prolate spheroidal wave functions and applications

Prolate Spheroidal Wave Functions and their PropertiesComputation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivativesApplications of the PSWFs

CIMPA School on Real and Complex Analysis with Applications,Buea Cameroun, 1–14 May 2011.

Prolate Spheroidal Wave functions andApplications.

Abderrazek Karoui in collaboration with Aline Bonami

University of Carthage, Department of Mathematics,Faculty of Sciences of Bizerte, Tunisia.

March 29, 2011

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.



Outline1 PSWFs and Properties

PSWFs and PDEdifferential and integral operators associated with PSWFsSome Properties of the PSWFs

2 Computation of the PSWFs

3 Uniform estimates of the PSWFs and their derivativesWKB method for the PSWFsUniform bounds of the PSWFs and their derivativesExponential decay of the eigenvalues associated with thePSWFs

4 Applications of the PSWFsPSWFs based spectral approximation in Sobolev spaces.Signal processing applications




Prolate Spheroidal Wave Functions from PDE point of ViewPSWFs as eigenfunctions of a differential and an integral operatorSome properties of the PSWFs

Spheroidal Coordinates

For a fixed a > 0, the elliptic coordinate system is given by

z = a coshµ cos ν,

y = a sinhµ sin ν, µ > 0, ν ∈ [0, 2π].

Figure: Graph from Wikipedia.Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.




The Prolate Spheroidal coordinate system is obtained by rotatingthe previous elliptic coordinates about the focal axis of the ellipse.This gives the following coordinates:

z = a coshµ cos ν,

y = a sinhµ sin ν sinφ,

x = a sinhµ sin ν cosφ µ > 0, ν ∈ [0, π], φ ∈ [0, 2π].

Let ξ = coshµ, η = cos ν, then the Spheroidal coordinates aregiven by

x = a√

(ξ2 − 1)(1− η2) cosφ,

y = a√

(ξ2 − 1)(1− η2) sinφ,

z = aξη, ξ > 1 η ∈ [−1, 1].





Spheroidal Coordinates

Figure: Graph from Wikipedia.Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.




Wave equation in Spheroidal coordinates

It is well known, see [Abramowitz], that the Helmotz Waveequation in spheroidal coordinates becomes

∆Φ + k2Φ =∂

∂ξ

[(ξ2 − 1)

∂Φ

∂ξ

]+

[(1− η2)

∂Φ

∂η

]+

ξ2 − η2

(ξ2 − 1)(1− η2)

∂2Φ

∂Φ2+ c2(ξ2 − η2)Φ = 0, c =

1

2ak.

If

Φ(ξ, η, φ) = Rmn(c , ξ)Smn(c, η)cossin

mφ,

then the radial and the angular solutions Rmn and Smn satisfy thefollowing ODEs





d

dξ

[(ξ2 − 1)

d

dξRmn(c , ξ)

]−(χmn − c2ξ2 +

m2

ξ2 − 1

)Rmn(c , ξ) = 0,

d

dη

[(1− η2)

d

dηSmn(c , η)

]+

(χmn − c2η2 − m2

1− η2

)Smn(c, η) = 0.

In the special case m = 0, the last ODE becomes

(1− x2)d2ψn,c(x)

dx2− 2x

dψn,c(x)

dx+ (χn(c)− c2x2)ψn,c(x) = 0.





Consider the finite convolution operator Tc , given by

Tc(ψ)(x) =

∫ 1

−1

sin c(x − y)

π(x − y)ψ(y) dy = λ ψ(x) ∀x ∈ R. (1).

D. Slepian has incedently discovered that

TcLc = LcTc

where

Lc(y) = (1− x2)d2y

dx2− 2x

dy

dx− c2x2y , (3).

=⇒ The ψn,c are the bounded eigenfunctions of Lc .• ψn,c is of the same parity as n.





In [F. Grunbaum et al, 1982], the authors have shown that if A,Bare two measurable sets and if A,B denote the restrictionoperators over A,B, respectively and if F denotes a Fouriertransform operator, then F−1BF acting on L2(A) is a convolutionoperator. In the special case whereF(ξ) = f (ξ) =

∫R f (x)e−ixξ dx , then it is easy to see that

(F−1BF

)f (x) =

∫A

∫B

e iξ(x−y)

2πdξf (y) dy =

∫A

KB(x , y)f (y) dy .





Let Dx be a second order differential operator satisfyingDxA = ADx = Dx , then by using integration by parts, one canshow that Dx commutes with the convolution operatorE ∗E = (AF−1B)(BFA) is equivalent to the condition

DxKB(x , y) = DyKB(x , y).

In the special case where A = [−1, 1], B = [−c , c], one gets

KB(x , y) =sin c(x − y)

π(x − y), then the differential operator Dx = Lc ,

defined by

Dx(ψ) = (1− x2)d2ψ

dx2− 2x

dψ

dx− c2x2ψ,

satisfies the commutativity condition DxKB(x , y) = DyKB(x , y).





1 Tc is a self-adjoint compact operator.

2 ρ(Tc) : the spectrum of Tc is infinite and countable.

ρ(Tc) = λn(c), n ∈ N; λ0(c) > λ1(c) > · · ·λn(c) > · · · .

and limn→+∞

λn(c) = 0.

3 If ψn,c denotes the eigenfunction associated with λn(c), thenψn,c , n ∈ N is an orthogonal basis of L2[−1, 1], anorthonormal basis of

Bc = f ∈ L2(R); Suppt f ⊆ [−c , c],

and an orthonormal system of L2(R).∫ 1

−1ψn,cψm,c = λn(c)δn,m,

∫Rψn,cψm,c = δn,m.





Fundamental property of the ψn,c

ψn,c(ξ) = (−i)n√

2π

c λnψn,c

(ξ

c

)1[−c,c](ξ).





Qc : L2[−1, 1]→ [−1, 1], f →∫ 1

−1e i c x y f (y) dy . (2).

Q∗(Qc f )(x) =2π

c

∫ 1

−1

sin c(x − y)

π(x − y)f (y) dy =

2π

cTc(f )(x).

Hence,

λn(c) =c

2π|µn(c)|2, µn(c) ∈ ρ(Qc), ∀n ∈ N.




Flammer’s Method

Let (Pk)k≥0 the set of the normalized Legendre polynomials.

• ∀ |x | ≤ 1, ψn,c(x) =∞∑

k=0,1

′

βnkPk(x), ou

(k + 1)(k + 2)

(2k + 3)√

(2k + 5)(2k + 1)c2βnk+2+

(k(k + 1) +

2k(k + 1)− 1

(2k + 3)(2k − 1)c2

)βnk

+k(k − 1)

(2k − 1)√

(2k + 1)(2k − 3)c2βnk−2 = χnβ

nk , k ≥ 0, χn ∈ ρ(Lx).




• Un(c) =∞∑

k=0,1

′

ik√

k + 1/2√

2π/c βnk jk+1/2(c),

where jk(·) denotes the normalized Bessel function of the first kindof order k .

• λn(c) =c

2π

[|Un(c)|∑∞

k=0,1′βnk√

k + 1/2

]2

.

• ∀|x | > 1,

ψn,c(x) =∞∑

k=0,1

′

(ik−n)(√

k + 1/2)(√

2π/c) βnk jk+1/2(c x).




Graphs of some PSWFs

Figure: Graphs of the PSWFs ψn,c , c = 100 and with (a) n = 0, (b)n = 1, (c) n = 15, (d) n = 30.Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.



WKB method for the PSWFsUniform bounds of the PSWFs and their derivativesExponential decay of the eigenvalues associated with the PSWFs

WKB method for the PSWFs

Let ψ be the n-th order PSWFs, then

d

dx

[(1− x2)ψ′(x)

]+ χn(1− qx2)ψ(x) = 0, x ∈ [−1, 1]. (1)

Consider the Elliptic integral

S(x) := Sq(x) =

∫ 1

x

√1− qt2

1− t2dt, x ∈ [0, 1). (2)

We look for ψ under the form

ψ(x) = ϕ(x)U(S(x)), ϕ(x) = (1− x2)−1/4(1− qx2)−1/4. (3)

The equation satisfied by U on the interval [0,+1) is written as

U ′′ + (χn + h1) U = 0, (4)

h1(S(x)) := ϕ(x)−1(1− qx2)−1 d

dx

[(1− x2)ϕ′(x)

].





Note that

h1(S(x)) =(1− q)−1

8(1− x)+ h2(S(x)),

with h2 S a rational function without poles on [0, 1].

Lemma

One has the following inequalities, valid on the interval [0, 1].

2(1− q)(1− x) ≤ S2q (x) ≤ 4(1− x). (5)

At 0 one has 1 ≤ Sq(0) ≤ π2 . Moreover

Sq(x)2

1−x extends into aholomorphic function in a neighborhood of 1 and takes the value2(1− q) at 1. Finally

0 ≤ Sq(x)−√

2(1− q)(1− x) ≤ 2

3

(1− x)3/2

(1− q)1/2. (6)





Lemma

For q < 1, there exists a function F := Fq that is continuous on[0, S(0)], satisfies the inequality

|F (s)| ≤ 3

(1− q)3, (7)

and such that U is a solution of the equation

U ′′(s) +

(χn +

1

4s2

)U(s) = F (s)U(s), s ∈ [0,S(0)]. (8)





The associated homogeneous equation has the two independentsolutions

U1(s) = χ1/4n√

sJ0(√χns), U2(s) = χ

1/4n√

sY0(√χns).

The Wronskian of U1 and U2 is given by

W (U1,U2)(s) = U1(s)U ′2(s)− U ′1(s)U2(s)

= sχn

[J0(√χns)Y ′0(

√χns)− J ′0(

√χns)Y0(

√χns)

]= =

2√χn

π

Then, the general solution of (8), is given by

U(s) = AU1(s) + BU2(s) +π

2√χn

×∫ s

0

√stχn [J0(

√χns)Y0(

√χnt)− J0(

√χnt)Y0(

√χns)] F (t)U(t)dt.(9)





Lemma

Let I =2√χn

∫ √χnSq(0)

0t (J0(t))2 dt. Then there exists a constant

C ′ independent of n and c, such that∣∣∣∣I − 2Sq(0)

π

∣∣∣∣ ≤ C ′√χn. (10)





Theorem

(A.Bonami, A.K. (2010)): There exist constants C ,C ′ with thefollowing properties. Assume that the parameters n, c are suchthat q = c2/χn(c) < 1. Then one can find a constantA := A(n, c) ≤ M such that, for 0 ≤ x ≤ 1,

ψn,c(x) = Aχn(c)1/4

√Sq(x)J0(

√χn(c)Sq(x))

(1− x2)1/4(1− qx2)1/4+ Rn,c(x) (11)

with

supx∈[0,1]

|Rn,c(x)| ≤ Cqχn(c)−1/2, Cq =C

(1− q)13/4. (12)





Next we state as a lemma the fact that, for q close from 0, theconstant A is close from 1.

Lemma

Let α < 1 and 0 < K < 1. Let C ′ be as defined by Lemma 4.There are constants H1 = H1(α,K ) and H2 = H2(α,K ) such that,

for q ≤ α and n satisfyingC ′√χn≤ K√

1− α, the constant A(n, c)

in Theorem 4 satisfies the inequality

|A2(n, c)− 1| ≤ H1(α,K )q + H2(α,K )χn(c)−1/2. (13)

As a consequence, under the same assumptions on q,

ψn,c(1)2 − n − 1

2≤ H3qn + H2. (14)





As a direct consequence of Theorem 1, we have the followingresult.

Corollary

There is a constant C such that, for Cq = C (1− q)−4, the twofollowing inequalities hold.

sup|x |≤1 |ψn,c(x)| ≤ Cqχ1/4n,c (15)

sup|x |≤1(1− x2)1/4|ψn,c(x)| ≤ Cq. (16)

Remark that the first bound is sharp since

ψn,c(1) = A(n, c)χn(c)1/4. (17)





Proposition

There exists a constant C depending only on α < 1 such that, for

q =c2

χn< α, then

supx∈[−1,1]

|ψ′(x)| ≤ Cχn5/4, (18)

supx∈[−1,1]

(1− x2)|ψ′(x)| ≤ C√χn. (19)





Proposition

There exists a constant C such that, for all n ≥ 0 and c ≥ 0,

supx∈[−1,1]

(1− x2)1/4|ψn,c(x)| ≤ (2χn(c))1/4. (20)

supx∈[−1,1]

(1− x2)|ψ′n,c(x)| ≤ C (c2 + χn(c))3/4. (21)





Proposition

For any integers n, k ≥ 0, satisfying k(k + 1) ≤ χn, we have∣∣∣ψ(k)n,c (0)

∣∣∣ ≤ (√χn)k |ψn,c(0)| , (22)

for n even and k even, and∣∣∣ψ(k)n,c (0)

∣∣∣ ≤ (√χn)k−1

∣∣ψ′n,c(0)∣∣ , (23)

for n odd and k odd. In particular, under the assumption that

q =c2

χn< 1, there exists a constant C , depending only on q and

such that for any positive integer k satisfying k(k + 1) ≤ χn, wehave ∣∣∣ψ(k)

n,c (0)∣∣∣ ≤ C (

√χn)k . (24)





We start from the well-known equality, [Slepian 1964, Rokhlin etal. (2007)], that for any positive integer n, we haveλn(c) = λ′ × λ′′, with

λ′ : =c2n+1(n!)4

2((2n)!)2(Γ(n + 3/2))2(25)

λ′′ : = exp

(2

∫ c

0

(ψn,τ (1))2 − (n + 1/2)

τdτ

). (26)

Proposition

Let α < 1. There exists a constant Mα with the following property.For all n and c ≥ 0 such that qn(c) ≤ α, then

supx∈[−1,1]

∣∣ψn,c(x)− Pn(x)∣∣ ≤ Mα

c2√n + 1/2

, (27)





Proposition

Let α < 1. Then there exists constants M1, M2 such that

λn(c) ≤ M1n

2(

e

4)2n

(c2

n2exp(M2

c2

n2)

)n

. (28)

Theorem

(A.Bonami, A.K. (2010)): Let δ > 0. There exists N and κ suchthat, for all c ≥ 0 and n ≥ max(N, κc),

λn(c) ≤ e−δ(n−κc).





Remark

Numerical evidence, see [Rokhlin et al. 2007], indicates that(ψn,τ )2 − (n + 1/2) ≤ 0, ∀ t ≥ 0. If we accept this assertion, then

we observe that the sequence λn(c) decays faster thanc

2

( ec

4n

)2n

so that the exponential decay has started at [ec/4].

Remark

∀c > 0, ∀ 0 < α < 1, N(α) = #λi (c); λi (c) > α is given by

N(α) =2c

π+

[1

π2log

(1− αα

)]log(c) + o(log(c)),

(Landau and Widom (1980)).





Graph of the λn(c) for different values of c and n




PSWFs based spectral approximation in Sobolev spaces.Signal Processing Applications of the PSWFs

Chen-Gottlieb-Hesthaven approach

Theorem

(Theorem 3.1 in [Chen et al. 2005]). Let f ∈ Hs(I ), s ≥ 0.Then

|aN(f )| ≤ C

N−2/3s‖f ‖Hs +

(√c2

χN(c)

)δN‖f ‖L2(I )

,

where C , δ are independent of f ,N and c.





Wang’s Approach

By considering the weighted Sobolev space H r (I ), associated withthe differential operator

Dcu = −(1− x2)u” + 2xu′ + c2x2u,

and given by

H r (I ) =

f ∈ L2(I ), ‖f ‖2Hr = ‖Dr/2

c f ‖2 =∑k≥0

(χk)r |fk |2 < +∞

,

the following result has been given in [L. Wang 2010].

Theorem

(Theorem3.3 in[Wang , 2010]). For any f ∈ H r (I ), with r ≥ 0, wehave

‖f − SN f ‖2 ≤ (χN(c))−r/2‖f ‖Hr ≤ N−r‖f ‖

Hr .





A.Bonami and A. K. approaches

Theorem

Let c ≥ 0 be a positive real number. Assume that f ∈ Hs(I ), forsome positive real number s > 0. Then for any integer N ≥ 1, wehave

‖f − SN f ‖2 ≤ K (1 + c2)−s/2‖f ‖Hs + K√λN(c)‖f ‖2. (29)

Here, the constant K depends only on s. Moreover it can be takenequal to 1 when f belongs to the space Hs

0(I ).





Remark

This should be compared with the results of [Wang, 2010], givenby Theorem 5. This has the advantage to give an error term for allvalues of c, while the first term in (29) is only small for c largeenough. On another side, Wang compares this specific Sobolevspace with the classical one and finds that

‖f ‖Hs

c≤ C (1 + c2)s/2‖f ‖Hs .

For large values of N we clearly have (1+c2)χN

(1 + c2)−1, but it

goes the other way around when χN and 1 + c2 are comparable.So it may be useful to have both kinds of estimates in mind fornumerical purpose and for the choice of the value of c .





Theorem

(A. Bonami, A. K. (2010)): Let s > 0, c > 0, be any positive realnumbers and let f ∈ Hs

per ([−1, 1]). Then for any integer N ≥ 1,

‖f−SN f ‖2 ≤√

(1/2 +π

4c)∑n≥N‖ψn,c‖2

∞λn(c)‖f ‖2+c−s‖f−f[c/π]‖Hsper.

(30)Here, f[c/π] is the truncated Fourier series expansion of f to theorder

[cπ

]. In particular, for any positive integer N satisfying

q = c2/χN < 1, we have

‖f −SN f ‖2 ≤ Kq

√∑n≥N

√χnλn(c)‖f ‖2 + c−s‖f − f[c/π]‖Hs

per, (31)

where Kq =√

(1/2 + π4c )Cq and Cq is as given by Corollary 1.





We consider the Weirstrass function

f (x) =∑k≥0

cos(2kπx)

2ks, −1 ≤ x ≤ 1, s = 1.4. (32)

f ∈ H1per ([−1, 1]), with ‖f ‖2

2 =∑k≥0

1

22ks. If c = 100, then we have

‖f ‖2 ≈ 1.0805838, ‖f − f[c/π]‖H1 ≈ 1.203854. Next, we have

found that EN =

[1

50

50∑k=−50

(f (k/50)− SN f (k/50))2

]1/2

. Note

that once N becomes larger than the critical value for the decay ofthe λn(c), which is Nc = [ec/4] = 67, the theoretical error boundbecomes very close to the actual error.





Figure: (a) graph of W3/4(x), (b) graph of W3/4,N(x),N = 90.





We let s > 0 be any positive real number and we consider theBrownian motion function Bs(x) given by as follows.

Bs(x) =∑k≥1

Xk

kscos(kπx), −1 ≤ x ≤ 1. (33)

Here, Xk is a Gaussian random variable. It is well known thatBs ∈ Hs([−1, 1]). For the special case s = 1, we consider theband-width c = 100, a truncation order N = 80 and compute B1,N

the approximation of B1 by its N−th terms truncated PSWFsseries expansion. The graphs of B1 and B1,N are given by thefollowing figure.





Figure: (a) graph of B1(x), (b) graph of B1,N(x),N = 80.





Approximation of band-limited functions

Lemma

Let f ∈ Bc be an L2 normalized function. Then∫ +1

−1|f − SN f |2dt ≤ λN(c). (34)





Approximation of almost band-limited functions

Let T and Ω de two measurable sets. A function pair (f , f ) is saidto be εT−concentrated in T and εΩ−concentrated in Ω if∫

T c

|f (t)|2 dt ≤ ε2T ,

∫Ωc

|f (ω)|2 dω ≤ ε2Ω.

Next we define the time-limiting operator PT and theband-limiting operator ΠΩ by:

PT (f )(x) = χT (x)f (x), ΠΩ(f )(x) =1

2π

∫Ω

e ixω f (ω) dω.





Approximation of almost band-limited functions

Proposition

If f is an L2 normalized function that is εT−concentrated inT = [−1,+1] and εΩ−band concentrated in Ω = [−c ,+c], thenfor any positive integer N, we have(∫ +1

−1|f − SN f |2dt

)1/2

≤ εΩ +√λN(c) (35)

and, as a consequence,

‖f − PTSN f ‖2 ≤ εT + εΩ +√λN(c). (36)





Exact reconstruction of band-limited functions withmissing data

In [Donoho-Stark 1989], the authors have shown the followinguncertainty principle:If ‖f ‖2 = ‖f ‖2 = 1 and (f , f ) is εT−concentrated on T andεΩ−concentrated on Ω, then

|Ω||T | ≥ (1− (εT + εΩ))2 .

Hence, if |Ω||T | < 1, then the following band-limitedreconstruction problem has a unique solution in BΩ.

Find S ∈ BΩ such that r(t) = χT c (t) (S(t) + η(t)) , η(·) ∈ L2.





The solution S is given by

S(t) = Qr(t) =∑n≥0

(PTPΩ)n r(t), t ∈ R.

‖S − Qr‖ ≤ C‖η‖, C ≤ (1−√|T ||Ω|)−1.

If T = [−τ, τ ], Ω = [−c , c], then

PΩPT (f )(x) =

∫ τ

−τ

sin 2πc(x − y)

π(x − y)f (y) dy , x ∈ R.

Hence

‖PTPΩ‖ ≤ λ0(c) < 1.

Consequently, the band-limited reconstruction problem has aband-limited solution no matter how large are T and Ω.





PSWFs versus Wavelts

we consider a real life 1-D discrete signal S = (sn)1≤n≤3850

corresponding to an electrical load consumption, measured minuteby minute. We have considered the bandwidth c = 50. Then, wehave divided S into 11 blocks S i , i = 0, . . . , 10, of equal length350. For a given block Si = (sj)1+350i≤j≤350(i+1), we have used 35uniformly sampled data points. Finally, the different data blocksare synthetized by the use of the following rule,

S i (sj) = sj =34∑n=0

αinΨn,c(tj), tj = −1+(j−1−350i)/175, 1+350i ≤ j ≤ 350(i+1), 0 ≤ i ≤ 10.

(37)We have found that

r =‖S − S‖2

‖S‖2=

[∑j |sj − sj |2∑

j s2j

]1/2

= 2.03E − 02. (38)





Next, we have considered the 8 taps Daubechies wavelet filters andapplied the DWT to analyze and synthetize the signal S , at thelevel 4. In this case, 241 wavelet coeficients have been used tosynthetize the signal. The graph of the reconstructed signal isgiven by Figure 4(c). Also, we have computed r ′, the relativeresidual of this second method, and we found thatr ′ = 1.42E − 02. Based on the previous numerical results, oneconcludes that the wavelet tool outperforms in term of accuracy,the PSWFs based tool.





Figure: (a) Original signal S , (b) reconstruction of S by the PSWFs, (c)reconstruction of S by wavelets.





A bandlimited example

We consider the signal given by

f (t) =sin(50t)

50t, −1 ≤ t ≤ 1.

A discrete signal S1 of length 2048, is obtained from f byuniformly sampling this later. We have considered the bandwidthc = 50 and computed the first 17 even indexed PSWFs analyzingcoefficients of S1. If S1 denotes the PSWFs synthetized signal,given by Figure 5(b), then we found that the relative residual isgiven by r1 = 5.26E − 03. Moreover, we have applied the samewavelet analysis/synthezis scheme of the previous example to S1.We found that 136 wavelet coefficients have been used to obtainthe synthetized signal S ′1, given by Figure 5(c). The correspondingrelative residual is given by r ′1 = 3.58E − 02.





Figure: (a) Original signal S1, (b) reconstruction of S1 by the PSWFs,(c) reconstruction of S1 by wavelets.





Figure: (a) Original noised signal, (b) reconstructed and denoised signalby the PSWFs





References[1] M. Abramowitz and I. Stegun, Handbook of MathematicalFunctions, Dover Publications, New York, 1970.[2] Aline Bonami and Abderrazek Karoui, Uniform Estimates of theProlate Spheroidal Wave Functions and Approximations in SobolevSpaces, hal-00547220, Arxiv:1012.3881, (2010).[3] John P. Boyd. Prolate spheroidal wavefunctions as analternative to Chebyshev and Legendre polynomials for spectralelement and pseudospectral algorithms. J. Comput. Phys., 199(2),(2004), 688716.[4] Q. Y. Chen, D. Gottlieb, and J. S. Hesthaven, Spectralmethods based on prolate spheroidal wave functions for hyperbolicPDEs. SIAM J. Numer. Anal., 43(5), (2005), 19121933.





[5] D. L. Donoho and P. B. Stark, Uncertainty principles and signalrecovery, SIAM Journal of Applied mathematics, 49, (1989),pp.906-931.[6] Li-Lian Wang, Analysis of Spectral Approximations usingProlate Spheroidal Wave Functions, Mathematics of Computation79 (2010), 807-827.[7] D. Slepian, H. O. Pollak, Prolate spheroidal wave functions,Fourier analysis, and uncertainty-I, Bell Syst. Tech. J. 40 (1961),43–63.[8] D. Slepian, Prolate spheroidal wave functions, Fourier analysisand uncertainty–IV: Extensions to many dimensions; generalizedprolate spheroidal functions, Bell System Tech. J. 43 (1964),3009–3057.





[9] H. J. Landau and H. Widom, Eigenvalue distribution of timeand frequency limiting, J. Math. Anal.Appl., 77, (1980), 469–481.[10] V. Rokhlin and H. Xiao, Approximate formulae for certainprolate spheroidal wave functions valid for large values of bothorder and band-limit, Appl. Comput. Harmon. Anal. 22, (2007),105–123.


prolate spheroidal wave functions and applications

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