Lebesgue Function Research Articles

On the Dirichlet problem for the elliptic equations with boundary values in variable Morrey-Lorentz spaces

Under a metric measure space setting, we show that a solution to the elliptic equation with a non-negative potential, defined on the upper half-space, is in the essentially-bounded-Morrey-Lorentz space with variable exponent if and only if it can be represented as the Poisson integral of a variable Morrey-Lorentz function, where the doubling property, the pointwise upper bound on the heat kernel, the mean value property and the Liouville property are assumed. This extends the known result of Stein-Weiss in 1971 from the Euclidean space to the metric measure space, and from the Lebesgue function to the Morrey-Lorentz function with variable exponent.

ФУНКЦИОНАЛЬНЫЕ УРАВНЕНИЯ КАК МАТЕМАТИЧЕСКИЕ МОДЕЛИ ЗАДАЧ СОПРЯЖЕНИЯ С ЦИКЛИЧЕСКИМ СДВИГОМ НА СЛОЖНЫХ КРИВЫХ

The paper considers linear functional equations with a shift function having a nonzero derivative satisfying the Helder condition on an arbitrary piecewise smooth curve. Such equations are studied in connection with the theory of boundary value problems for analytical functions, which are a mathematical tool in the study of mathematical models of elasticity theory in which the conjugation conditions contain a boundary shift. The shift function acts cyclically on a set of simple curves forming a given curve, and except the ends of simple curves, there are no periodic points relative to the shift function. The purpose of the study is to find conditions for the existence and uniqueness of a solution (and in the case of non–uniqueness of the cardinality of the set of solutions) of such equations in the classes of Helder and primitive Lebesgue functions with a coefficient and the right part of the same classes.

On a two-dimensional analogue of the Lebesgue function for Fourier-Chebyshov sums

This article considers the problem of approximating a function of two variables f(x,y) by Fourier sums over Chebyshev polynomials orthogonal on a discrete grid. The paper shows that if Tα,βn(x, N) (α,β > -1, n = 0,1,2,...) are classical Chebyshev polynomials, orthogonal on a discrete grid, then the system of polynomials of two variables {Zα,βm,n(x, y)}km,n=0 = {Tα,βm(x, N)}km,n=0, where k = m + n ≤ N −1 is orthogonal on the set ΩN × N = {(xi, yj)}i,j=1N - 1. For an arbitrary function f(x,y) continuous on [−1,1]2, partial Fourier Chebyshev sums Sα,βm,n,N(f, x,y) are constructed over the system of polynomials τα,βm,n,N(x,y) orthonormal on the grid ΩN × N = {(xi, yj)}i,j=1N - 1. The task is to estimate the partial sum Sα,βm,n,N(f, x,y) of the Fourier series of the function f(x,y) from the system of polynomials τα,βm,n,N(x,y) from the function itself f ∈ C[-1, 1]2, in the case when(x,y) ∈ [−1,1]×[−1,1], which in turn reduces to the problem of estimating the Lebesgue function Wα,βm,n,N(x,y). The result of the work is a theorem in which it is proved that the order of the Lebesgue constants ‖Sα,βm,n,N‖ of the indicated discrete sums under certain conditions is O((mn)q+1/2), where q = max{α,β}. As a consequence of the obtained result, some approximation properties of discrete sums Sα,βm,n,N(f, x,y) are considered.

On the stability of the representation of finite rank operators

The stability of the representation of finite rank operators in terms of a basis is analyzed. A conditioning is introduced as a measure of the stability properties. This conditioning improves some other conditionings because it is closer to the Lebesgue function. Improved bounds for the conditioning of the Fourier sums with respect to an orthogonal basis are obtained, in particular, for Legendre, Chebyshev, and disk polynomials. The Lagrange and Newton formulae for the interpolating polynomial are also considered.

ОБРАТНАЯ ЗАДАЧА КОШИ ДЛЯ СТОЕЧНО-БАЛОЧНОЙ КОНСТРУКТИВНОЙ СИСТЕМЫ

The object of this study is the building frames with rigid beam-to-column assemblies. The aim of the study is to assess the impact on the accuracy of the solution of the problem of the error of the input data and the number of given coefficients of the deflection equation. The studies were carried out using analytical and experimental methods of regularization, reduction of measurements, solutions on a measuring compact, polynomial approximation, linear Lagrangian interpolation and numerical differentiation. Rigid coupling of a beam with a rack is modeled analytically and by a full-scale experiment. For a quantitative assessment of the effectiveness of solving the problem, the values of the target parameter and the optimization criterion are determined through the minimum of the Lebesgue function. It is proposed to use the obtained results of solving the inverse Cauchy problem in experimental and theoretical studies of rack-and-beam structures.

СВОЙСТВА И ОПИСАНИЕ МНОЖЕСТВ РЕШЕНИЙ ЛИНЕЙНЫХ ФУНКЦИОНАЛЬНЫХ УРАВНЕНИЙ НА ПРОСТОЙ ГЛАДКОЙ КРИВОЙ

The article investigates linear functional equations given in the field of complex numbers on simple smooth curves with a shift function of infinite order. The shift function has a nonzero derivative satisfying the Helder condition, and fixed points only at the ends of the curve. The paper gives a complete description of the sets of solutions of such equations in the classes of continuous, Helder, and primitive Lebesgue functions with a coefficient and the right side of the same classes, depending on the values of the coefficient of the equation at the ends of the curve. Sufficient conditions have been established for the solutions to belong to the specified functional classes.

The essential spectrum of periodically stationary pulses in lumped models of short‐pulse fiber lasers

AbstractIn modern short‐pulse fiber lasers, there is significant pulse breathing over each round trip of the laser loop. Consequently, averaged models cannot be used for quantitative modeling and design. Instead, lumped models, which are obtained by concatenating models for the various components of the laser, are required. As the pulses in lumped models are periodic rather than stationary, their linear stability is evaluated with the aid of the monodromy operator obtained by linearizing the round‐trip operator about the periodic pulse. Conditions are given on the smoothness and decay of the periodic pulse that ensure that the monodromy operator exists on an appropriate Lebesgue function space. A formula for the essential spectrum of the monodromy operator is given, which can be used to quantify the growth rate of continuous wave perturbations. This formula is established by showing that the essential spectrum of the monodromy operator equals that of an associated asymptotic operator. Since the asymptotic monodromy operator acts as a multiplication operator in the Fourier domain, it is possible to derive a formula for its spectrum. Although the main results are stated for a particular experimental stretched pulse laser, the analysis shows that they can be readily adapted to a wide range of lumped laser models.

Stable interpolation with exponential-polynomial splines and node selection via greedy algorithms

In this work we extend some ideas about greedy algorithms, which are well-established tools for, e.g., kernel bases, and exponential-polynomial splines whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we develop some results on theoretically optimal interpolation points. Moreover, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals (f-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function (lambda-greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e., the spline nodes. While the f-greedy selection is tailored to one specific target function, the lambda-greedy algorithm enables us to define target-data-independent interpolation nodes.

On subsequences of Lebesgue functions of general uniformly bounded ONS

On subsequences of Lebesgue functions of general uniformly bounded ONS

Series expansions among weighted Lebesgue function spaces and applications to positive definite functions on compact two-point homogeneous spaces

Geodesically isotropic positive definite functions on compact two-point homogeneous spaces of dimension d have series representation as members of weighted Lebesgue spaces L1w([−1,1]), where the weight w(x)=wα,β(x)=(1−x)α(1+x)β is the one related to the Jacobi orthogonal polynomials P(α,β)(x) in [−1,1], and the exponents α and β are related to the dimension d. We derive some recurrence relations among the coefficients of the series representations under different exponents, and we apply them to prove inheritance of positive definiteness between dimensions. Additionally, we give bounds on the curvature at the origin of such positive definite functions with compact support, extending the existing solutions from d-dimensional spheres to compact two-point homogeneous spaces.

Chebyshev Alternance when Approximating Initial Conditions of the Inverse Cauchy Problem

Purpose of research. The work is devoted to a range of questions related to Cauchy problem on the segment of real axis with the application of the inverse Cauchy problem, in which real constants are initial conditions which are optimally restored according to experimental or tabular values of the solution of the differential equation. The object of the study is an information-measuring system, in which approximate values of initial conditions are calculated from discrete function values of Cauchy problem solving.Methods. The following problems are solved for this purpose: parameters of measuring section placement on the investigated object and approximation grid on measuring section are developed. Characteristics of recovery accuracy of initial conditions of the task are formulated.Results. An experimental-calculated method of determining initial conditions in the inverse Cauchy problem is proposed. It is based on the concept of objective function of regularization of the problem. Task regularization parameter in the form of minimum value by Lebesgue function is proposed.Conclusion. The reaction of uniformly approximating method of the initial conditions of the inverse Cauchy problem to the deviation of the approximation grid coordinates nodes from the coordinates of Chebyshev alternance was described. Graphs of method reaction to deviation of grid pitch from optimal pitch are given.

ВОССТАНОВЛЕНИЕ НАЧАЛЬНЫХ ПАРАМЕТРОВ БАЛКИ ПРИ ЗАДАННЫХ МЛАДШИХ КОЭФФИЦИЕНТАХ УРАВНЕНИЯ ПРОГИБОВ

Mathematical modeling and experimental study of the inverse Cauchy problem of restoring the initial parameters of the elastic line of a beam element of a building structure for given minor coefficients of the beam deflection equation are considered. With a uniform continuous absolute norm of error for measuring deflections by interpolating with a Lagrange polynomial, the distribution of deflection meters over the beam is obtained, which minimizes the error in restoring the initial parameters by the criterion of minimum of the Lebesgue function.

Backward and forward stability analysis of Neville’s algorithm for interpolation and a pyramid algorithm for the computation of Lebesgue functions

Backward and forward stability analysis of Neville’s algorithm for interpolation and a pyramid algorithm for the computation of Lebesgue functions

Lebesgue Function for Higher Order Hermite-Fej´er Interpolation Polynomials with Exponential-Type Weights

Lebesgue Function for Higher Order Hermite-Fej´er Interpolation Polynomials with Exponential-Type Weights

A fixed-point characterization of weakly compact sets in L1(μ) spaces

Let (Ω,Σ,μ) be a σ-finite measure space and consider the Lebesgue function space L1(μ) endowed with its standard norm. We obtain a characterization of weak compactness for closed bounded convex subsets of L1(μ) in terms of the existence of fixed points for certain classes of eventually affine, uniformly Lipschitzian mappings.

Approximation by the Third-Order Splines on Uniform and Non-uniform Grids and Image Processing

This work is one of a series of papers that is devoted to the further investigation of polynomial splines and trigonometric splines of the third order approximation. Polynomial basis splines are better known and therefore more commonly used. However, the use of trigonometric basis splines often provides a smaller approximation error. In some cases, the use of the trigonometric approximations is preferable to the polynomial approximations. Here we continue to compare these two types of approximation. The Lebesgue functions and constants are discussed for the polynomial splines and the trigonometric splines. The examples of the applications of the splines to image enlargement are given.

Linear operators with infinite entropy

We examine the chaotic behavior of certain continuous linear operators on infinite-dimensional Banach spaces, and provide several equivalent characterizations of when these operators have infinite topological entropy. For example, it is shown that infinite topological entropy is equivalent to non-zero topological entropy for translation operators on weighted Lebesgue function spaces. In particular, finite non-zero entropy is impossible for this class of operators, which answers a question raised by Yin and Wei.

On the structure of variable exponent spaces

The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) Lp(⋅)(Ω). In the second part strictly singular and disjointly strictly singular operators between spaces Lp(⋅)(Ω) are studied. New results on the disjoint strict singularity of the inclusions Lp(⋅)(Ω)↪Lq(⋅)(Ω) are given.

Приближение функций частными суммами ряда Фурье по многочленам, ортогональным на произвольных сетках

for arbitrary continuous function f(t) on the segment [−1,1] we construct discrete Fourier sums Sn,N(f,t) on a system of polynomials forming an orthonormal system on non-uniform grids $${T_N} = \left\{{{t_j}} \right\}\matrix{{N - 1} \cr {j = 0} \cr}$$ of N points from segment [−1,1] with weight Δtj=tj+1−tj. Approximation properties of the constructed partial sums Sn,N(f,t) of order n ≤ N−1 are investigated. A two-sided pointwise estimate is obtained for the Lebesgue function Ln,N(t) of considered discrete Fourier sums for $$n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right),{\delta _N} = {\max _0} \le j \le N - 1\Delta {t_j}$$. The question of the convergence of Sn,N(f,t) to f(t) is also investigated. In particular, we obtain the deflection estimation of a partial sum Sn,N(f,t) from f(t) for $$n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right)$$, which depends on n and the position of a point t ∊ [−1,1].

Pointwise and Uniform Convergence of Fourier Extensions

Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the L^2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the L^2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.