Wavelets and polylogarithms of negative integer order
T.E.Krenkel ∗ , E.T.Krenkel ∗ , K.O.Egiazarian ∗∗ , and J.T.Astola ∗∗
∗
Moscow Technical University for Communications and Informatics,
Department of Computer Science, Russia, E-mail: [email protected]
∗∗
Signal Processing Laboratory, Tampere University of Technology,
Tampere, Finland, E-mail: [email protected], [email protected]
Abstract
Wavelets are objects on the real line which are stable under the action of an affine
group (dilations and translations). In the Fourier domain they can be designed with
the polylogarithms
ls (x) =
∞
X
e2πinx /ns
for x ∈ R/Z and Re(s) > 1,
n=1
satisfying (for every positive integer m) the following relation (Kubert identity)[1]:
φ(mx) = m
s−1
m−1
X
φ(x + m−1 j)
(∗s ).
j=0
Using the periodic polylogarithms of the negative integer order s = k =
−1, −2, −3, . . . we demonstrate how orthonormalized wavelets of Battle-Lemarie
([2] and [3]) can be designed.
1
Introduction
Recently [4] we have demonstrated that using the polylogarithms [5] of the negative
integer order , we can design the subfamily of even Battle-Lemarie wavelets. This
example leads us to a general method of reduction of the wavelet design problems
to the problems of number theory.
2
Periodic zeta function
The function ls (x) is also known as a periodic polylogarithm function or periodic
zeta function [5]. Evidently, ls (x) satisfies the Kubert identities (∗s ) as well as the
identity
∂ls (x)/∂x = 2πils−1 .
(1)
1
If Re(s) > 1, we can write
X
X
ls (x) =
cos(2πnx)/ns + i
sin(2πnx)/ns ,
where the summands on the right side are the even and odd parts of ls (x). If s is
real, these can be identified with the real and imaginary parts of ls (x).
The polylogarithm may be defined for Re(s) > 0 and|z| < 1 as the function
(Jonquiere function):
Z ∞ s−1
z
t
Ls (z) =
dt, s > 0.
Γ(s) 0 et − z
Derivatives of the polylogarithm are specified as
z
d
Ls (z) = Ls−1 (z),
dz
(z
d m
) Ls (z) = Ls−m (z).
dz
From the first equation we have
L1 (z) = −ln(1 − z).
Hence, by the second equation,
L0 (z) =
z
,
1−z
L−1 (z) =
z
,
(1 − z)2
L−2 =
z(z + 1)
,
(1 − z)3
L−3 =
z(z 2 + 4z + 1)
....
(1 − z)4
In general, L−m (z) is easy to compute for integer m ≥ 1 :
L−m (z) =
m
X
1
am,k z k , for m > 0,
(1 − z)m+1
k=1
where the coefficients ( Eulerian numbers) can be obtained by the recurrence equation:
am,k = (m + 1 − k)am−1,k−1 + kam−1,k .
The numerator in the equation of the polylogarithm is equal to a corresponding
Euler-Frobenius polynomial.
It is evident that for negative integer values of the parameter s, the functions
Ls (z) are rational, with rational coefficients, holomorphic in z except for a pole at
z = 1. It is also obvious that
ls (x) = Ls (e2πix )
for x ∈ R/Z, x 6= 0, and for all complex s.
For the function l0 (x) = e2πix /(1 − e2πix ), a brief computation shows that
l0 (x) = (−1 + i cot(πx))/2.
Differentiating this expression, we obtain corresponding formulas for the periodic
polylogarithmic functions of negative integer order l−1 (x), l−2 (x), . . . . Note, in particular, that ls (x) is either an odd or an even function respectively, as s − 1 is odd
or even, for every negative integer s.
2
3 Zhamalov-Weiss polynomials and BattleLemarie wavelets
The first five of the Zhamalov polynomials are as follows [4]:
A1 (w) = w,
A2 (w) = 6w2 − 4w,
A3 (w) = 120w3 − 120w2 + 16w,
A4 (w) = 5040w4 − 6720w3 + 2016w2 − 64w,
A5 (w) = 362880w5 − 604800w4 + 282240w3 − 32640w2 + 256w.
where ω is frequency and is a substitute for x in the previous text and
w=
1
.
sin2 πω
It is evident from the above that
l1−2M (ω)/2 =
Γ(2M )
AM (w),
π 2M
M = 1, 2, 3, . . . .
(2)
Remark 1 . Zhamalov was the first to introduce in the early 70-ies even polynomials in the process of finding a fundamental solution of a polyharmonic equation on
R/Z by inverting the polyharmonic operator in the Fourier domain. He also derived
a reccurent formula for the calculation of even polynomials [6].
Remark 2 . It is also noteworthy that Weil in his book [7] also introduced even
and odd functions in the following form:
ζ(2M, ω) + ζ(2M, 1 − ω) =
1
{Ψ(2M − 1, ω) + Ψ(2M − 1, 1 − ω)}
(2M − 1)!
and
ζ(2M − 1, ω) − ζ(2M − 1, 1 − ω) =
1
{−Ψ(2M − 2, ω) + Ψ(2M − 2, 1 − ω)},
(2M )!
where Ψ(t, ω) is the t-th polygamma function, which is t-th derivative of the
digamma function Ψ(ω).
It is known that
X
n∈Z
1
1
=
.
2
2
(ω + 2nπ)
4sin (ω/2)
Differentiating twice both sides of this equality we get for every ω ∈ R
X
n∈Z
1
1
2
=
{1 − sin2 (ω/2)}
(ω + 2nπ)4
16sin4 (ω/2)
3
which is clearly the second even Zhamalov polynomial in variable
w=
1
.
sin (ω/2)
2
3
The technique of design of the even Battle-Lemarie wavelets using the radicals of
Zhamalov polynomials is described in [4].
If one wants to design the even and odd Battle-Lemarie wavelets in a unified
manner, the technique described by Hernandez and Weiss [8] [see Ch.4.2] should be
used.
Let us assume the validity for the case t ≥ 2 of the following equality:
X
n∈Z
1
Pt−1 (ω/2)
=
,
(ω + 2nπ)t
(2 sin(ω/2))t
where Pt−1 (ω) is a trigonometric polynomial. Differentiating both sides of this equality we obtain
X
1
Pt (ω/2)
=
,
t+1
(ω + 2nπ)
(2 sin(ω/2))t+1
n∈Z
where Pt (ω) is a Weiss trigonometric polynomial (i. e. , a polynomial in sin ω and
cos ω). Put P1 ≡ 1, then the recurrent formula for Weiss polynomials looks like:
0
Pt (ω) = (cos ω)Pt−1 (ω) − 1/t(sin ω)Pt−1
(ω).
The first five of the Weiss polynomials are as follows:
P1 (ω) = 1,
P2 (ω) = cos ω,
P3 (ω) = 1 − 32 sin2 ω = 23 + 13 cos(2ω),
P4 (ω) = 13 cos3 ω + 23 cos ω,
1
8
P5 (ω) = 30
cos2 (2ω) + 13
30 cos(2ω) + 15 .
In the Fourier domain we have the following equations for the design of the
scaling functions with t = 2M + 1:

M +1
sin(ω/2)
1

√

if M is odd,
ω/2
P2M +1 (ω/2)
M
ϕ̂ (ω) =
M +1
(3)

1
 e−iω/2 sin(ω/2)
√
if
M
is
even,
ω/2
P2M +1 (ω/2)
low-pass filters
ϕˆM (2ω)
mM
= eiαM (ω) (cos(ω/2))M +1
0 (ω) =
ϕˆM (ω)
where
αM (ω) =
s
P2M +1 (ω/2)
,
P2M +1 (ω)
(4)
0 if M is odd,
−ω/2 if M is even,
and wavelets
M
ψ̂ (2ω) =



2M +2
q
P2M +1 (ω/2+π/2)
P
2M
+1 (ω)P2M +1 (ω/2)
q
P2M +1 (ω/2+π/2)
(sin(ω/2))2M +2
iω
−ie
P2M +1 (ω)P2M +1 (ω/2)
(ω/2)M +1
eiω (sin(ω/2))
(ω/2)M +1
if M is odd,
(5)
if M is even.
As a result of the above, a technique for the design of the Battle-Lemarie wavelets
in the Fourier domain using polylogarithms of negative integer order was obtained.
4
References
[1] D. Kubert, The universal ordinary distribution, Bull. Soc. math. France 107
(1979), 179-202.
[2] G. Battle, A block spin construction of ondolettes. Part I: Lemarie functions,
Commun. Math. Phys. 110 (1987), 601-615.
[3] P.-G. Lemarie, Ondolettes a localisation exponentielles, J. Math. pures et appl.
67 (1988), 227-236.
[4] T. Krenkel, E. Krenkel, K. Egiazarian and J. Astola, Sobolev-Zhamalov
wavelets: Battle-Lemarie wavelets revisited I, (unpublished report).
[5] L. Lewin, Polylogarithms and Associated Functions, North Holland, 1981.
[6] S. L. Sobolev. The Theory of Cubature Formulas, Kluwer, 1997.
[7] A. Weil, Elliptic functions according to Eisenstein and Kronecker, Springer,
New York, 1976.
[8] E. Hernandez, G. Weiss, A first course on wavelets, CRC Press, Boca Raton,
1996.
5