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Application of Carleson’s theorem to wavelet inversion. (English) Zbl 0816.42018

Inversion formula for continuous wavelet transforms on the real line is proved to be valid pointwise almost everywhere for functions \(f\in L^ p\), \(1< p<\infty\). The related condition for mother wavelets is given in terms of their Fourier transforms.
Remark: Close results for \(L^ p\)-functions including convergence of the inversion formula in \(L^ p\)-norm and in a.e.-sense have been obtained by the reviewer and E. Shamir in Integral Equations and Oper. Theory (to appear) and by the reviewer [see “On Calderón’s reproducing formula”, Inst. of Math., Hebrew Univ. of Jerusalem, Preprint No. 15, 1993/94]. According to these papers the condition for wavelet functions can be given just in terms of these functions without using the Fourier transform.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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