2 one can see the decay of the coefficients of the expansion in the case

m = 1, L = 1 and b = 0 for the eigenfunctions G5,G10, G30,G80; m = 4, L = 7 andb = −1/2 for the eigenfunction G5,G10, G30,G80.

In figure 3 eigenfunctions Gk of D are presented, for k = 1, 5, 10, 20, 35, 50, L = 5,

m = 2, b = 1/2.

2.5. Construction of generalized Gaussian quadratures for the eigenfunctions of D

In this section, an algorithm for constructing the generalized Gaussian quadrature rule for theeigenfunctions of D is described and results of some numerical experiments are presented.

Functions Gn(x) form a complete orthonormal basis for the space L2[b, 1], and aquadrature rule for computing integrals of the form

∫ 1b

f (x)Gn(x) dx efficiently for variousfunctions f (x) is needed. In this instance, an ‘efficient quadrature rule’ refers to a quadraturesuch that the error decreases at least exponentially as a function of the number of nodes usedin the integration.

The eigenfunctions Gn form an extended Chebyshev system, and according to the principalresult of [16], there exists a unique n-point generalized Gaussian quadrature rule with weightW : [b, 1] → R+. Nodes (x1, . . . , xn) and weights (w1, . . . , wn) of the quadrature satisfy asystem of nonlinear equations

n∑i=1

wiGj (xi) =∫ 1

b

Gj (x)W(x) dx; j = 0, . . . , 2n − 1. (10)

Newton’s method for this system of equations is always quadratically convergent, since the factthat eigenfunctions form an extended Chebyshev system implies the Jacobian of the systembeing always nonsingular ([15, lemma 2.6]).

In order to provide a good starting point for Newton’s method we use the continuationscheme suggested in [15]. A numerical algorithm for obtaining weights and nodes of n-pointgeneralized Gaussian quadrature is described below.

(i) As a starting point of the algorithm take the n roots of Gn, denote them as r = (r1, . . . , rn).(ii) Use a continuation scheme, i.e. let the weight functions ω: [0, 1] × [b, 1] → R+ be

defined by the formula

ω(α, x) = αW(x) + (1 − α)

n∑j=1

δ(x − rj ),

where δ denotes the Dirac delta function. Observe that when α = 1, the weight functionis equal to the desired weight function W(x), and when α = 0 the Gaussian weights andnodes are wi = 1, xi = ri for i = 1, . . . , n.


0 50 100 150 200 250 300 350 40010

–25

10–20

10–15

10–10

10–5

100

10–25

10–20

10–15

10–10

10–5

100

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 40010

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10–20

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10–10

10–5

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10–25

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10–10

10–5

100

0 50 100 150 200 250 300 350 40010

–20

10–18

10–16

10–14

10–12

10–10

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10–6

10–4

10–2

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0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 40010

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10–15

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10–5

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0 50 100 150 200 250 300 350 40010

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10–10

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10– 2

100

0 50 100 150 200 250 300 350 40010

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10–16

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10–12

10–10

10–8

10–6

10–4

10–2

100

Figure 2. Top four plots: L = 1,m = 1, b = 0; bottom four plots: L = 7, m = 4, b =−1/2; graphs correspond to absolute values of the first 350 coefficients of the expansions foreigenfunctions G5, G10, G30, G80 versus their index.

(iii) Damped Newton’s method (at every iteration of Newton’s method search for anappropriate step size along the direction prescribed by regular Newton’s method) is usedto solve the system on every step of the continuation scheme.

888 L Miranian

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1–1.8

–1.6

–1.4

–1.2

–1

–0.8

–0.6

–0.4

–0.2

0

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1–3

–2

–1

0

1

2

3

4

5

6

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1–3

–2

–1

0

1

2

3

4

5

6

7

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1–5

0

5

10

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1–6

–4

–2

0

2

4

6

8

10

12

14

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1–10

–5

0

5

10

15

20

Figure 3. Eigenfunctions Gk of D for k = 1, 5, 10, 20, 35, 50; b = 1/2, L = 5, m = 2.

In the numerical experiments conducted using MATLAB, the quadrature rule (10) for variousweight functions was obtained. In particular, quadratures with W(x) = 1,

√x − b were

constructed and the resulting weights and nodes are presented in tables 1 and 2.To see the accuracy of the quadratures, the test function f (x) = sin5(x) cos(7x)(1−x2)2,

for instance, is integrated with relative and absolute errors of 7.08 × 10−10 and 2.40 × 10−16

with a 15-point quadrature, where W(x) = 1. The test function f (x) = x2(x−b)3/2(1−x2)5/2

is integrated with relative and absolute errors of 5.65×10−10 and 1.04×10−11 with a 10-pointquadrature, where W(x) = √

x − b.


Table 1. Nodes and weights of a n = 5-, 10-, 20-point generalized Gaussian quadrature;L = 4;m = 2; b = 0;W(x) = 1.

N Nodes xi Weights wi

5 3.061 430 650 969 785 × 10−2 7.837 570 652 165 048 × 10−2

1.591 475 526 561 901 × 10−1 1.767 635 421 198 835 × 10−1

3.761 709 472 998 213 × 10−1 2.506 589 661 618 823 × 10−1

6.408 517 480 864 147 × 10−1 2.658 094 013 325 130 × 10−1

8.780 822 138 328 971 × 10−1 1.932 312 576 068 380 × 10−1

10 8.604 975 547 632 166 × 10−3 2.207 769 315 114 833 × 10−2

4.527 940 410 382 819 × 10−2 5.123 752 625 510 280 × 10−2

1.108 701 381 097 156 × 10−1 7.975 363 584 427 851 × 10−2

2.041 512 383 172 495 × 10−1 1.063 292 519 876 557 × 10−1

3.221 379 242 071 772 × 10−1 1.286 970 636 980 550 × 10−1

4.590 563 702 382 297 × 10−1 1.435 765 250 777 324 × 10−1

6.055 072 094 196 928 × 10−1 1.471 171 930 360 809 × 10−1

7.483 505 046 653 010 × 10−1 1.358 982 670 708 245 × 10−1

8.718 162 722 550 230 × 10−1 1.083 234 916 072 349 × 10−1

9.600 009 126 676 842 × 10−1 6.587 759 900 056 947 × 10−2

20 2.313 389 565 053 658 × 10−3 5.936 750 445 069 434 × 10−3

1.218 752 444 025 983 × 10−2 1.381 563 354 931 116 × 10−2

2.994 246 439 820 815 × 10−2 2.169 029 360 224 803 × 10−2

5.555 435 092 225 135 × 10−2 2.952 299 561 392 022 × 10−2

8.895 761 623 636 096 × 10−2 3.726 213 368 537 990 × 10−2

1.300 203 295 751 197 × 10−1 4.482 511 676 477 427 × 10−2

1.785 092 814 999 323 × 10−1 5.209 093 302 886 898 × 10−2

2.340 488 118 981 268 × 10−1 5.889 506 175 213 475 × 10−2

2.960 759 639 570 856 × 10−1 6.502 739 863 025 273 × 10−2

3.637 955 855 843 940 × 10−1 7.023 445 222 659 594 × 10−2

4.361 403 335 253 513 × 10−1 7.422 717 345 129 071 × 10−2

5.117 418 243 587 686 × 10−1 7.669 557 687 323 547 × 10−2

5.889 200 757 075 352 × 10−1 7.733 072 212 747 030 × 10−2

6.656 984 698 634 406 × 10−1 7.585 359 103 842 103 × 10−2

7.398 503 206 783 987 × 10−1 7.204 896 745 102 966 × 10−2

8.089 804 217 861 422 × 10−1 6.580 079 947 124 912 × 10−2

8.706 406 686 725 344 × 10−1 5.712 405 693 982 534 × 10−2

9.224 733 629 316 755 × 10−1 4.618 724 004 952 186 × 10−2

9.623 698 984 922 738 × 10−1 3.331 967 754 066 058 × 10−2

9.886 262 004 970 869 × 10−1 1.899 524 010 577 561 × 10−2

On average, the algorithm described above performs only one step of the continuationscheme. On every step of the continuation scheme it does about seven steps of Newton’siteration with around three step-size adjustments per each iteration.

The functions Gn eimφ form a complete, orthonormal basis for functions on the sphericalcap, 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 aswell, so the final quadrature rule is∫ 2π

0

∫ 1

b

f (x, φ) dx dφ =nφ∑i=1

wφ

i

(n∑

k=1

f (xk, φi)wxk

),

890 L Miranian

Table 2. Nodes and weights of a n = 5-, 10-, 20-point generalized Gaussian quadrature;L = 3;m = 1; b = 0;W(x) = √

x − b.

N Nodes xi Weights wi

5 5.171 882 969 714 534 × 10−2 2.341 505 038 088 086 × 10−2

2.031 623 105 912 188 × 10−1 8.884 898 607 298 555 × 10−2

4.356 411 421 025 461 × 10−1 1.714 654 253 019 344 × 10−1

7.002 764 501 137 072 × 10−1 2.137 749 099 088 692 × 10−1

9.160 789 294 435 172 × 10−1 1.544 365 124 946 195 × 10−1

10 1.470 060 286 401 878 × 10−2 3.562 626 303 314 735 × 10−3

5.868 587 471 593 844 × 10−2 1.417 300 251 017 768 × 10−2

1.314 023 014 034 694 × 10−1 3.141 851 068 881 919 × 10−2

2.312 259 684 709 282 × 10−1 5.404 249 071 427 261 × 10−2

3.545 272 566 781 712 × 10−1 7.926 234 023 750 671 × 10−2

4.946 547 898 729 779 × 10−1 1.022 669 162 191 473 × 10−1

6.412 444 375 669 494 × 10−1 1.164 567 246 852 908 × 10−1

7.803 824 426 328 084 × 10−1 1.149 571 587 384 354 × 10−1

8.960 668 473 279 682 × 10−1 9.335 975 293 941 803 × 10−2

9.729 912 080 571 931 × 10−1 5.258 717 715 892 774 × 10−2

20 3.959 614 057 989 759 × 10−3 4.979 655 538 890 171 × 10−4

1.582 745 120 856 827 × 10−2 1.990 269 851 064 446 × 10−3

3.559 214 977 062 290 × 10−2 4.471 865 425 313 305 × 10−3

6.321 587 478 637 355 × 10−2 7.927 664 681 909 845 × 10−3

9.861 524 371 928 763 × 10−2 1.232 329 088 245 827 × 10−2

1.416 316 287 947 248 × 10−1 1.759 239 754 490 365 × 10−2

1.919 943 624 001 203 × 10−1 2.362 155 104 951 021 × 10−2

2.492 785 459 883 913 × 10−1 3.023 415 310 220 783 × 10−2

3.128 601 705 648 356 × 10−1 3.717 573 173 420 895 × 10−2

3.818 724 965 396 798 × 10−1 4.410 387 041 811 009 × 10−2

4.551 689 741 193 452 × 10−1 5.058 684 026 339 364 × 10−2

5.312 991 690 114 278 × 10−1 5.611 528 812 727 251 × 10−2

6.085 048 417 924 119 × 10−1 6.013 067 280 364 542 × 10−2

6.847 430 949 754 737 × 10−1 6.207 213 941 364 178 × 10−2

7.577 419 536 990 212 × 10−1 6.144 003 961 120 213 × 10−2

8.250 906 475 975 498 × 10−1 5.786 968 919 983 928 × 10−2

8.843 622 938 401 178 × 10−1 5.120 418 486 594 962 × 10−2

9.332 611 653 960 123 × 10−1 4.155 170 611 239 503 × 10−2

9.697 812 394 998 446 × 10−1 2.931 237 128 126 203 × 10−2

9.923 585 381 315 836 × 10−1 1.516 351 454 911 496 × 10−2

where nodes and weights xk,wxk are products of the scheme presented before, and nodes and

weights φi, wφ

i are those that correspond to the quadrature rule for longitudinal variable withnφ being the desired number of nodes.

Figures 4 and 5 show the nodes of the 40-point and 10-point quadratures on the sphere,where L = 1,m = 1, b = −1/2 and L = 5,m = 2, b = 1/2 correspondingly. The nodestend to concentrate more at the bottom of the spherical cap rather than around the pole. Aconsequence of this fact is that the quadrature scheme described in this section does notproduce the familiar ‘north pole oversampling’ problem.


–1

–0.5

0

0.5

1

–1

–0.5

0

0.5

1–1

–0.5

0

0.5

1

Figure 4. Nodes for 40-point quadrature, b = 1/2, L = 1,m = 1.

–1

–0.5

0

0.5

1

–1

–0.5

0

0.5

1–1

–0.5

0

0.5

1

Figure 5. Nodes for 10-point quadrature, b = 1/2, L = 5,m = 2.

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 basisfunctions form an orthonormal set, as well as an extended Chebyshev system, which guaranteesthe existence of a generalized Gaussian quadrature rule. An algorithm for computing weightsand nodes for such a quadrature rule is presented. The nodes produced tend to concentratenot at the pole of the sphere, but at the bottom of the spherical cap, not exhibiting a commonproblematic effect called ‘north pole oversampling’.

892 L Miranian

Acknowledgments

The author is very grateful to Professor F A Grunbaum for many invaluable discussions. Theauthor would also like to thank Professor W Kahan for a discussion on the Gaussian quadraturerule and for providing very useful references on the subject.

References

[1] Grunbaum F A, Longhi L and Perlstadt M 1982 Differential operators commuting with finite convolutionintegral operators: some non-Abelian examples SIAM J. Appl. Math. 42 941–55

[2] Slepian D and Pollak H O 1961 Prolate spheroidal wave functions, Fourier analysis and uncertainty, I Bell Syst.Tech. J. 40 43–64

[3] Landau H J and Pollak H O 1961 Prolate spheroidal wave functions, Fourier analysis and uncertainty, II BellSyst. Tech. J. 40 65–84

[4] Landau H J and Pollak H O 1962 Prolate spheroidal wave functions, Fourier analysis and uncertainty, III BellSyst. Tech. J. 41 1295–336

[5] Slepian D 1964 Prolate spheroidal wave functions, Fourier analysis and uncertainty, IV Bell Syst. Tech. J. 433009–58

[6] Slepian D 1983 Some comments on Fourier analysis, uncertainty and modeling SIAM Rev. 25 379–93[7] Slepian D 1976 On bandwidth Proc. IEEE 63 292–300[8] Slepian D 1978 Prolate spheroidal wave functions, Fourier analysis and uncertainty, V Bell Syst. Tech. J. 57

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