Wavelet analysis of the localized Weierstrass function Текст научной статьи по специальности «Математика»

Вестник Челябинского государственного университета. 2009. № 8 (146). Физика. Вып. 4. С. 25-27.

МАТЕМАТИЧЕСКАЯ ФИЗИКА

C. Cattani

wavelet analysis of the localized weierstrass function

This paper attempts to characterize a localized (in space) fractal function by using complex wavelets.

It will be shown that a good representation can be obtained by a very few instances of the wavelet family. Moreover, the reconstruction is independent on scale (according to the property of scale invariance of fractals). A suitable reconstruction of derivatives will be also given for the localized Weierstrass function.

Keywords: Harmonic wavelets, Weierstrass function, fractals.

1. Introduction. Wavelets are special functions ynk (x) [1; 2] which depend on two parameters: n and k. These functions fulfill the fundamental axioms of multiresolution analysis so that by a suitable choice of these parameters one is able to easily and quickly approximate any function (even tabular) with decay to infinity.

Harmonic wavelets are some bases with sharp localization in frequency domain [3]. It has been shown [3-7] that they are one of the most expedient tool for studying processes which are localized in Fourier domain. Also some kind of highly oscillating functions and random-like functions [5; 6] can be easily reconstructed by using harmonic wavelets. Their analytical form enable us to easily compute also the corresponding derivatives, up to any order, with some kind of hypergeometric series [4; 7]. They will be used, in the following, to study the regularity of a slightly modified definition of the Weierstrass function [1; 8], which is a fractal [9-12]. It will be shown that a finite sum of the Weierstrass function can be represented in wavelet space, with respect to the harmonic wavelet basis with only few coefficients. Thus showing that harmonic wavelets can be an expedient tool to study properties of fractals. Indeed the Weierstrass function is periodic and therefore a modification is needed in order to obtain a reconstruction, by harmonic wavelet series, which converges in L2 (M). Some attempts to formally approach the derivative of fractals (which are nowhere differentiable functions) were done also by using the fractional calculus [13; 14], thus modifying the concept of derivative. However, it seems to be more expedient to connect the scale invariance of fractals [6; 11; 12] with harmonic wavelets [1; 4] which have a localized high oscillation.

The derivative of the localized Weierstrass function will be obtained by using the connection coefficients and it will be shown the good representation.

In the following we will discuss in details a slightly modified version of the Weierstrass function which, for fixed limits, is localized in space (see fig. 1a)

¥

f (x) = ^ a—kH cos bk x (a > 1, b >1, 0 £ H £ 1) (1)

k=—¥

Although we have a linear combination of elementary functions (cos bk x) this function is nowhere differentiable.

2. Harmonic Wavelet Basis and Wavelet reconstruction. The dilated and translated instances of the harmonic wavelets are

UCI

j" (x)= 2"

<2" )

2 pi (2" x — k )

1 def /Ч2" х—к)_г— x——

1, y; w=e (2)

2 pi (2" x — — )

with n, k E Z. They are a basis for the L2 (M,C) functions, with inner product

def 00 00

'...............''ill! - , (3)

for any f (x)eL2(M,C), g(x)e L2(M,C). The bar stands for complex conjugation. Harmonic wavelets are orthonormal functions [3; 4; 7], being (ykn (x), ym (x) = dnmdhk, where dnm (Shk) is the Kronec-ker symbol. Analogous computations give the inner product of the remaining elements of the basis.

Let f (x)e L2 (M, C) be a complex function which decays in space variable; its reconstruction in terms of harmonic wavelets is (see e. g. [4-7])

f (x)=

¥ ¥

^ atj°t (x)+^ ^ b"ty" (x)

:=—¥ "=0 k=—¥

E a* j k М+Ё Ё b * k

k=—¥

"=0 k=—¥

(4)

which involves the basis and, for a complex function, its conjugate basis.

Due to the orthonormality of the basis the coefficients can be computed as

¥ ¥

ak = J f (x)j0 (^^ = J f (x) j0 (x)dx;

— ¥ —¥

¥ ¥

bk = J f (x)y; (x)dx, b‘n = / f (x)yn (x)dx.

(5)

The wavelet approximation is obtained by fixing an upper limits in the series expansion (4), so that with N < rc>, M < o> we have

f (x)

E avl (x)+E E by (x)

n=0 k=—M

N M

E jk (x )+E E b *"n y k (x)

(6)

For instance for, finite limits, the Weierstrass function:

f (x)= E 2—M k cos 2 kx (7)

k=—5

belongs to f (x)e L2 (M, C) and we have a good wavelet representation (see fig. 1b, with N = 0, M = 30). It can be noticed that, thanks to their localized oscillations, harmonic wavelets can fit very well with the high oscillation of the Weierstrass function (the approximation error will be considered in a following paper)

Fig. 1. A) The localized Weierstrass function (7); b) its harmonic wavelet representation

3. Derivatives of functions by the connection coefficients. In order to define the any order differential properties of harmonic wavelets, we need the preliminary results that for a given m E Z and £ E N, it is [4]

J (ix )£ eim x d x =i 1 (i— \m (m))

*£+1

£ +1

—im(m)eimxE(-i)f1+m(m)(2£—s+1 + cst„ £ > 1 (8)

(£—s+i)i | m

where ^ (x )=

—1, x < 0; 1, x > 0; 0, x = 0.

The any order connection coefficients of the harmonic wavelets, are defined as

A(£)kh ^ j0 (x), j0 (x)), L(£)mh (x), £ m;

g kh =

dx

Wm f dl

kh \ dx1' j

jd

5 kh ' \dxe

It can be shown [4-7] that the n-order derivatives of the wavelet basis can be easily computed in the Fourier domain. So that the only non trivial connection coefficients X(£\h, y are given by [47] (£ > 1):

\0 = —

kh 2p

i' (1—m(h—k ))2+-

—im(h—k )E(—1

h+m(h—k )](2£—5+1)

,/2 £!(2ip)'

(' — 5 +1)! |h — s|‘

g(£)nm = I kh ~

—(n+m) 2

2p

.£+1/7 [1+(«+m)/2](£+1) /_ £+1

(1 — \m(h—k))

f p'+121

(2'+1 —1)

' + 1

+1

— im(h — k )xE(—1

j1+M(h—k )](2 £—5+1) 2

£!(ip)

,£—5+1

5=1 (£ — 5 +1)!| h —

x 12[1+(”+”)2](£+1)—2s (2^ +1 2s )j d nm

The fundamental connection coefficients enable us to compute any order derivative of the wavelet basis:

d 1 jk (x) _ dxe d yn (x). dxe

M

E A(£)khj0 (x)

h=—M N M

E E g(')lгyг (x)

m=0 h=—M

An approximation is obtained by a finite value of the limits M < +a>, N < +rc>.

To show the importance of the connection coefficients let us compute the first derivative of the Weierstrass function (7) directly (see fig. 2a) and by deriving its wavelet reconstruction through the connection coefficients (see fig. 2b with N = 30, M = 0).

Fig. 2. A) Derivative of the Weiertrass function and b) its derivative based on connection coefficients (right)

—¥

—¥

5=1

5=1

References

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To appear on Telecommunication Systems. 2009.

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