### 321-polygon-avoiding permutations and Chebyshev polynomials.

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The construction of nonseparable and compactly supported orthonormal wavelet bases of L 2(R n); n ≥ 2, is still a challenging and an open research problem. In this paper, we provide a special method for the construction of such wavelet bases. The wavelets constructed by this method are dyadic wavelets. Also, we show that our proposed method can be adapted for an eventual construction of multidimensional orthogonal multiwavelet matrix masks, candidates for generating multidimensional multiwavelet...

Let ${{A}_{k}}_{k=0}^{+\infty}$ be a sequence of arbitrary complex numbers, let α,β > -1, let Pₙα,βn=0+∞$betheJacobipolynomialsanddefinethefunctions$$H\u2099(\alpha ,z)...$

Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by ${\left(\varphi \u2098\right)}_{m\in \mathbb{N}\u2080}$. The system ${\left(\varphi \u2098\right)}_{m\in \mathbb{N}\u2080}$ is defined in the space C(ℤₚ,ℂₚ) of ℂₚ-valued continuous functions on ℤₚ. J. Rutkowski has also considered some questions concerning expansions of functions from C(ℤₚ,ℂₚ) with respect to ${\left(\varphi \u2098\right)}_{m\in \mathbb{N}\u2080}$. This paper is a remark to Rutkowski’s paper. We define another system ${\left(h\u2099\right)}_{n\in \mathbb{N}\u2080}$ in C(ℤₚ,ℂₚ), investigate its properties and compare it to the system defined by Rutkowski. The system...

This is a survey of results in a particular direction of the theory of strong approximation by orthogonal series, related mostly with author's contributions to the subject.