On Fourier Transforms of Wavelet Packets

  • R. Kumar

    Jamia Millia Islama, New Delhi, India
  • K. Ahmad

    Jamia Millia Islama, New Delhi, India
  • L. Debnath

    University of Central Florida, Orlando, USA


This paper deals with the Fourier transform ω^n\hat{\omega}_n of wavelet packets ωnL2(R)\omega_n \in L^2 (\mathbb R) relative to the scaling function φ=ω0\varphi = \omega_0. Included there are proofs of the following statements:
(i) ω^n(0)\hat{\omega}_n (0)) = 0 for all nNn \in \mathbb N.
(ii) ω^n(4nkπ)=0\hat{\omega}_n (4nk\pi) = 0 for all kZ,n=2jk \in \mathbb Z, n = 2j for some jN0j \in \mathbb N_0, provided φ^,m0|\hat{\varphi}|, |m_0| are continuous.
(iii)ω^n(ξ)2=s=02r1ω^2rn+s(2rξ)2|\hat{\omega}_n (\xi)|^2 = \sum^{2^r–1}_{s=0} |\hat{\omega}_{2^r n+s} (2^r \xi)|^2 for r2Nr 2\in \mathbb N.
(iv) j=1s=02r1kZω^n(2j+r(ξ+2kπ))2=1\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. ξR\xi \in \mathbb R where r=1,2,...,jr = 1, 2,...,j.
Moreover, several theorems including a result on quadrature mirror filter are proved by using the Fourier transform of wavelet packets.

Cite this article

R. Kumar, K. Ahmad, L. Debnath, On Fourier Transforms of Wavelet Packets. Z. Anal. Anwend. 20 (2001), no. 3, pp. 579–588

DOI 10.4171/ZAA/1032