We introduce a framework for calculating sparse approximations to signals based on elements of continuous wavelet systems. The method is based on an extension of the continuous wavelet theory. In the new theory, the signal space is embedded in larger abstract signal space, which we call the window-signal space. There is a canonical extension of the wavelet transform on the window-signal space, ...

In this paper, we study the Plancherel measure of a class of non-connected nilpotent groups which is of special interest in Gabor theory. Let G be a time-frequency group. That is G = ⟨ Tk,Ml : k ∈ Z, l ∈ BZ ⟩ , where Tk, Ml are translations and modulations operators acting in L(R), and B is a non-singular matrix. We compute the Plancherel measure of the left regular representation of G which is...

This paper uses some basic concepts and results on harmonic analysis associated with Heckman-Opdam theory \(\mathbb {R}^{d}\) to study types of wavelet packets the corresponding transforms. More precisely, we two transforms related hypergeometric Fourier transform W-invariant functions, prove Plancherel, Calderón, reconstruction formulae for these

This note introduces a quantum random walk in R and proves the weak convergence of its rescaled n-step densities. Quantum walks of the type we consider in this note were introduced in [1], which defined and analyzed the Hadamard quantum walk on Z, and a “new type of convergence theorem” for such quantum walks on Z was discovered by Konno [4, 5]. A much simpler proof of Konno’s theorem has recen...

In this paper, we are interested in the asymptotic size of the rows and columns of a random Young diagram under a natural deformation of the Plancherel measure coming from Hecke algebras. The first lines of such diagrams are typically of order n, so it does not fit in the context of the work of P. Biane and P. Śniady. Using the theory of polynomial functions on Young diagrams of Kerov and Olsha...