On Fourier Transforms of Wavelet Packets

This paper deals with the Fourier transform b ωn of wavelet packets ωn ∈ L(R) relative to the scaling function φ = ω0. Included there are proofs of the following statements: (i) b ωn(0) = 0 for all n ∈ N. (ii) b ωn(4nkπ) = 0 for all k ∈ Z, n = 2 for some j ∈ N0, provided |b φ|, |m0| are continuous. (iii) |b ωn(ξ)| = P2r−1 s=0 |b ω2rn+s(2ξ)| for r ∈ N. (iv) P∞ j=1 P2r−1 s=0 P k∈Z |b ωn(2(ξ + 2kπ))| = 1 for a.a. ξ ∈ 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.