Wavelet Transform of Periodic Generalized Functions

The aim of this paper is to define the wavelet transform for spaces of periodic functions, then extend this definition to spaces of generalized functions larger than the space of periodic Schwartz distributions, such as spaces of periodic Beurling ultradistributions and hyperfunctions on the unit circle. It is shown that the wavelet transforms of such generalized functions are nice and smooth functions defined on an infinite cylinder, provided that the analyzing wavelet is also nice and smooth. For example, it is shown that the growth rate of the derivatives of the wavelet transform is almost as good as that of the analyzing wavelet. More precisely, if the mother wavelet g satisfies Supx ∈ R|xkg(q)(x)| ≤ CAkBqkkβqqα (k, q = 0, 1, 2, …), then the wavelet transform Wg(ƒ) of a periodic Beurling ultradistribution ƒ satisfies sup(r,θ) ∈ Yϵ |rk ∂pθ ∂qrWg(ƒ)(r, θ)| ≤ DAkkαkBpCqppαqq(α + β); k, p, q ≥ 0, where Yϵ = {(r, θ): r ≥ ϵ > 0, θ ∈ T}.