Related papers: Rational Orthonormal Littlewood-Paley Wavelet Basi…

The orthonormal basis generated by a wavelet of $L^2(\mathbb R)$ has poor frequency localization. To overcome this disadvantage Coifman, Meyer, and Wickerhauser constructed wavelet packets. We extend this concept to the higher dimensions…

We show the existence of smooth band-limited multiresolution analysis (MRA) for any expansive dilation with real entries in any spatial dimension. We then prove the existence of orthonormal Meyer wavelets, which have smooth and compactly…

In this paper we show how to construct a certain class of orthonormal bases in $L^2({\bf R}^d)$ starting from one or more Gabor orthonormal bases in $L^2({\bf R})$. Each such basis can be obtained acting on a single function…

We present a construction of a wavelet-type orthonormal basis for the space of radial $L^2$-functions in $\R^3$ via the concept of a radial multiresolution analysis. The elements of the basis are obtained from a single radial wavelet by…

Let a sequence $(P_n)$ of polynomials in one complex variable satisfy a recurre ce relation with length growing slowlier than linearly. It is shown that $(P_n) $ is an orthonormal basis in $L^2_{\mu}$ for some measure $\mu$ on $\C$, if and…

In this work, we prove that certain L^2-unbounded transformations of orthogonal wavelet bases generate vaguelets. The L^2-unbounded functions involved in the transformations are assumed to be quasi-homogeneous at high frequencies. We…

Adapting the recently developed randomized dyadic structures, we introduce the notion of spline function in geometrically doubling quasi-metric spaces. Such functions have interpolation and reproducing properties as the linear splines in…

Let $G$ be a connected, simply connected nilpotent group and $\pi$ be a square-integrable irreducible unitary representation modulo its center $Z(G)$ on $L^2(\mathbf{R}^d)$. We prove that under reasonably weak conditions on $G$ and $\pi$…

A variety of different orthogonal wavelet bases has been found in L_2(R) for the last three decades. It appeared that similar constructions also exist for functions defined on some other algebraic structures, such as the Cantor and Vilenkin…

The article proves an assertion analogous to the Littlewood-Paley theorem for the orthoprojectors onto wavelet subspaces corresponding to the multidimensional multiresolution analysis generated as tensor product of smooth finite scaling…

A version of Littlewood-Paley-Rubio de Francia inequality for the two-parameter Walsh system is proved: for any family of disjoint rectangles $I_k = I_k^1 \times I_k^2$ in ${\mathbb{Z}_+ \times \mathbb{Z}_+}$ and a family of functions $f_k$…

Let $\mu$ be a Borel measure on $R^d$ which may be non doubling. The only condition that $\mu$ must satisfy is $\mu(B(x,r))\leq C r^n$ for all $x\in R^d$, $r>0$, and for some fixed $0<n\leq d$. In this paper, we develop Littlewood-Paley…

We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar group $$BS(1,2)=< u,t | utu^{-1}=t^2>.$$ We give a precise description of this representation in some…

We give a characterization of a class of band-limited wavelets of $L^2({\mathbb R})$ and show that none of these wavelets come from a multiresolution analysis (MRA). For each $n\geq 2$, we construct a subset $S_n$ of ${\mathbb R}$ which is…

A \emph{rational orthogonal matrix} $Q$ is an orthogonal matrix with rational entries, and $Q$ is called \emph{regular} if each of its row sum equals one, i.e., $Qe = e$ where $e$ is the all-one vector. This paper presents a method for…

A generalization of Mallat's classic theory of multiresolution analysis based on the theory of spectral pairs was considered by Gabardo and Nashed (J. Funct. Anal. 158, 209-241, 1998). In this article, we introduce the notion of…

The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property [6]. They are an orthogonal basis of both $L^2(-1,1)$ and the Paley-Wiener space of bandlimited…

The definition for the Slater-type orbitals is generalized. Transformation between an orthonormal basis function and the Slater-type orbital with non-integer principal quantum numbers is investigated. Analytical expressions for the linear…

In this paper, we obtain an analogue of the discrete Calderon condition and prove that this condition is sufficient for an orthonormal twisted wavelet system to be complete in L^{2}(R^{2}).

We construct a uniformly bounded orthonormal almost greedy basis for $L_p([0,1])$, $1<p<\infty$. The example shows that it is not possible to extend Orlicz's theorem, stating that there are no uniformly bounded orthonormal unconditional…