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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, K. Ahmad and L. Debnath

(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