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Zeitschrift für Analysis und ihre Anwendungen


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Volume 20, Issue 3, 2001, pp. 579–588
DOI: 10.4171/ZAA/1032

Published online: 2001-09-30

On Fourier Transforms of Wavelet Packets

R. Kumar[1], K. Ahmad[2] and L. Debnath[3]

(1) Jamia Millia Islama, New Delhi, India
(2) Jamia Millia Islama, New Delhi, India
(3) University of Central Florida, Orlando, USA

This paper deals with the Fourier transform $\hat{\omega}_n$ of wavelet packets $\omega_n \in L^2 (\mathbb R)$ relative to the scaling function $\varphi = \omega_0$. Included there are proofs of the following statements:
(i) $\hat{\omega}_n (0)$) = 0 for all $n \in \mathbb N$.
(ii) $\hat{\omega}_n (4nk\pi) = 0$ for all $k \in \mathbb Z, n = 2j$ for some $j \in \mathbb N_0$, provided $|\hat{\varphi}|, |m_0|$ are continuous.
(iii)$|\hat{\omega}_n (\xi)|^2 = \sum^{2^r–1}_{s=0} |\hat{\omega}_{2^r n+s} (2^r \xi)|^2$ for $r 2\in \mathbb N$.
(iv) $\sum^\infty_{j=1} \sum^{2^r–1}_{s=0} \sum_{k \in \mathbb Z} |\hat{\omega}_n (2^{j+r} (\xi + 2k\pi))|^2 = 1$ for a.a. $\xi \in \mathbb R$ where $r = 1, 2,...,j$.
Moreover, several theorems including a result on quadrature mirror filter are proved by using the Fourier transform of wavelet packets.

Keywords: Wavelet packets, multi-resolution analysis, Fourier transform, quadrature mirror filter

Kumar R., Ahmad K., Debnath L.: On Fourier Transforms of Wavelet Packets. Z. Anal. Anwend. 20 (2001), 579-588. doi: 10.4171/ZAA/1032