-
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 Grünbaum 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
0266-5611/04/030877+16$30.00 © 2004 IOP Publishing Ltd Printed
in the UK 877
http://stacks.iop.org/ip/20/877
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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 = ddx
[(1 − x2)(b − x) d
dx
]− L(L + 2)x − m
2(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
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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∫ 2π0
∫ 1−1
Ylm(x, φ)Ylm(x, φ) dx dφ = 1,
and
Ylm(u) = Ylm(cos θ, φ) =√
2l + 1
4π
(l − m)!(l + m)!
P ml (cos θ) eimφ,
or
Ylm(x, φ) =√
2l + 1
4π
(l − m)!(l + m)!
P ml (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 Hmand 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.
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880 L Miranian
Proof. Let us fix L and m̄, and see what the integral operator
looks like on Hm̄, i.e. takeg(u) = f (cos θ) eim̄φ = f (x) eim̄φ
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
∫ 1b
(L∑
l=0
l∑m=−l
Ylm(x, φ)Ylm(x ′, φ′)
)f (x ′) eim̄φ
′dx ′ dφ′
=∫ 2π
0
∫ 1b
(L∑
l=0
l∑m=−l
2l + 1
4π
(l − m)!(l + m)!
P ml (x) eimφP ml (x
′) e−imφ′)
f (x ′) eim̄φ′dx ′ dφ′
=∫ 1
b
L∑l=0
l∑m=−l
2l + 1
4π
(l − m)!(l + m)!
P ml (x)Pml (x
′)f (x ′)(
eimφ∫ 2π
0eiφ
′(−m+m̄) dφ′)
dx ′
= eim̄φ∫ 1
b
L∑l=|m̄|
2l + 1
2π
(l − |m̄|)!(l + |m̄|)! P
m̄l (x)P
m̄l (x
′)f (x ′) dx ′.
To simplify the notation denote
K(m, x, x ′) =L∑
l=|m|
2l + 1
2π
(l − |m|)!(l + |m|)! P
ml (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.
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Slepian functions on the sphere, generalized Gaussian quadrature
rule 881
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).
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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 = ddx
[(1 − x2)(b − x) d
dx
]− L(L + 2)x − m
2(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 + 1
Sn+1 +b2n
2n + 1Sn−1;
(ii) derivative
(1 − x)(b − x)S ′n = b2n(n + 1)
2n + 1(Sn+1 − Sn−1);
(iii) normalized shifted Legendre polynomials
S̄k ≡ Sk√
(2k + 1)/2b2.
2.3. Computation of the eigenproblem DFn = λnFn: case m = 0In
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)
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Slepian functions on the sphere, generalized Gaussian quadrature
rule 883
Let(an0 , a
n1 , a
n2 , . . .
)be the coefficients of the expansion of Fn(x) in the basis of
shifted
Legendre polynomials
Fn =∞∑
k=0ank Sk.
After substituting Fn(x) into (1), using properties (i), (ii)
and the linear independence of Skone obtains a recursion
relation
Ck−1ak−1 + Bk+1ak+1 + Akak − λnak = 0, (2)where
Ak = k(k + 1)(1 + b1) − L(L + 2)b1Bk = k
2k + 1b2[(k − 1)(k + 1) − L(L + 2)]
Ck = k + 12k + 1
b2[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
Mān = λnān, (3)where ān = (ān0 , ān1 , . . . )T are the
coefficients of the expansion in the basis of the normalizedshifted
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 ān obtainedfrom (3), we express Fn =
∑∞k=0 ā
nk S̄k . The numerical evidence strongly suggests that the
coefficients ānk decay very fast. In practice, this allows us
to compute Fn =∑N
k=0 ānk S̄k , for
certain large values of N .
2.4. Computation of the eigenproblem DGn = µnGn: case m > 0In
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 − x2 Gn = µnGn. (4)Introduce the following simplifying
notation:
(i) Matrix S = (S̄0, S̄1, S̄3, . . .), where S̄k are normalized
shifted Legendre polynomials.(ii) Matrix A with columns Ak = (āk0,
āk1, āk2, . . . )T , where ākj are the coefficients of the
expansion of Fk(x) in the basis of normalized shifted Legendre
polynomials.
(iii) Fk =∞∑
j=0S̄j ā
kj = SAk .
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884 L Miranian
(iv) Also,
xFk =∞∑
j=0xSja
kj =
∞∑j=0
akj
(b1Sj +
b2(j + 1)
2j + 1Sj+1 +
b2j
2j + 1Sj−1
)
=∞∑
j=0ākj
(b1S̄j +
b2(j + 1)√(2j + 1)(2j + 3)
S̄j+1 +b2j√
(2j + 1)(2j − 1) S̄j−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 =
(cn0 , cn1 , cn2 , . . . )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=0cnkFk(x) =
∞∑k=0
cnkSAk = SAcn =
∞∑k=0
γ nk S̄k(x), (5)
where γ n = (γ n0 , γ n1 , . . .)T = Acn. After substituting (5)
into (4) and using the fact that Fksatisfies (1) one obtains∑
cnk
(λkFk − m
2(b − x)1 − x2 Fk − µnFk
)= 0,
or∞∑
k=0cnk (λkFk(1 − x2) − m2(b − x)Fk − µnFk(1 − x2)) = 0. (6)
Using the notation above, (6) can be written as∞∑
k=0cnk (λkS(I − Q2)Ak − m2S(b − Q)Ak − µnS(I − Q2)Ak)
= S∞∑
k=0cnk (λk(I − Q2)Ak − m2(b − Q)Ak − µn(I − Q2)Ak) = 0.
Since the functions Sk are linearly independent, one obtains
0 =∞∑
k=0cnk (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,
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Slepian functions on the sphere, generalized Gaussian quadrature
rule 885
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 γnk S̄k(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 cnkFk for
some coefficients cnk 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
Q̃3(k, k) = −b2(2k + 3)(2k − 1) ,
Q̃3(k + 1, k) = −Q̃3(k, k + 1) = b1(2k + 3)(2k − 1) ,
Q̃3(k, k + 2) = −b2(k + 1)(2k + 3)
√(2k + 1)(2k + 5)
,
Q̃3(k + 2, k) = b2(k + 2)(2k + 3)
√(2k + 1)(2k + 5)
,
E(k, k) = b2k(k + 1), Q3 = Q̃3Efor k = 0, 1, 2, 3, . . . ;
matrices Q and M were defined before, and Acn = αn. Observe
that
gn(x) =∞∑
k=0cnkFk(x) =
∞∑k=0
cnkSAk = S
∞∑k=0
cnkAk = SAcn = Sαn,
which means that while cn are coefficients of the expansion of
gn(x) in the basis ofFk(x)
′ s, ᾱn 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 α
nk S̄k , and obtain
Gn(x) = (1 − x2)m/2gn(x).Experiments suggest that coefficients
of the expansions of gn =
∑∞k=0 α
nk S̄k decay
very rapidly. Moreover, it is not hard to see that if Gn(x) = (1
− x2)m/2gn(x) =(1 − x2)m/2 ∑∞k=0 αnk S̄k is an eigenfunction of K
that corresponds to a non-zero eigenvalue,then ᾱnp = 0 for all p
> L − m. In order to observe this recall that on the subspace Hm
kernelof 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
S̄p(x)Gn(x)
(1 − x2)m/2 dx∣∣∣∣ =
∣∣∣∣∫ 1
b
S̄p(x)
(1 − x2)m/21
ξn
(∫ 1b
K(m, x, x ′)Gn(x ′) dx ′)
dx
∣∣∣∣= 1|ξn|
∣∣∣∣∫ 1
b
Gn(x′)
(∫ 1b
S̄p(x)
(1 − x2)m/2 K(m, x, x′) dx
)dx ′
∣∣∣∣
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886 L Miranian
� 1|ξn|
√∫ 1b
(∫ 1b
K(m, x, x ′)(1 − x2)m/2 S̄p(x) dx
)2dx ′
= 1|ξn|
√∫ 1b
(∫ 1b
(1 − x ′2)m/2ZL−m(x, x ′)S̄p(x) dx)2
dx ′ = 0
for all p > L − m, since the Legendre polynomial S̄p(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 computedefficiently. It is very important
to sort eigenvalues (along with corresponding eigenvectors) of(9)
in the ascending order before computing the sum gn =
∑Nk=0 ā
nk S̄k 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=1δ(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.
-
Slepian functions on the sphere, generalized Gaussian quadrature
rule 887
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
–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 40010
–20
10–18
10–16
10–14
10–12
10–10
10–8
10–6
10–4
10–2
100
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350
40010
–25
10–20
10–15
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
10–8
10–6
10–4
10– 2
100
0 50 100 150 200 250 300 350 40010
–18
10–16
10–14
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 wereconstructed 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 × 10−16with a 15-point
quadrature, where W(x) = 1. The test function f (x) =
x2(x−b)3/2(1−x2)5/2is 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.
-
Slepian functions on the sphere, generalized Gaussian quadrature
rule 889
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 × 10−21.591
475 526 561 901 × 10−1 1.767 635 421 198 835 × 10−13.761 709 472
998 213 × 10−1 2.506 589 661 618 823 × 10−16.408 517 480 864 147 ×
10−1 2.658 094 013 325 130 × 10−18.780 822 138 328 971 × 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 ×
10−24.527 940 410 382 819 × 10−2 5.123 752 625 510 280 × 10−21.108
701 381 097 156 × 10−1 7.975 363 584 427 851 × 10−22.041 512 383
172 495 × 10−1 1.063 292 519 876 557 × 10−13.221 379 242 071 772 ×
10−1 1.286 970 636 980 550 × 10−14.590 563 702 382 297 × 10−1 1.435
765 250 777 324 × 10−16.055 072 094 196 928 × 10−1 1.471 171 930
360 809 × 10−17.483 505 046 653 010 × 10−1 1.358 982 670 708 245 ×
10−18.718 162 722 550 230 × 10−1 1.083 234 916 072 349 × 10−19.600
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 ×
10−31.218 752 444 025 983 × 10−2 1.381 563 354 931 116 × 10−22.994
246 439 820 815 × 10−2 2.169 029 360 224 803 × 10−25.555 435 092
225 135 × 10−2 2.952 299 561 392 022 × 10−28.895 761 623 636 096 ×
10−2 3.726 213 368 537 990 × 10−21.300 203 295 751 197 × 10−1 4.482
511 676 477 427 × 10−21.785 092 814 999 323 × 10−1 5.209 093 302
886 898 × 10−22.340 488 118 981 268 × 10−1 5.889 506 175 213 475 ×
10−22.960 759 639 570 856 × 10−1 6.502 739 863 025 273 × 10−23.637
955 855 843 940 × 10−1 7.023 445 222 659 594 × 10−24.361 403 335
253 513 × 10−1 7.422 717 345 129 071 × 10−25.117 418 243 587 686 ×
10−1 7.669 557 687 323 547 × 10−25.889 200 757 075 352 × 10−1 7.733
072 212 747 030 × 10−26.656 984 698 634 406 × 10−1 7.585 359 103
842 103 × 10−27.398 503 206 783 987 × 10−1 7.204 896 745 102 966 ×
10−28.089 804 217 861 422 × 10−1 6.580 079 947 124 912 × 10−28.706
406 686 725 344 × 10−1 5.712 405 693 982 534 × 10−29.224 733 629
316 755 × 10−1 4.618 724 004 952 186 × 10−29.623 698 984 922 738 ×
10−1 3.331 967 754 066 058 × 10−29.886 262 004 970 869 × 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
∫ 1b
f (x, φ) dx dφ =nφ∑i=1
wφ
i
(n∑
k=1f (xk, φi)w
xk
),
-
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 × 10−22.031
623 105 912 188 × 10−1 8.884 898 607 298 555 × 10−24.356 411 421
025 461 × 10−1 1.714 654 253 019 344 × 10−17.002 764 501 137 072 ×
10−1 2.137 749 099 088 692 × 10−19.160 789 294 435 172 × 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 ×
10−35.868 587 471 593 844 × 10−2 1.417 300 251 017 768 × 10−21.314
023 014 034 694 × 10−1 3.141 851 068 881 919 × 10−22.312 259 684
709 282 × 10−1 5.404 249 071 427 261 × 10−23.545 272 566 781 712 ×
10−1 7.926 234 023 750 671 × 10−24.946 547 898 729 779 × 10−1 1.022
669 162 191 473 × 10−16.412 444 375 669 494 × 10−1 1.164 567 246
852 908 × 10−17.803 824 426 328 084 × 10−1 1.149 571 587 384 354 ×
10−18.960 668 473 279 682 × 10−1 9.335 975 293 941 803 × 10−29.729
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 ×
10−41.582 745 120 856 827 × 10−2 1.990 269 851 064 446 × 10−33.559
214 977 062 290 × 10−2 4.471 865 425 313 305 × 10−36.321 587 478
637 355 × 10−2 7.927 664 681 909 845 × 10−39.861 524 371 928 763 ×
10−2 1.232 329 088 245 827 × 10−21.416 316 287 947 248 × 10−1 1.759
239 754 490 365 × 10−21.919 943 624 001 203 × 10−1 2.362 155 104
951 021 × 10−22.492 785 459 883 913 × 10−1 3.023 415 310 220 783 ×
10−23.128 601 705 648 356 × 10−1 3.717 573 173 420 895 × 10−23.818
724 965 396 798 × 10−1 4.410 387 041 811 009 × 10−24.551 689 741
193 452 × 10−1 5.058 684 026 339 364 × 10−25.312 991 690 114 278 ×
10−1 5.611 528 812 727 251 × 10−26.085 048 417 924 119 × 10−1 6.013
067 280 364 542 × 10−26.847 430 949 754 737 × 10−1 6.207 213 941
364 178 × 10−27.577 419 536 990 212 × 10−1 6.144 003 961 120 213 ×
10−28.250 906 475 975 498 × 10−1 5.786 968 919 983 928 × 10−28.843
622 938 401 178 × 10−1 5.120 418 486 594 962 × 10−29.332 611 653
960 123 × 10−1 4.155 170 611 239 503 × 10−29.697 812 394 998 446 ×
10−1 2.931 237 128 126 203 × 10−29.923 585 381 315 836 × 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 andweights φi, w
φ
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.
-
Slepian functions on the sphere, generalized Gaussian quadrature
rule 891
–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 Grünbaum 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.
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