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We provide constructive necessary and sufficient conditions for a family of periodic wavelets to be a Parseval wavelet frame. The criterion generalizes unitary and oblique extension principles. The case of one wavelet generator and refinable functions being trigonometric polynomials is discussed in details., Comment: 21 pages

In the paper we design a Parseval wavelet frame with a compact support. The corresponding refinement mask uniformly approximates an arbitrary continuous periodic function $f$, $f(0)=1$, $|f(x)|^2+|f(x+\pi)|^2\le 1$. The refinable function has stable integer shifts., Comment: 6 pages. arXiv admin note: substantial text overlap with arXiv:2207.08252

In the paper we design a Parseval wavelet frame with a compact support and many vanishing moments. The corresponding refinement mask approximates an arbitrary continuous periodic function $f$, $f(0)=1$. The refinable function has stable integer shifts., Comment: 8 pages

Matched wavelets interpolating equidistant data are designed. These wavelets form Riesz bases. Meyer wavelets that interpolate data on a particular uniform lattice are found., Comment: 7 pages

We study a localization of functions defined on Vilenkin groups. To measure the localization we introduce two uncertainty products $UP_{\lambda}$ and $UP_{G}$ that are similar to the Heisenberg uncertainty product. $UP_{\lambda}$ and $UP_{G}$ differ from each other by the metric used for the Vilenkin group $G$. We discuss analogs of a quantitative uncertainty principle. Representations for $UP_{\lambda}$ and $UP_{G}$ in terms of Walsh and Haar basis are given., Comment: 12 pages

We compare frameworks of nonstationary nonperiodic wavelets and periodic wavelets. We construct one system from another using periodization. There are infinitely many nonstationary systems corresponding to the same periodic wavelet. Under mild conditions on periodic scaling functions, among these nonstationary wavelet systems, we find a system such that its time-frequency localization is adjusted with an angular-frequency localization of an initial periodic wavelet system. Namely, we get the following equality $ \lim_{j\to \infty} UC_B(\psi^P_j) = \lim_{j\to \infty}UC_H(\psi^N_j), $ where $UC_B$ and $UC_H$ are the Breitenberger and the Heisenberg uncertainty constants, $\psi^P_j \in L_2(\mathbb{T})$ and $\psi^N_j\in L_2(\mathbb{R})$ are periodic and nonstationary wavelet functions respectively., Comment: 15 pages

Recently, it has been proven [R. Soc. Open Sci. 1 (2014) 140124] that the continuous wavelet transform with non-admissible kernels (approximate wavelets) allows for an existence of the exact inverse transform. Here we consider the computational possibility for the realization of this approach. We provide modified simpler explanation of the reconstruction formula, restricted on the practical case of real valued finite (or periodic/periodized) samples and the standard (restricted) Morlet wavelet as a practically important example of an approximate wavelet. The provided examples of applications includes the test function and the non-stationary electro-physical signals arising in the problem of neuroscience., Comment: 18 pages, 4 figures

An explicit description of all Walsh polynomials generating tight wavelet frames is given. An algorithm for finding the corresponding wavelet functions is suggested, and a general form for all wavelet frames generated by an appropriate Walsh polynomial is described. Approximation properties of tight wavelet frames are also studied. In contrast to the real setting, it appeared that a wavelet tight frame decomposition has an arbitrary large approximation order whenever all wavelet functions are compactly supported., Comment: 19 pages

An inequality refining the lower bound for a periodic (Breitenberger) uncertainty constant is proved for a wide class of functions. A connection of uncertainty constants for periodic and non-periodic functions is extended to this class. A particular minimization problem for a non-periodic (Heisenberg) uncertainty constant is studied., Comment: 14 pages

The application of the continuous wavelet transform to study of a wide class of physical processes with oscillatory dynamics is restricted by large central frequencies due to the admissibility condition. We propose an alternative reconstruction formula for the continuous wavelet transform, which is applicable even if the admissibility condition is violated. The case of the transform with the standard Morlet wavelet, which is an important example of such analyzing functions, is discussed., Comment: 6 pages

We introduce a notion of localization for dyadic functions, i.e. functions defined on the Cantor group. Localization is characterized by functional $UC_d$ similar to the Heisenberg uncertainty constant used for real-line functions. We are looking for dyadic analogs of quantitative uncertainty principles. To justify definition we use some test functions including dyadic scaling and wavelet functions., Comment: 14 pages

A family of Parseval periodic wavelet frames is constructed. The family has optimal time-frequency localization (in the sense of the Breitenberger uncertainty constant) with respect to a family parameter and it has the best currently known localization with respect to a multiresolution analysis parameter., Comment: 10 pages, 2 figures