Class Lp Research Articles

Seminormed approximation of functions by operators based on their Fourier series

This paper deals with the degree of approximation by an operator based on the partial sum of the Fourier series of a function of class Lp,p≥1. We use modulus of continuity of the function satisfying weaker conditions than those used by Deǧer and Bayindir [On the degree of approximation by the Woronoi-Nörlund and Riesz type means in the GHM, Azerb. J. Math., 8 (2018), 122-141]. The results of our paper also improve those of Singh and Sonker [Degree of approximation of function in Hpw in generalized Hölder metric by matrix means, Mathematical modelling and scientific computation, 1-10, Commun. Comput. Inf. Sci., 283, Springer, Heidelberg, 2012].

On the Dirichlet problem for beltrami equations with sources in simply connected domains

In this paper, we present our recent results on the solvability of the Dirichlet problem Reω(z) → φ (ζ) as z → ζ, z∈ D, ζ∈ ∂D, with continuous boundary data φ: ∂D ???? R for degenerate Beltrami equations ωz =µ(z)ω2 + σ(z), |µ(z) ˂ 1 a.e., with sources σ: D → C that belong to the class Lp (D), p ˃ 2, and have compact supports in D. In the case of locally uniform ellipticity of the equations, we formulate, in arbitrary simply connected domains D of the complex plane C a series of eff ective integral criteria of the type of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz on singularities of the equations at the boundary for existence of locally Hölder continuous solutions in the class W1.2loc (D) of the Dirichlet problem with their representation through the so-called generalized analytic functions with sources.

When does a hypergeometric function [formula omitted] belong to the Laguerre–Pólya class LP+?

I show that a hypergeometric function Fqp(a1,…,ap;b1,…,bq;⋅) with p≤q belongs to the Laguerre–Pólya class LP+ for arbitrarily large bp+1,…,bq>0 if and only if, after a possible reordering, the differences ai−bi are nonnegative integers. This result arises as an easy corollary of the case p=q proven two decades ago by Ki and Kim. I also give explicit examples for the case F21.

On boundary-value problems for semi-linear equations in the plane

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Hölder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, $$ where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that $$ {\partial}_{\overline{z}}f=g $$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $$ {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). $$ On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

Approximation by the Kλ means of Fourier series and conjugate series of functions in Hα,p

In this paper, we consider two operators based on the K? means of the Fourier series of and conjugate series of functions of class Lp, p > 1. We study the degree of convergence of these operators to the functions whose Fourier series and Allied series are used.

Quasiuniversal Fourier–Walsh Series for the Classes Lp[0, 1], p > 1

It is proved that, for each number p > 1, there exists a function L1[0, 1] whose Fourier–Walsh series is quasiuniversal with respect to subseries-signs in the class Lp[0, 1] in the sense of Lp-convergence.

КРИТЕРИЙ ПРИНАДЛЕЖНОСТИ КЛАССУ Wl p ОБОБЩЕННОГО ИЗ КЛАССА Lp РЕШЕНИЯ ВОЛНОВОГО УРАВНЕНИЯ

КРИТЕРИЙ ПРИНАДЛЕЖНОСТИ КЛАССУ Wl p ОБОБЩЕННОГО ИЗ КЛАССА Lp РЕШЕНИЯ ВОЛНОВОГО УРАВНЕНИЯ

Rate of approximation by $q$ -Durrmeyer operators in $L_{p}([0,1])$ , $1\leq p\leq\infty$

We obtain global rates of approximation by q-Durrmeyer operators Dn,q(f;x) for the functions in the class Lp([0,1]),1≤p≤∞. First, rates of approximation in terms of the norms of f and f' and in terms of the ordinary modulus of smoothness are obtained. Subsequently, we obtain rates of approximation in terms of the generalized modulus of smoothness ωφ(f,δ).

On the universal function for the class L[0,1], p∈ (0,1)

It is shown that for each number p∈(0,1) there exists a function g(x)∈L1[0,1] with monotonically decreasing Fourier–Walsh coefficients, which is universal for the class Lp[0,1] with respect to signs of those coefficients.

Necessary and sufficient condition for the generalized solution of a mixed problem for the wave equation to belong to the class L p for p ≥ 1

In terms of requirements imposed on the boundary function, we obtain a necessary and sufficient condition for the generalized solution of a mixed problem for the wave equation with zero initial conditions and with boundary conditions of the first kind to belong to the class Lp.

Maximal inequality and ergodic theorems for Markov groups

The paper shows that for actions of Markov semigroups, in particular, of finitely generated word hyperbolic groups, the Cesaro means of spherical averages converge almost everywhere for any function from the class Lp, p > 1.

Non-real zeroes of real entire derivatives

A real entire function belongs to the Laguerre-Polya class LP if it is the limit of a sequence of real polynomials with real zeroes. By building upon results that resolved a long-standing conjecture of Wiman, a number of conditions are established under which a real entire function f must belong to the class LP, or to one of the related classes U2p*. These conditions typically involve the non-real zeroes of f and its derivatives, or those of the differential polynomial f f″−a(f′)2.

Approximation of Functions in the Spaces L p (ℝ) by Means of Shifts and Contractions of One Function

Some approximation problems for functions of class Lp are considered. Bibliography: 4 titles.

Evolution of measures in the phase space of nonlinear Hamiltonian systems

We establish the existence of weak limits of solutions (in the class Lp, p ≥ 1) of the Liouville equation for nondegenerate quasihomogeneous Hamilton equations. We find the limit probability distributions in the configuration space. We give conditions for a uniform distribution of Gibbs ensembles for geodesic flows on compact manifolds.

Characterization of the Points of ϕ-Strong Summability of Fourier–Laplace Series for Functions of the Class Lp(Sm), p > 1

We consider the behavior of the ϕ-strong means of Fourier–Laplace series for functions that belong to Lp(Sm), p > 1, on a set of points of full measure on an m-dimensional sphere Sm.

Reducible Linear Quasi-periodic Systems with Positive Lyapunov Exponent and Varying Rotation Number

A linear system in two dimensions is studied. The coefficients are 2π-periodic in three angles, θj, j=1, 2, 3, and these angles are linear with respect to time, with incommensurable frequencies. The system has positive Lyapunov coefficients and the rotation number changes in a continuous way when some parameter moves. A lift to T3×R2, however, is only of class Lp, for any p<2.

Free interpolation in some spaces of analytic functions described in terms of fourier coefficients

Interpolation subsets of a disk are described for some classes of analytic functions whose Fourier coefficients belong to the weighted class lp. For nontangent sets with a single condensation point the necessary and sufficient conditions for interpolation are obtained. For such spaces, it is proved that the natural description of the trace space is different. An explicit construction of the extension operator from functions defined on the interpolation set to the corresponding space of analytic functions is given. The obtained construction can be viewed as a generalization of Newton interpolation polynomials. Bibliography: 10 titles.

A class of meromorphic univalent functions with positive and fixed finitely many coefficients

The object of the present paper is to derive several interesting properties of the class Lp, ck (a, Ş, y) consisting of regular and univalent meromorphic functions with positive and fixed finitely many coefficients. These incinde coefficient estimates, closure theorems, and radius of convexity for functions belonging to the class Sp, ck (a, P.Y).

Criterion for holomorphic continuability of the functions of class LP defined on a portion of the Shilov boundary of a circular strongly starlike domain

Criterion for holomorphic continuability of the functions of class LP defined on a portion of the Shilov boundary of a circular strongly starlike domain

A theorem for Fourier coefficients of a function of class Lp

This paper deals with the Fourier coefficients of a function of class Lp. We give a necessary and sufficient condition for function to be of class Lp for p greater than one.