Two-wavelet theory in Weinstein setting

Abstract

In this paper we introduce the notion of a Weinstein two-wavelet. Then we establish and prove the resolution of the identity formula for the Weinstein continuous wavelet transform. Next, we give results on Calderón's type reproducing formula in the context of the Weinstein two-wavelet.

Time-frequency Analysis of two-wavelet theory in Weinstein setting

“…Theorem 2.11. (Parseval's formula) [35] Let (ϕ, ψ) be a Weinstein two-wavelet. Then for all ϕ and ψ in L 2 α (R d+1 + ), we have the following Parseval type formula…”

“…Corollary 2.12. [35] Let (ϕ, ψ) be a Weinstein two-wavelet. Then we have the following assertion: If the Weinstein two-wavelet constant…”

“…Using the harmonic analysis associated with the Weinstein operator (generalized translation operators, generalized convolution, Weinstein transform, ...) and the same idea as for the classical case, we study the localisations operators associated with the Weinstein two-wavlet [35] and we prove that under suitable conditions on the symbols and two Weinstein wavelets, the boundedness and compactness of these localization operators. Our main results for the boundedness and compactness of the Weinstein two wavelet localisation operators, with different symbols and windows, are summarized in the following table.…”