Volume 12, Issue 4

The purpose of this paper is to study the class of polynomial sets which are at the same time 𝑑-orthogonal and 𝑞-Appell. By a linear change of variable, the resulting set reduces to 𝑞-Al-Salam–Carlitz polynomials, for 𝑑 = 1. Various properties of the obtained polynomials are singled out: a generating function, a recurrence relation of order 𝑑 + 1. We also explicitly express a 𝑑-dimensional functional for which the d-orthogonality holds.

We solve the functional equation where 𝐺 is a locally compact Abelian group, {𝑘 0 = 𝑖𝑑 𝐺 , 𝑘 1 , . . . , 𝑘 𝑁–1 } is a finite group of continuous automorphisms of 𝐺, 𝑓 is a bounded continuous complex-valued function on 𝐺 and 𝑔 is an almost periodic function on 𝐺.

The main aim of this paper is to prove that the logarithmic means of the double Walsh–Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of 𝐿log 𝐿(𝐼 2 ), the set of functions for which quadratic logarithmic means of the double Walsh–Fourier series converge in measure is of first Baire category.

A general representation of solutions by six harmonic functions is obtained for a system of homogeneous equations of statics of two-component mixtures. The problems are investigated when the normal components of partial displacement vectors and the tangent components of partial rotation vectors are given on the boundary. Uniqueness theorems are proved. Solutions are obtained in terms of absolutely and uniformly convergent series.

This paper deals with wavelet frames in anisotropic Besov spaces , 𝑠 ∈ ℝ, 0 < 𝑝, 𝑞 ≤ ∞, and 𝑎 = (𝑎 1 , . . . , 𝑎 𝑛 ) is an anisotropy, with 𝑎 𝑖 > 0, 𝑖 = 1, . . . , 𝑛, 𝑎 1 + . . . + 𝑎 𝑛 = 𝑛. We present sub-atomic and wavelet decompositions for a large class of distributions. To some extent our results can be regarded as anisotropic counterparts of those recently obtained in [Triebel, Studia Math. 154: 59–88, 2003].

We extend the concept of 𝑅-subweakly commuting maps due to Shahzad [J. Math. Anal. Appl. 257: 39–45, 2001] to the case of non-starshaped domains and obtain common fixed point results for this class of maps on non-starshaped domains in the setup of Fréchet spaces. As applications, we establish Brosowski–Meinardus type approximation theorems. Our results unify and extend the results of Al-Thagafi, Dotson, Habiniak, Jungck and Sessa, Sahab, Khan and Sessa and Shahzad.

The notion of compatibility of enumerations of countable (in particular, finite) sets is introduced and investigated. Some purely combinatorial problems connected with this notion are considered and several conditions are presented for the existence of mutually coherent enumerations of all members of a given family of countable sets.

For a scaled family of centered nonsingular Gaussian measures in 𝑘 𝑘 the 𝐿 1 -distance between any two densities is compared with its Kraft upper bound.

We prove the continuity properties of multilinear Marcinkiewicz operators for the extreme cases of 𝑝.

An approximation order of the optimal stopping problem for multidimensional diffusion processes by the corresponding semidiscretization is considered.

We study the problem posed by Nörlund in terms of dual sequences. We determine the functional equation fulfilled by the canonical form of any generalized Bernoulli sequence. Surprisingly these canonical forms are positive definite. Some results are given for an Euler sequence.

Let 𝐻 be a separable infinite dimensional complex Hilbert space, and let 𝔹(𝐻) denote the algebra of all bounded linear operators on 𝐻. Let 𝐴, 𝐵 be operators in 𝔹(𝐻). We define the generalized derivation δ 𝐴, 𝐵 : 𝔹(𝐻) ↦ 𝔹(𝐻) by δ 𝐴, 𝐵 (𝑋) = 𝐴𝑋 – 𝑋𝐵. In this paper we consider the question posed by Turnsek [Publ. Math. Debrecen 63: 293–304, 2003], when ? We prove that this holds in the case where 𝐴 and 𝐵 satisfy the Fuglede–Putnam theorem. Finally, we apply the obtained results to double operator integrals.

The purpose of this work is to study the nonexistence of weak solutions for 𝑝-degenerate subelliptic inequalities formed by the generalized Baouendi–Grushin vector fields. Also, we give the 𝑝-volume and 𝑝-area of the ball induced by the these vector fields.

Let 𝑆 stand for the usual class of univalent regular functions in the unit disk 𝑈 = {𝑧 : |𝑧| < 1} normalized by 𝑓(𝑧) = 𝑧 + 𝑎 2 𝑧 2 + . . . in 𝑈, and let 𝑆 𝑀 be its subclass defined by restricting |𝑓(𝑧)| < 𝑀 in 𝑈, 𝑀 ≥ 1. We consider two classical problems: Bombieri's coefficient problem for the class 𝑆 and the sharp estimate of the fourth coefficient of a function from 𝑆 𝑀 . Using Löwner's parametric representation and the optimal control method we propose a way to finding initial Bombieri's numbers and derive a sharp constant 𝑀 0 such that for all 𝑀 ≥ 𝑀 0 the Pick function gives the local maximum to |𝑎 4 |. Numerical approximation is given.

Let 𝐶 = 𝐶[0, 1] denote the Banach space of continuous real functions on [0, 1] with the sup norm and let 𝐶* denote the topological subspace of 𝐶 consisting of functions with values in [0, 1]. We investigate the preimages of residual sets in 𝐶 under the operation of superposition Φ : 𝐶 × 𝐶* → 𝐶, Φ(𝑓, 𝑔) = 𝑓 ○ 𝑔. Their behaviour can be different. We show examples when the preimages of residual sets are nonresidual of second category, or even nowhere dense, and examples when the preimages of nontrivial residual sets are residual.

We study the existence of oscillatory solutions of the even order superlinear ordinary differential equations where 𝑛 is an even integer ≥ 2, α > 1 and 𝑝(𝑡) ∈ 𝐶[𝑡 0 , ∞), 𝑡 0 > 0 and 𝑝(𝑡) > 0. When 𝑛 = 2, our results reduce to those of Jasny and Kurzweil, and Erbe and Muldowney for (). When 𝑛 = 4, our result becomes that of Kura for equation (). We present here new techniques suitable for the study of oscillation and nonoscillation of solutions of the general equations of even order () and ().