Using the techniques of approximation and factorization of convolution operators we study the problem of irregular sampling of band-limited functions on a locally compact Abelian group $G$. The results of this paper relate to earlier work by Feichtinger and Gröchenig in a similar way as Kluvánek’s work published in 1969 relates to the classical Shannon Sampling Theorem. Generally speaking we claim that reconstruction is possible as long as there is sufficient high sampling density. Moreover, the iterative...

Ullrich, Grubb and Moore proved that a lacunary trigonometric series satisfying Hadamard's gap condition is recurrent a.e. We prove the existence of a recurrent trigonometric series with bounded gaps.

A method is given to find a recurrence relation for the coefficients of the series expansion of a function f with respect to classical orthogonal polynomials of a discrete variable, which follows from a linear difference equation satisfied by f.

We show that polynomials defined by recurrence relations with periodic coefficients may be represented with the help of Chebyshev polynomials of the second kind.

Let $P_k$ be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients $a_k$ in $f={\sum }_k{a}_k{P}_k$. A systematic use of the basic properties (including some nonstandard ones) of the polynomials $P_k$ results in obtaining a low order of the recurrence.

The aim of this article is to present “refined” Hardy-type inequalities. Those inequalities are generalisations of the usual Hardy inequalities, their additional feature being that they are invariant under oscillations: when applied to highly oscillatory functions, both sides of the refined inequality are of the same order of magnitude. The proof relies on paradifferential calculus and Besov spaces. It is also adapted to the case of the Heisenberg group.

It has been proved recently that the two-direction refinement equation of the form $f\left(x\right)={\sum }_{n\in }{c}_{n,1}f(kx-n)+{\sum }_{n\in \mathbb{Z}}{c}_{n,-1}f(-kx-n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f\left(x\right)={\sum }_{n\in \mathbb{Z}}cₙf(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f\left(x\right)={\int }_{ℝ}c\left(y\right)f(kx-y)dy$ has also various interesting applications....

The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by regular mappings (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-regular mappings. These mappings are the same as ordinary regular mappings when s = t, but otherwise they behave somewhat...

The concepts of statistical convergence of single and double sequences of complex numbers were introduced in [1] and [7], respectively. In this paper, we introduce the concept indicated in the title. A double sequence $x_{jk}:(j,k)\in \mathbb{N}²$ is said to be regularly statistically convergent if (i) the double sequence $x_{jk}$ is statistically convergent to some ξ ∈ ℂ, (ii) the single sequence $x_{jk}:k\in \mathbb{N}$ is statistically convergent to some ${\xi }_j\in ℂ$ for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence $x_{jk}:j\in \mathbb{N}$ is statistically convergent to some ${\eta }_k\in ℂ$ for...

Nous reprenons la construction des bases orthonormées d'ondelettes à partir des filtres miroirs en quadrature tel qu'elle apparaît dans [4]. Nous montrons que leur régularité est liée à une mesure invariante pour la transformation ω → 2ω mod-2π. Cette méthode permet d'obtenir le facteur exact qui relie asymptotiquement la régularité des ondelettes constriutes dans [4] à la taille de leur support.