p-Adic Pseudodifferential Operators and p-Adic Wavelets

We introduce a new wide class of p-adic pseudodifferential operators. We show that the basis of p-adic wavelets is the basis of eigenvectors for the introduced operators.     

Some applications of projection operators in wavelets

By rewriting the projection operatorP ₀ in wavelets in another formula, we obtain a characterization of dimJ _V0 (x) where V₀ is a -shift-invariant subspace ofL ²(R ⁿ ) derived from a dual wavelet basis and prove that there does not exist a wavelet function L ²(R) such that has compact support and _k(supp +4k) =R up to a zero subset ofR.     

p-Adic probability theory and its applications. The principle of statistical stabilization of frequencies

The development ofp-adic quantum mechanics has made it necessary to construct a probability theory in which the probabilities of events arep-adic numbers. The foundations of this theory are developed here. The frequency definition of probability is used. A general principle of statistical stabilization of relative frequencies is formulated. By virtue of this principle, statistical stabilization of relative frequencies, which are, like all experimental data, rational numbers, can be considered not only in the real topology but also inp-adic topologies.Institute of Electronic Technology, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 3, pp. 348–363, December, 1993.     

On cardinal wavelets with compact support and their duals

Some people try to construct an orthonormal wavelet such that the corresponding scaling function φ(t) has the cardinal property,i.e. ϕ(n)= σn0, since such wavelets have many good applications. Unfortunately it is impossible to do so, except for a trivial case^[1]. In this work, a family of non-orthogonal cardinal wavelets with compact support is constructed and their duals are investigated. This work is supported by the project of new stars of Beijing     

p-Adic nonconvex compactoids

A bounded subset X of a Banach space over a non-archimedean field ϰ is a compactoid if and only if each basic sequence in X tends to zero (Theorem 2). As a consequence the notions “weakly precompact’ and ‘precompact’ are identical for members of a wide class of ϰ-Banach spaces (Theorem 3).     

p-Adic multi-zeta values

Let X be the projective line minus 0,1, and ∞ over . The aim of the following is to give a series representations of the p-adic multi-zeta values in the depth two quotient. The approach is to use the lifting of the frobenius which is not a good choice near 1, but which gives simple formulas away from 1, and to relate the action of frobenius on the de Rham path from 0 to ∞ and on the one from 0 to 1. Also some relations between the p-adic multi-zeta values of depth two are obtained.     

p-Adic valued quantization

This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ? _p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ? _p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.     

Log-Syntomic Regulators and p-Adic Polylogarithms

The purpose of this paper is to show that under the syntomic regulator map from the motivic cohomology to the syntomic cohomology of the integer ring of a cyclotomic field, the image of Beilinson's element is a special value of the p-adic polylogarithm.     

p-Adic dynamics and angle doubling

We consider the squaring map over the p-adic numbers for an odd prime p, and study its symbolic dynamics on the unit circle in ? _p , the p-adic integers. When the map is restricted to the set of squares, we show an equivalence to angle doubling (mod 1) for rational angles. For primes p ≡ 3 (mod 4), this map may be represented as a unitary permutation matrix of the type used in quantum phase estimation.     

Lefschetz Formulae for p-Adic Groups

In this paper,Lefschetz formulae for torus actions on p-adic groups are proven. These are similar to comparable formulae for real Lie groups.Applications lie in the realm of dynamical zeta functions.     

One-Dimensional p-Adic Subanalytic Sets

In this paper we extend two theorems from [2] on p-adic subanalyticsets, where p is a fixed prime number, Q_p is the field of p-adicnumbers and Z_p is the ring of p-adic integers. One of thesetheorems [2, 3.32] says that each subanalytic subset of Z_p issemialgebraic. This is extended here as follows.     

p-Adic orthogonal wavelet bases

We describe all MRA-based p-adic compactly supported wavelet systems forming an orthogonal basis for L ²(ℚ _p ). The text was submitted by the authors in English.     

p-Adic Multiresolution Analysis and Wavelet Frames

We study p-adic multiresolution analyses (MRAs). A complete characterization of test functions generating an MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate the Haar MRA. We also suggest a method for constructing sets of wavelet functions and prove that any set of wavelet functions generates a p-adic wavelet frame.     

Pointwise convergence of some conjugate convolution operators with applications to wavelets

In this paper, we discuss the pointwise convergence of conjugate convolution operators with some applications to wavelets. Some criteria of convergence at (C, 1) continuous points, Lebesgue points and almost everywhere are established.