Uncertainty principles and optimally sparse wavelet transforms

In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First, we distinguish between the group generators, and the operators that measure localization (called observables). Second, we define the uncertainty of a signal transform as a whole, instead of defining the uncertainty of an individual window. We show that the uncertainty of a window function, in the signal space, is closely related to the localization of the reproducing kernel of the wavelet transform, in phase space. As a result, we show that using uncertainty minimizing window functions, results in representations which are optimally sparse in some sense.     

Homogeneous approximation property for continuous wavelet transforms

The homogeneous approximation property (HAP) for frames is useful in practice and has been developed recently. In this paper, we study the HAP for the continuous wavelet transform. We show that every pair of admissible wavelets possesses the HAP in L² sense, while it is not true in general whenever pointwise convergence is considered. We give necessary and sufficient conditions for the pointwise HAP to hold, which depends on both wavelets and functions to be reconstructed.     

On the boundedness result of wavelet transform associated with fractional Hankel transform

This article is concerned with the study of the continuity of wavelet transform involving fractional Hankel transform on certain function spaces. The n-dimensional boundedness property of the fractional wavelet transform is also discussed on Sobolev type space. Particular cases are also considered.     

On the Sobolev boundedness results of the product of pseudo-differential operators involving a couple of fractional Hankel transforms

This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed.     

Index transforms with the squares of Bessel functions

New index transforms, involving the squares of Bessel functions of the first kind as the kernel, are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution to the initial value problem for the third-order partial differential equation, involving the Laplacian, is obtained.     

A Bessel function multiplier

We obtain nearly sharp estimates for the norms of certain convolution operators.

     

Wavelets associated with Hankel transform and their Weyl transforms

The Hankel transform is an important transform. In this paper, we study thewavelets associated with the Hankel transform, then define the Weyl transform of thewavelets. We give criteria of its boundedness and compactness on the L~p-spaces.     

Fast algorithms for discrete polynomial transforms

Consider the Vandermonde-like matrix , where the polynomials satisfy a three-term recurrence relation. If are the Chebyshev polynomials , then coincides with . This paper presents a new fast algorithm for the computation of the matrix-vector product in arithmetical operations. The algorithm divides into a fast transform which replaces with and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes with almost the same precision as the Clenshaw algorithm, but is much faster for .

     

Integrals involving products of Airy functions, their derivatives and Bessel functions

A new integral representation of the Hankel transform type is deduced for the function Fn(x,Z)=Zn−1Ai(x−Z)Ai(x+Z) with x∈R, Z>0 and n∈N. This formula involves the product of Airy functions, their derivatives and Bessel functions. The presence of the latter allows one to perform various transformations with respect to Z and obtain new integral formulae of the type of the Mellin transform, K-transform, Laplace and Fourier transform. Some integrals containing Airy functions, their derivatives and Chebyshev polynomials of the first and second kind are computed explicitly. A new representation is given for the function ²|Ai(z)| with z∈C.     

离散超小波变换下双正交小波谱分析

我们研究发现，在离散超小波变换下双正交小波谱是有界的．并且任何一个双正交小波变换的谱不可能分布在1附近的某个区间内，并给出了该区间的一个估计．     

Weighted Laplace transforms and Bessel functions on Hermitian symmetric spaces

This paper defines -weighted Laplace transforms on the spaces of -covariant functions. By the inverse Laplace transform we define operator-valued Bessel functions. We also study the holomorphic discrete series of the automorphism group of a Siegel domain of type II.

     

Continuity of Pseudo-differential Operators Associated with the Bessel Operator in Some Gevrey Spaces

The pseudo-differential operator (p.d.o) h w , a associated with the Bessel operator d 2 / dx 2 + (1 m 4 w 2 )/4 x 2 involving the symbol a ( x , y ) whose derivatives satisfy certain growth conditions depending on some increasing sequences, is studied on certain Gevrey spaces (ultradifferentiable function spaces). It is shown that the operator h w , a is a continuous linear map of one Gevrey space into another Gevrey space. A special p.d.o. called the Gevrey-Hankel potential is defined and some of its properties are investigated. A variant H w , a of h w , a is also discussed.     

Inverse radon transform with one-dimensional wavelet transform

1. IntroductionSince Radon obtained the inverse formula of Radon transform in 1917, different inversemethods such as Fourier inversion, convolution back-projection inversion etc. have beeninvestigated['l']. Wavelet as a useful tool is interested in the inversion of Radon transformin recent years['--']. The application of wavelet analysis to Radon transform was proposedin I4] and [5]. An inversion formula based on continuous wavelet transform was derivedin [6] and [7]. This formula was based…     

Numerical stability of biorthogonal wavelet transforms

Biorthogonal wavelets are essential tools for numerous practical applications. It is very important that wavelet transforms work numerically stable in floating point arithmetic. This paper presents new results on the worst-case analysis of roundoff errors occurring in floating point computation of periodic biorthogonal wavelet transforms, i.e. multilevel wavelet decompositions and reconstructions. Both of these wavelet algorithms can be realized by matrix–vector products with sparse structured matrices. It is shown that under certain conditions the wavelet algorithms can be remarkably stable. Numerous tests demonstrate the performance of the results.      

Uncertainty principles for the continuous wavelet transform in the Hankel setting

Shapiro’s dispersion and Umbrella theorems are proved for the continuous Hankel wavelet transform. As a side results, we extend local uncertainty principles for set of finite measure to the latter transform.     

The quadratic-phase Fourier wavelet transform

In this paper, we define the quadratic-phase Fourier wavelet transform (QPFWT) and discuss its basic properties including convolution for QPFWT. Further, inversion formula and the Parseval relation of QPFWT are also discussed. Continuity of QPFWT on some function spaces are studied. Moreover, some applications of quadratic-phase Fourier transform (QPFT) to solve the boundary value problems of generalized partial differential equations.     

Basic definition and properties of Bessel multipliers

This paper introduces the concept of Bessel multipliers. These operators are defined by a fixed multiplication pattern, which is inserted between the analysis and synthesis operators. The proposed concept unifies the approach used for Gabor multipliers for arbitrary analysis/synthesis systems, which form Bessel sequences, like wavelet or irregular Gabor frames. The basic properties of this class of operators are investigated. In particular the implications of summability properties of the symbol for the membership of the corresponding operators in certain operator classes are specified. As a special case the multipliers for Riesz bases are examined and it is shown that multipliers in this case can be easily composed and inverted. Finally the continuous dependence of a Bessel multiplier on the parameters (i.e., the involved sequences and the symbol in use) is verified, using a special measure of similarity of sequences.     

Relating transplantation and multipliers for Dunkl and Hankel transforms

By expressing the Dunkl transform of order α of a function f in terms of the Hankel transforms of orders α and α + 1 of even and odd parts of f, respectively, we show that a considerable part of harmonic analysis of the Dunkl transform on the real line may be reduced to known results for the Hankel transform. In particular, defining the modified Dunkl transform and then considering the Dunkl transplantation operator we transfer known multiplier results for the Hankel transform to the Dunkl transform setting. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A classification of continuous wavelet transforms in dimension three

This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups $H<\mathrm{GL}(3,\mathbb{R})$ that give rise to a continuous wavelet transform with associated irreducible quasi-regular representation. For each group in this class, coorbit theory allows to consistently define spaces of sparse signals, and to construct atomic decompositions converging simultaneously in a whole range of these spaces. As an application of the classification, we investigate the existence of compactly supported admissible vectors and atoms for the groups.