AB-wavelet frames in L2(Rn)

Abstract

In order to provide a unified treatment for the continuum and digital realm of multivariate data, Guo, Labate, Weiss and Wilson [Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 78-87] introduced the notion of AB-wavelets in the context of multiscale analysis. We continue and extend their work by studying the frame properties of AB-wavelet systems D A D B T k ψ (k ∈ Z n ; 1 L) in L 2 (R n). More precisely, we establish four theorems giving sufficient conditions under which the AB-wavelet system constitutes a frame for L 2 (R n). The proposed conditions are stated in terms of the Fourier transforms of the generating functions.

A family of Mexican hat wavelet transforms associated with an isometry in the heat equation

“…With a view to motivating the readers toward further researches on the topics of wavelets and wavelet transforms as well as on other related topics, we have chosen to include citations of a number of recent works (see, e.g., other studies [19][20][21][22][23][24] ) dealing with many different aspects of these subjects. In fact, usages of the wavelet theory in the mathematical modelling and analysis of various fractional-order and other applied problems can be found in several recent developments including previous studies [25][26][27][28] (see also Kilbas et al 29 and Srivastava 30 ).…”

The composition of linear canonical wavelet transforms on generalized function spaces

“…The earlier work on the linear canonical wavelet transform was published by authors [15,30,43,44]. The wavelet transform has major applications in the field of image and signal processing, mathematical analysis, communications, radar and other [7,13,19,21,37,39,40]. Let (W A ψ 1 ϕ)(β, α) and (W A ψ 2 ϕ)(ρ, γ) be two LCWT of a function ϕ ∈ L 2 (R) w.r.t.…”

Two-sided quaternion wave-packet transform and the quantitative uncertainty principles

“…The major development in the realm of time-frequency analysis came in the form of short-time Fourier transform (STFT) or Gabor transform (see [12]), which is reliant upon analysing functions determined by the fundamental operations of translation and modulation acting on a given window function. Although the Gabor representations are quite handy, however, such representations are not adequate for signals having high frequency components for shorter durations and low frequency components for longer durations, leading to the birth of time-scale integral transform, often known as the wavelet transform [11,26,33,36]. As of now, several generalizations of the classical wavelet transform have been reported in recent years including the fractional wavelet transform [32,34,35], linear canonical wavelet transform [28,29], quadratic-phase and special affine wavelet transform [30].…”