Title

Wavelet Transform of Periodic Generalized Functions

Abstract

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(f(hook)) of a periodic Beurling ultradistribution f(hook) satisfies sup(r,θ) ∈ Yε(lunate) |rk ∂pθ ∂qrWg(f(hook))(r, θ)| ≤ DAkkαkBpCqpp αqq(α + β); k, p, q ≥ 0, where Yε(lunate) = ((r, θ): r ≥ ε(lunate) > 0, θ ∈ T). © 1994 Academic Press, Inc.

Publication Date

1-1-1994

Publication Title

Journal of Mathematical Analysis and Applications

Volume

183

Issue

2

Number of Pages

391-412

Document Type

Article

Identifier

scopus

Personal Identifier

scopus

DOI Link

https://doi.org/10.1006/jmaa.1994.1150

Socpus ID

50749134887 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/50749134887

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