p-Adic wavelet transform and quantum physics

Enhanced Symmetry of the $p$-adic Wavelets

Wavelet analysis has been extended to the $p$-adic line $\mathbb{Q}_p$. The $p$-adic wavelets are complex valued functions with compact support. As in the case of real wavelets, the construction of the basis functions is recursive, employing scaling and translation. Consequently, wavelets form a representation of the affine group generated by scaling and translation. In addition, $p$-adic wavelets are eigenfunctions of a pseudo-differential operator, as a result of which they turn out to have a larger symmetry group. The enhanced symmetry of the $p$-adic wavelets is demonstrated.

Frames of p-adic wavelets and orbits of the affine group

The general construction of frames of p-adic wavelets is described. We consider the orbit of a mean zero generic locally constant function with compact support (mean zero test function) with respect to the action of the p-adic affine group and show that this orbit is a uniform tight frame. We discuss relation of this result to the multiresolution wavelet analysis.

Multidimensional basis of p-adic wavelets and representation theory

A multidimensional basis of p-adic wavelets is constructed. The relation of the constructed basis to a system of coherent states (i.e. orbit of action) for some $p$-adic group of linear transformations is discussed. We show that the set of products of the vectors from the constructed basis and p-roots of one is the orbit of the corresponding p-adic group of linear transformations.

Clifford-wavelet Transform and the uncertainty principle

In this paper we derive a Heisenberg-type uncertainty principle for the continuous Clifford wavelet transform. A brief review of Clifford algebra/analysis, wavelet transform on $\mathbb{R}$ and Clifford-Fourier transform and their proprieties has been conducted. Next, such concepts have been applied to develop an uncertainty principle based on Clifford wavelets.

Wavelet Transforms Associated With the Basic Bessel Operator

This paper aims to study the $q$-wavelet and the $q$-wavelet transforms, associated with the $q$-Bessel operator for a fixed $q\in ]0, 1[$. As an application, an inversion formulas of the $q$-Riemann-Liouville and $q$-Weyl transforms using $q$-wavelets are given. For this purpose, we shall attempt to extend the classical theory by giving their $q$-analogues.

Continuous wavelet transform with the Shannon wavelet from the point of view of hyperbolic partial differential equations

We identify the result of the continuous wavelet transform with the difference of solutions of two hyperbolic partial differential equations, for which wavelet's shift and scale are considered as independent variables on 2D plane. The characteristic property, which follows from the introduced representation is is the fact that the transform's values inside the triangle defined by two characteristics (a=const, b=const) and crossecting them slopped line on a scale-shift plane (a,b) are completely and uniquely defined by the value of transform and its derivatives along the last mentioned line.

A continuous spherical wavelet transform for~$\mathcal C(\mathcal S^n)$

In the present paper, a wavelet family over the $n$-dimensional sphere is constructed such that for each scale the wavelet is a polynomial and the inverse wavelet transform of a continuous function converges in the supremum norm.

The Quaternionic Affine Group and Related Continuous Wavelet Transforms on Complex and Quaternionic Hilbert Spaces

By analogy with the real and complex affine groups, whose unitary irreducible representations are used to define the one and two-dimensional continuous wavelet transforms, we study here the quaternionic affine group and construct its unitary irreducible representations. These representations are constructed both on a complex and a quaternionic Hilbert space. As in the real and complex cases, the representations for the quaternionic group also turn out to be square-integrable. Using these representations we constrct quaternionic wavelets and continuous wavelet transforms on both the complex and…

连续线性正则 Dunkl 小波变换：性质与应用

Continous linear canonical Dunkl wavelet transform: properties and applications

本文建立线性正则 Dunkl 小波变换，引入广义平移与卷积，研究其性质并推导不确定性不等式。

Discretization of quaternionic continuous wavelet transforms

A scheme to form a basis and a frame for a Hilbert space of quaternion valued square integrable function from a basis and a frame, respectively, of a Hilbert space of complex valued square integrable functions is introduced. Using the discretization techniques for 2D-continuous wavelet transform of the $SIM(2)$ group, the quaternionic continuous wavelet transform, living in a complex valued Hilbert space of square integrable functions, of the quaternion wavelet group is discretized, and thereby, a discrete frame for quaternion valued Hilbert space of square integrable functions is obtained.

Clifford-小波变换的对数不确定性原理

Logarithmic uncertainty principle for Clifford-wavelet transform

本文在Clifford小波变换框架下，将小波变体与对数变体相结合，提出并建立了一个新的对数不确定性原理，拓展了经典不确定性原理的研究。

Continuous wavelet transforms on $n$-dimensional spheres

In this paper, we are concerned with $n$-dimensional spherical wavelets derived from the theory of approximate identities. For nonzonal bilinear wavelets introduced by Ebert \emph{et al.} in 2009 we prove isometry and Euclidean limit property. Further, we develop a theory of linear wavelets. In the end, we discuss the relationship to other wavelet constructions.

A sharp Clifford-wavelet Heisenberg-type uncertainty principle

In the present work we are concerned with the development of a new uncertainty principle based on wavelet transform in the Clifford analysis/algebras framework. We precisely derive a sharp Heisenberg-type uncertainty principle for the continuous Clifford wavelet transform.

Wavelet transform on the torus: a group theoretical approach

We construct a Continuous Wavelet Transform (CWT) on the torus $\mathbb T^2$ following a group-theoretical approach based on the conformal group $SO(2,2)$. The Euclidean limit reproduces wavelets on the plane $\mathbb R^2$ with two dilations, which can be defined through the natural tensor product representation of usual wavelets on $\mathbb R$. Restricting ourselves to a single dilation imposes severe conditions for the mother wavelet that can be overcome by adding extra modular group $SL(2,\mathbb Z)$ transformations, thus leading to the concept of \emph{modular wavelets}. We define modular-…

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner functions

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. The wavelet transform of a signal is an $L^2$-function on an appropriately chosen group, while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps $L^2$-functions on a group unitarily to fields of Hilbert-Schmidt operat…

Wavelet transform and diffusion equations: applications to the processing of the "Cassini" spacecraft observations

We show that continuous transform with the complex Morlet wavelet is easily performed if we replace the integration of the fast-oscillation function by the solution of the diffusion differential equations. The most important advantage of this approach is that the initial data can be represented by non-uniform sample of an arbitrary node number. We apply the proposed method to the processing of the image of the Saturn A-ring obtained from the Cassini spacecraft. Also we have got via the wavelet transform using PDE, the local correlation coefficient of the signal and the harmonic function.