Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
CIMPA School on Real and Complex Analysis with Applications,
Buea Cameroun, 1–14 May 2011.
Prolate Spheroidal Wave functions and
Applications.
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.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Outline
1
PSWFs and Properties
PSWFs and PDE
differential and integral operators associated with PSWFs
Some Properties of the PSWFs
2
Computation of the PSWFs
3
Uniform estimates of the PSWFs and their derivatives
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the
PSWFs
4
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal processing applications
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Prolate Spheroidal Wave Functions from PDE point of View
PSWFs as eigenfunctions of a differential and an integral operator
Some properties of the PSWFs
Spheroidal Coordinates
For a fixed a > 0, the elliptic coordinate system is given by
z
= a cosh µ cos ν,
y
= a sinh µ sin ν,
µ > 0,
ν ∈ [0, 2π].
Figure: Graph from Wikipedia.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Prolate Spheroidal Wave Functions from PDE point of View
PSWFs as eigenfunctions of a differential and an integral operator
Some properties of the PSWFs
The Prolate Spheroidal coordinate system is obtained by rotating
the previous elliptic coordinates about the focal axis of the ellipse.
This gives the following coordinates:
z
= a cosh µ cos ν,
y
= a sinh µ sin ν sin φ,
x
= a sinh µ sin ν cos φ
Let ξ = cosh µ,
given by
µ > 0,
ν ∈ [0, π],
φ ∈ [0, 2π].
η = cos ν, then the Spheroidal coordinates are
y
q
(ξ 2 − 1)(1 − η 2 ) cos φ,
q
= a (ξ 2 − 1)(1 − η 2 ) sin φ,
z
= aξη,
x
= a
ξ>1
Abderrazek Karoui in collaboration with Aline Bonami
η ∈ [−1, 1].
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Prolate Spheroidal Wave Functions from PDE point of View
PSWFs as eigenfunctions of a differential and an integral operator
Some properties of the PSWFs
Spheroidal Coordinates
Figure: Graph from Wikipedia.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Prolate Spheroidal Wave Functions from PDE point of View
PSWFs as eigenfunctions of a differential and an integral operator
Some properties of the PSWFs
Wave equation in Spheroidal coordinates
It is well known, see [Abramowitz], that the Helmotz Wave
equation in spheroidal coordinates becomes
∂Φ
∂
2
2 ∂Φ
2
(ξ − 1)
+ (1 − η )
∆Φ + k Φ =
∂ξ
∂ξ
∂η
2
2
2
ξ −η
∂ Φ
+
+ c 2 (ξ 2 − η 2 )Φ = 0,
2
2
(ξ − 1)(1 − η ) ∂Φ2
1
c = ak.
2
If
Φ(ξ, η, φ) = Rmn (c, ξ)Smn (c, η)
cos
mφ,
sin
then the radial and the angular solutions Rmn and Smn satisfy the
following ODEs
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Prolate Spheroidal Wave Functions from PDE point of View
PSWFs as eigenfunctions of a differential and an integral operator
Some properties of the PSWFs
d
d
m2
2
2 2
(ξ − 1) Rmn (c, ξ) − χmn − c ξ + 2
Rmn (c, ξ) = 0,
dξ
dξ
ξ −1
d
m2
2 d
2 2
(1 − η ) Smn (c, η) + χmn − c η −
Smn (c, η) = 0.
dη
dη
1 − η2
In the special case m = 0, the last ODE becomes
(1 − x 2 )
dψn,c (x)
d 2 ψn,c (x)
− 2x
+ (χn (c) − c 2 x 2 )ψn,c (x) = 0.
2
dx
dx
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Prolate Spheroidal Wave Functions from PDE point of View
PSWFs as eigenfunctions of a differential and an integral operator
Some properties of the PSWFs
Consider the finite convolution operator Tc , given by
Z 1
sin c(x − y )
Tc (ψ)(x) =
ψ(y ) dy = λ ψ(x) ∀x ∈ R.
−1 π(x − y )
(1).
D. Slepian has incedently discovered that
Tc Lc = Lc Tc
where
Lc (y ) = (1 − x 2 )
dy
d 2y
− 2x
− c 2x 2y ,
dx 2
dx
(3).
=⇒ The ψn,c are the bounded eigenfunctions of Lc .
• ψn,c is of the same parity as n.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Prolate Spheroidal Wave Functions from PDE point of View
PSWFs as eigenfunctions of a differential and an integral operator
Some properties of the PSWFs
In [F. Grunbaum et al, 1982], the authors have shown that if A, B
are two measurable sets and if A, B denote the restriction
operators over A, B, respectively and if F denotes a Fourier
transform operator, then F −1 BF acting on L2 (A) is a convolution
operator. In theRspecial case where
F(ξ) = b
f (ξ) = R f (x)e −ixξ dx, then it is easy to see that
F
−1
Z Z
BF f (x) =
A
B
e iξ(x−y )
dξf (y ) dy =
2π
Abderrazek Karoui in collaboration with Aline Bonami
Z
KB (x, y )f (y ) dy .
A
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Prolate Spheroidal Wave Functions from PDE point of View
PSWFs as eigenfunctions of a differential and an integral operator
Some properties of the PSWFs
Let Dx be a second order differential operator satisfying
Dx A = ADx = Dx , then by using integration by parts, one can
show that Dx commutes with the convolution operator
E ∗ E = (AF −1 B)(BFA) is equivalent to the condition
Dx KB (x, y ) = Dy KB (x, y ).
In the special case where A = [−1, 1], B = [−c, c], one gets
sin c(x − y )
, then the differential operator Dx = Lc ,
KB (x, y ) =
π(x − y )
defined by
Dx (ψ) = (1 − x 2 )
d 2ψ
dψ
− 2x
− c 2 x 2 ψ,
2
dx
dx
satisfies the commutativity condition Dx KB (x, y ) = Dy KB (x, y ).
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
1
2
Prolate Spheroidal Wave Functions from PDE point of View
PSWFs as eigenfunctions of a differential and an integral operator
Some properties of the PSWFs
Tc is a self-adjoint compact operator.
ρ(Tc ) : the spectrum of Tc is infinite and countable.
ρ(Tc ) = {λn (c), n ∈ N; λ0 (c) > λ1 (c) > · · · λn (c) > · · · }.
and
3
lim λn (c) = 0.
n→+∞
If ψn,c denotes the eigenfunction associated with λn (c), then
{ψn,c , n ∈ N} is an orthogonal basis of L2 [−1, 1], an
orthonormal basis of
Bc = {f ∈ L2 (R); Supp t b
f ⊆ [−c, c]},
and an orthonormal system of L2 (R).
Z 1
Z
ψn,c ψm,c = λn (c)δn,m ,
ψn,c ψm,c = δn,m .
−1
Abderrazek Karoui in collaboration with Aline Bonami
R
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Prolate Spheroidal Wave Functions from PDE point of View
PSWFs as eigenfunctions of a differential and an integral operator
Some properties of the PSWFs
Fundamental property of the ψn,c
r
ψbn,c (ξ) = (−i)
n
2π
ψn,c
c λn
Abderrazek Karoui in collaboration with Aline Bonami
ξ
1[−c,c] (ξ).
c
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Prolate Spheroidal Wave Functions from PDE point of View
PSWFs as eigenfunctions of a differential and an integral operator
Some properties of the PSWFs
Z
2
1
Qc : L [−1, 1] → [−1, 1], f →
e i c x y f (y ) dy .
(2).
−1
2π
Q (Qc f )(x) =
c
∗
Z
1
−1
sin c(x − y )
2π
f (y ) dy =
Tc (f )(x).
π(x − y )
c
Hence,
λn (c) =
c
|µn (c)|2 ,
2π
Abderrazek Karoui in collaboration with Aline Bonami
µn (c) ∈ ρ(Qc ), ∀n ∈ N.
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Flammer’s Method
Let (Pk )k≥0 the set of the normalized Legendre polynomials.
∞ 0
X
• ∀ |x| ≤ 1, ψn,c (x) =
βkn Pk (x), où
k=0,1
(k + 1)(k + 2)
2k(k + 1) − 1 2
2 n
p
c βk+2 + k(k + 1) +
c β
(2k + 3)(2k − 1)
(2k + 3) (2k + 5)(2k + 1)
+
k(k − 1)
n
p
c 2 βk−2
= χn βkn ,
(2k − 1) (2k + 1)(2k − 3)
Abderrazek Karoui in collaboration with Aline Bonami
k ≥ 0, χn ∈ ρ(Lx ).
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
∞ 0 p
X
p
• Un (c) =
i k k + 1/2 2π/c βkn jk+1/2 (c),
k=0,1
where jk (·) denotes the normalized Bessel function of the first kind
of order k.
"
#2
c
|Un (c)|
.
• λn (c) =
P∞ 0 n p
2π
k + 1/2
k=0,1 βk
• ∀|x| > 1,
∞ 0
X
p
p
ψn,c (x) =
(i k−n )( k + 1/2)( 2π/c) βkn jk+1/2 (c x).
k=0,1
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Graphs of some PSWFs
Figure: Graphs of the PSWFs ψn,c , c = 100 and with (a) n = 0,
n = 1, (c) n = 15, (d) n = 30.
Abderrazek Karoui in collaboration with Aline Bonami
(b)
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
WKB method for the PSWFs
Let ψ be the n-th order PSWFs, then
d (1 − x 2 )ψ 0 (x) + χn (1 − qx 2 )ψ(x) = 0, x ∈ [−1, 1].
dx
Consider the Elliptic integral
s
Z 1
1 − qt 2
S(x) := Sq (x) =
dt, x ∈ [0, 1).
1 − t2
x
(1)
(2)
We look for ψ under the form
ψ(x) = ϕ(x)U(S(x)),
ϕ(x) = (1 − x 2 )−1/4 (1 − qx 2 )−1/4 . (3)
The equation satisfied by U on the interval [0, +1) is written as
U 00 + (χn + h1 ) U = 0,
d (1 − x 2 )ϕ0 (x) .
h1 (S(x)) := ϕ(x)−1 (1 − qx 2 )−1
dx
Abderrazek Karoui in collaboration with Aline Bonami
(4)
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Note that
h1 (S(x)) =
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
(1 − q)−1
+ h2 (S(x)),
8(1 − 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) ≤ Sq2 (x) ≤ 4(1 − x).
(5)
S (x)2
q
At 0 one has 1 ≤ Sq (0) ≤ π2 . Moreover 1−x
extends into a
holomorphic function in a neighborhood of 1 and takes the value
2(1 − q) at 1. Finally
0 ≤ Sq (x) −
p
2 (1 − x)3/2
2(1 − q)(1 − x) ≤
.
3 (1 − q)1/2
Abderrazek Karoui in collaboration with Aline Bonami
(6)
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
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
and such that U is a solution of the equation
1
00
U (s) + χn + 2 U(s) = F (s)U(s), s ∈ [0, S(0)].
4s
Abderrazek Karoui in collaboration with Aline Bonami
(7)
(8)
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
The associated homogeneous equation has the two independent
solutions
√
√
1/4 √
1/4 √
U1 (s) = χn
sJ0 ( χn s), U2 (s) = χn
sY0 ( χn s).
The Wronskian of U1 and U2 is given by
W (U1 , U2 )(s) = U1 (s)U20 (s) − U10 (s)U2 (s)
√
√
√
√
= sχn J0 ( χn s)Y00 ( χn s) − J00 ( χn s)Y0 ( χn s)
√
2 χn
= =
π
Then, the general solution of (8), is given by
π
U(s) = AU1 (s) + BU2 (s) + √
2 χn
Z s
√
√
√
√
√
×
stχn [J0 ( χn s)Y0 ( χn t) − J0 ( χn t)Y0 ( χn s)] F (t)U
0
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
Lemma
Z √χn Sq (0)
2
Let I = √
t (J0 (t))2 dt. Then there exists a constant
χn 0
C 0 independent of n and c, such that
0
2S
(0)
q
≤ √C .
I −
(10)
π χn
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
Theorem
(A.Bonami, A.K. (2010)): There exist constants C , C 0 with the
following properties. Assume that the parameters n, c are such
that q = c 2 /χn (c) < 1. Then one can find a constant
A := A(n, c) ≤ M such that, for 0 ≤ x ≤ 1,
p
p
χn (c)1/4 Sq (x)J0 ( χn (c)Sq (x))
+ Rn,c (x) (11)
ψn,c (x) = A
(1 − x 2 )1/4 (1 − qx 2 )1/4
with
sup |Rn,c (x)| ≤ Cq χn (c)−1/2 ,
x∈[0,1]
Abderrazek Karoui in collaboration with Aline Bonami
Cq =
C
.
(1 − q)13/4
(12)
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
Next we state as a lemma the fact that, for q close from 0, the
constant A is close from 1.
Lemma
Let α < 1 and 0 < K < 1. Let C 0 be as defined by Lemma 4.
There are constants H1 = H1 (α, K ) and H2 = H2 (α, K ) such that,
√
C0
for q ≤ α and n satisfying √ ≤ K 1 − α, the constant A(n, c)
χn
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 −
Abderrazek Karoui in collaboration with Aline Bonami
1
≤ H3 qn + H2 .
2
(14)
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
As a direct consequence of Theorem 1, we have the following
result.
Corollary
There is a constant C such that, for Cq = C (1 − q)−4 , the two
following inequalities hold.
sup|x|≤1 |ψn,c (x)|
sup|x|≤1 (1 −
x 2 )1/4 |ψn,c (x)|
1/4
≤ Cq χn,c
(15)
≤ Cq .
(16)
Remark that the first bound is sharp since
ψn,c (1) = A(n, c)χn (c)1/4 .
Abderrazek Karoui in collaboration with Aline Bonami
(17)
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
Proposition
There exists a constant C depending only on α < 1 such that, for
c2
< α, then
q=
χn
sup |ψ 0 (x)| ≤ C χn 5/4 ,
(18)
x∈[−1,1]
√
sup (1 − x 2 )|ψ 0 (x)| ≤ C χn .
(19)
x∈[−1,1]
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
Proposition
There exists a constant C such that, for all n ≥ 0 and c ≥ 0,
sup (1 − x 2 )1/4 |ψn,c (x)| ≤ (2χn (c))1/4 .
(20)
x∈[−1,1]
0
sup (1 − x 2 )|ψn,c
(x)| ≤ C (c 2 + χn (c))3/4 .
(21)
x∈[−1,1]
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
Proposition
For any integers n, k ≥ 0, satisfying k(k + 1) ≤ χn , we have
√ k
(k) ψ
(0)
n,c ≤ ( χn ) |ψn,c (0)| ,
(22)
for n even and k even, and
√ k−1 0
(k) ψn,c (0) ,
ψn,c (0) ≤ ( χn )
(23)
for n odd and k odd. In particular, under the assumption that
c2
q=
< 1, there exists a constant C , depending only on q and
χn
such that for any positive integer k satisfying k(k + 1) ≤ χn , we
have
√
(k) (24)
ψn,c (0) ≤ C ( χn )k .
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
We start from the well-known equality, [Slepian 1964, Rokhlin et
al. (2007)], that for any positive integer n, we have
λn (c) = λ0 × λ00 , with
c 2n+1 (n!)4
2((2n)!)2 (Γ(n + 3/2))2
Z c
(ψn,τ (1))2 − (n + 1/2)
00
λ : = exp 2
dτ .
τ
0
λ0 : =
(25)
(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
c2
,
sup ψn,c (x) − Pn (x) ≤ Mα p
n + 1/2
x∈[−1,1]
Abderrazek Karoui in collaboration with Aline Bonami
(27)
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
Proposition
Let α < 1. Then there exists constants M1 , M2 such that
n
n e 2n c 2
c2
λn (c) ≤ M1 ( )
exp(M2 2 ) .
2 4
n2
n
(28)
Theorem
(A.Bonami, A.K. (2010)): Let δ > 0. There exists N and κ such
that, for all c ≥ 0 and n ≥ max(N, κc),
λn (c) ≤ e −δ(n−κc) .
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
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
c ec 2n
we observe that the sequence λn (c) decays faster than
2 4n
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
2c
1
1−α
N(α) =
+ 2 log
log(c) + o(log(c)),
π
π
α
(Landau and Widom (1980)).
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
WKB method for the PSWFs
Uniform bounds of the PSWFs and their derivatives
Exponential decay of the eigenvalues associated with the PSWFs
Graph of the λn (c) for different values of c and n
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
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 ∈ H s (I ), s ≥ 0.
Then


s
!δN
2
c
|aN (f )| ≤ C N −2/3s kf kH s +
kf kL2 (I )  ,
χN (c)
where C , δ are independent of f , N and c.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
Wang’s Approach
e r (I ), associated with
By considering the weighted Sobolev space H
the differential operator
Dc u = −(1 − x 2 )u” + 2xu 0 + c 2 x 2 u,
and given by




X
r b 2
e r (I ) = f ∈ L2 (I ), kf k2 = kDcr /2 f k2 =
H
(χ
)
|
f
|
<
+∞
,
k
k
er
H


k≥0
the following result has been given in [L. Wang 2010].
Theorem
e r (I ), with r ≥ 0, we
(Theorem 3.3 in[Wang , 2010]). For any f ∈ H
have
kf − SN f k2 ≤ (χN (c))−r /2 kf kHe r ≤ N −r kf kHe r .
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
A.Bonami and A. K. approaches
Theorem
Let c ≥ 0 be a positive real number. Assume that f ∈ H s (I ), for
some positive real number s > 0. Then for any integer N ≥ 1, we
have
p
kf − SN f k2 ≤ K (1 + c 2 )−s/2 kf kH s + K λN (c)kf k2 .
(29)
Here, the constant K depends only on s. Moreover it can be taken
equal to 1 when f belongs to the space H0s (I ).
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
Remark
This should be compared with the results of [Wang, 2010], given
by Theorem 5. This has the advantage to give an error term for all
values of c, while the first term in (29) is only small for c large
enough. On another side, Wang compares this specific Sobolev
space with the classical one and finds that
kf kHfs ≤ C (1 + c 2 )s/2 kf kH s .
c
2
)
For large values of N we clearly have (1+c
(1 + c 2 )−1 , but it
χN
goes the other way around when χN and 1 + c 2 are comparable.
So it may be useful to have both kinds of estimates in mind for
numerical purpose and for the choice of the value of c.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
Theorem
(A. Bonami, A. K. (2010)): Let s > 0, c > 0, be any positive real
s ([−1, 1]). Then for any integer N ≥ 1,
numbers and let f ∈ Hper
s
π X
s .
kf −SN f k2 ≤ (1/2 + )
kψn,c k2∞ λn (c)kf k2 +c −s kf −f[c/π] kHper
4c
n≥N
(30)
Here, f[c/π]
is
the
truncated
Fourier
series
expansion
of
f
to
the
order πc . In particular, for any positive integer N satisfying
q = c 2 /χN < 1, we have
sX
√
s , (31)
χn λn (c)kf k2 + c −s kf − f[c/π] kHper
kf − SN f k2 ≤ Kq
n≥N
where Kq =
p
(1/2 +
π
4c )Cq
and Cq is as given by Corollary 1.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
We consider the Weirstrass function
f (x) =
X cos(2k πx)
k≥0
2ks
,
1 ([−1, 1]), with kf k2 =
f ∈ Hper
2
−1 ≤ x ≤ 1,
s = 1.4.
(32)
X 1
. If c = 100, then we have
22ks
k≥0
kf k2 ≈ 1.0805838, kf − f[c/π] kH 1 ≈ 1.203854. Next, we have
"
#1/2
50
1 X
2
found that EN =
(f (k/50) − SN f (k/50))
. Note
50
k=−50
that once N becomes larger than the critical value for the decay of
the λn (c), which is Nc = [ec/4] = 67, the theoretical error bound
becomes very close to the actual error.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Figure: (a) graph of W3/4 (x),
Abderrazek Karoui in collaboration with Aline Bonami
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
(b) graph of W3/4,N (x), N = 90.
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
We let s > 0 be any positive real number and we consider the
Brownian motion function Bs (x) given by as follows.
Bs (x) =
X Xk
k≥1
ks
cos(kπx),
−1 ≤ x ≤ 1.
(33)
Here, Xk is a Gaussian random variable. It is well known that
Bs ∈ H s ([−1, 1]). For the special case s = 1, we consider the
band-width c = 100, a truncation order N = 80 and compute B1,N
the approximation of B1 by its N−th terms truncated PSWFs
series expansion. The graphs of B1 and B1,N are given by the
following figure.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
Figure: (a) graph of B1 (x),
Abderrazek Karoui in collaboration with Aline Bonami
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
(b) graph of B1,N (x), N = 80.
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
Approximation of band-limited functions
Lemma
Let f ∈ Bc be an L2 normalized function. Then
Z +1
|f − SN f |2 dt ≤ λN (c).
(34)
−1
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
Approximation of almost band-limited functions
Let T and Ω de two measurable sets. A function pair (f , b
f ) is said
to be T −concentrated in T and Ω −concentrated in Ω if
Z
Z
|f (t)|2 dt ≤ 2T ,
|b
f (ω)|2 dω ≤ 2Ω .
Tc
Ωc
Next we define the time-limiting operator PT and the
band-limiting operator ΠΩ by:
Z
1
PT (f )(x) = χT (x)f (x),
ΠΩ (f )(x) =
e ixω b
f (ω) dω.
2π Ω
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
Approximation of almost band-limited functions
Proposition
If f is an L2 normalized function that is T −concentrated in
T = [−1, +1] and Ω −band concentrated in Ω = [−c, +c], then
for any positive integer N, we have
Z
+1
|f − SN f |2 dt
1/2
≤ Ω +
p
λN (c)
(35)
−1
and, as a consequence,
kf − PT SN f k2 ≤ T + Ω +
Abderrazek Karoui in collaboration with Aline Bonami
p
λN (c).
(36)
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
Exact reconstruction of band-limited functions with
missing data
In [Donoho-Stark 1989], the authors have shown the following
uncertainty principle:
If kf k2 = kb
f k2 = 1 and (f , b
f ) is T −concentrated on T and
Ω −concentrated on Ω, then
|Ω||T | ≥ (1 − (T + Ω ))2 .
Hence, if |Ω||T | < 1, then the following band-limited
reconstruction problem has a unique solution in BΩ .
Find S ∈ BΩ such that r (t) = χT c (t) (S(t) + η(t)) , η(·) ∈ L2 .
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
The solution S is given by
S(t) = Qr (t) =
X
(PT PΩ )n r (t),
t ∈ R.
n≥0
kS − Qr k ≤ C kηk,
C ≤ (1 −
p
|T ||Ω|)−1 .
If T = [−τ, τ ], Ω = [−c, c], then
Z τ
sin 2πc(x − y )
f (y ) dy ,
PΩ PT (f )(x) =
π(x − y )
−τ
x ∈ R.
Hence
kPT PΩ k ≤ λ0 (c) < 1.
Consequently, the band-limited reconstruction problem has a
band-limited solution no matter how large are T and Ω.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
PSWFs versus Wavelts
we consider a real life 1-D discrete signal S = (sn )1≤n≤3850
corresponding to an electrical load consumption, measured minute
by minute. We have considered the bandwidth c = 50. Then, we
have divided S into 11 blocks S i , i = 0, . . . , 10, of equal length
350. For a given block Si = (sj )1+350i≤j≤350(i+1) , we have used 35
uniformly sampled data points. Finally, the different data blocks
are synthetized by the use of the following rule,
e i (sj ) = e
S
sj =
34
X
αni Ψn,c (tj ), tj = −1+(j−1−350i)/175,
1+350i ≤ j ≤
n=0
(37)
We have found that
e 2
kS − Sk
=
r=
kSk2
"P
j
|sj − e
sj |2
P 2
j sj
Abderrazek Karoui in collaboration with Aline Bonami
#1/2
= 2.03E − 02.
(38)
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
Next, we have considered the 8 taps Daubechies wavelet filters and
applied the DWT to analyze and synthetize the signal S, at the
level 4. In this case, 241 wavelet coeficients have been used to
synthetize the signal. The graph of the reconstructed signal is
given by Figure 4(c). Also, we have computed r 0 , the relative
residual of this second method, and we found that
r 0 = 1.42E − 02. Based on the previous numerical results, one
concludes that the wavelet tool outperforms in term of accuracy,
the PSWFs based tool.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
Figure: (a) Original signal S, (b) reconstruction of S by the PSWFs, (c)
reconstruction of S by wavelets.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
A bandlimited example
We consider the signal given by
f (t) =
sin(50t)
, −1 ≤ t ≤ 1.
50t
A discrete signal S1 of length 2048, is obtained from f by
uniformly sampling this later. We have considered the bandwidth
c = 50 and computed the first 17 even indexed PSWFs analyzing
e1 denotes the PSWFs synthetized signal,
coefficients of S1 . If S
given by Figure 5(b), then we found that the relative residual is
given by r1 = 5.26E − 03. Moreover, we have applied the same
wavelet analysis/synthezis scheme of the previous example to S1 .
We found that 136 wavelet coefficients have been used to obtain
e 0 , given by Figure 5(c). The corresponding
the synthetized signal S
1
relative residual is given by r10 = 3.58E − 02.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
Figure: (a) Original signal S1 , (b) reconstruction of S1 by the PSWFs,
(c) reconstruction of S1 by wavelets.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
Figure: (a) Original noised signal, (b) reconstructed and denoised signal
by the PSWFs
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
References
[1] M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, Dover Publications, New York, 1970.
[2] Aline Bonami and Abderrazek Karoui, Uniform Estimates of the
Prolate Spheroidal Wave Functions and Approximations in Sobolev
Spaces, hal-00547220, Arxiv:1012.3881, (2010).
[3] John P. Boyd. Prolate spheroidal wavefunctions as an
alternative to Chebyshev and Legendre polynomials for spectral
element and pseudospectral algorithms. J. Comput. Phys., 199(2),
(2004), 688716.
[4] Q. Y. Chen, D. Gottlieb, and J. S. Hesthaven, Spectral
methods based on prolate spheroidal wave functions for hyperbolic
PDEs. SIAM J. Numer. Anal., 43(5), (2005), 19121933.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
[5] D. L. Donoho and P. B. Stark, Uncertainty principles and signal
recovery, SIAM Journal of Applied mathematics, 49, (1989),
pp.906-931.
[6] Li-Lian Wang, Analysis of Spectral Approximations using
Prolate Spheroidal Wave Functions, Mathematics of Computation
79 (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 analysis
and uncertainty–IV: Extensions to many dimensions; generalized
prolate spheroidal functions, Bell System Tech. J. 43 (1964),
3009–3057.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.
Prolate Spheroidal Wave Functions and their Properties
Computation of the PSWFs by Flammer’s method
Uniform estimates of the PSWFs and their derivatives
Applications of the PSWFs
PSWFs based spectral approximation in Sobolev spaces.
Signal Processing Applications of the PSWFs
[9] H. J. Landau and H. Widom, Eigenvalue distribution of time
and frequency limiting, J. Math. Anal.Appl., 77, (1980), 469–481.
[10] V. Rokhlin and H. Xiao, Approximate formulae for certain
prolate spheroidal wave functions valid for large values of both
order and band-limit, Appl. Comput. Harmon. Anal. 22, (2007),
105–123.
Abderrazek Karoui in collaboration with Aline Bonami
Prolate Spheroidal Wave functions and Applications.