On Fourier Transforms of Wavelet Packets

Abstract

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 $\phi ={\omega }_0$. 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 $|\hat{\phi }|,|m_0|$ are continuous.
(iii)$|{\hat{\omega }}_n(\xi ){|}^2={\sum }_{s=0}^{2^r–1}|{\hat{\omega }}_{2^{r}n+s}(2^r\xi ){|}^2$ for $r2\in \mathbb{N}$.
(iv) ${\sum }_{j=1}^{\infty }{\sum }_{s=0}^{2^r–1}{\sum }_{k\in \mathbb{Z}}|{\hat{\omega }}_n(2^{j+r}(\xi +2k\pi )){|}^2=1$ 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.