Class Of Continuous Functions Research Articles

Riemann problem for multiply connected domain in Besov spaces

In this paper, we obtain conditions of the solvability of the Riemann boundary value problem for sectionally analytic functions in multiply connected domains in Besov spaces embedded into the class of continuous functions. We indicate a new class of Cauchy-type integrals, which are continuous on a closed domain with continuous (not Hölder) density in terms of Besov spaces, and for which the Sokhotski–Plemelj formulas are valid.

On a composition of Ismail–May operator

In this paper, we address a discrete operator based on the composition of Ismail–May operator and Szász–Mirakjan operator and study its approximation properties for various classes of continuous functions using Taylor's formula. We estimate its moments and further give direct theorems employing Peetre's K‐functional and the modulus of continuity. Furthermore, Voronovskaja‐type asymptotic theorems are given. Additionally, we provide further compositions and give a comparison among their approximation properties.

Existence Theorems for Hybrid Fractional Differential Equations with ψ -Weighted Caputo–Fabrizio Derivatives

In this study, two classes of hybrid boundary value problems involving ψ -weighted Caputo–Fabrizio fractional derivatives are considered. Based on the properties of the given operator, we construct the hybrid fractional integral equations corresponding to the hybrid fractional differential equations. Then, we establish and extend the existence theory for given problems in the class of continuous functions by Dhage’s fixed point theory. Furthermore, as special cases, we offer further analogous and comparable conclusions. Finally, we give two examples as applications to illustrate and validate the results.

Analysis of a Fractional-Order Quadratic Functional Integro-Differential Equation with Nonlocal Fractional-Order Integro-Differential Condition

Here, we center on the solvability of a fractional-order quadratic functional integro-differential equation with a nonlocal fractional-order integro-differential condition in the class of continuous functions. The maximal and minimal solutions will be discussed. The continuous dependence of the solutions on a few parameters will be examined. Finally, the problems of conjugate orders and integer orders, and some other problems and remarks will be discussed and presented.

Approximation of classes of Poisson integrals by Fejer means

The paper is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series.
 The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, many studies devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer sums have been widely studied recently. One of the important problems in this area is the study of asymptotic behavior of the sharp upper bounds over a given class of functions of deviations of the trigonometric polynomials.
 In the paper, we study upper asymptotic estimates for deviations between a function and the Fejer means for the Fourier series of the function. The asymptotic behavior is considered for the functions represented by the Poisson integrals of periodic functions of a real variable. The mentioned classes consist of analytic functions of a real variable. These functions can be regularly extended into the corresponding strip of the complex plane.An asymptotic equality for the upper bounds of Fejer means deviations on classes of Poisson integrals was obtained.

Approximation of classes of Poisson integrals by Fejer means

The work is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, a many studies is devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer means have been widely studied in the last time. One of the important problems in this field is the study of asymptotic behavior of the upper bounds over a fixed classes of functions of deviations of the trigonometric polynomials. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals of continuous functions by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. The asymptotic behavior of the upper bounds on classes of Poisson integrals of periodic functions of the real variable of deviations of linear means of Fourier series, which are defined by applying the Fejer summation method is studied. The mentioned classes consist of analytic functions of a real variable, which are narrowing of bounded harmonic in unit disc functions of complex variable. In the work, asymptotic equalities for the upper bounds of deviations of Fejer means on classes of Poisson integrals were obtained.

A global in time existence and uniqueness result of a multidimensional inverse problem

ABSTRACT This study is the multidimensional inverse problem for a viscoelasticity equation. This inverse problem aims to reconstruct the displacement and convolutional kernel of the integral term simultaneously, from the measurement described as the overdetermination condition. The main result is the theorem of global unique solvability of the inverse problem in the class of functions continuously in the time variable t and analytic in the space variable. The methods of scales of Banach spaces of real analytic functions of variable and weight norms in the class of continuous functions are applied.

A Generalization of Pisier Homogeneous Banach Algebra

In 1979, Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra P strictly contained in C(T), the class of continuous functions on the unit circle T and strictly containing the classical Wiener algebra A(T), that is, A(T)⫋P⫋C(T). This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper, we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in C(T). Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series ∑ n∈Zfˆ(n)ξnexp(2πint) in the general setting of dependent random variables (ξn).

Generalized \(\mathcal{I}_\mathcal{p}\)-Closed Sets in Ideal Topological Spaces

The main focus of this study is to introduce a new category of generalized closed sets, referred to as \(\mathcal{I}_\mathcal{p}\)- closed sets, within the framework of ideal topological spaces. By using a few instances, we demonstrate \(\mathcal{I}_\mathcal{p}\)- closed sets and establish some fundamental properties of \(\mathcal{I}_\mathcal{p}\)-closed sets. We also investigate the relationship between \(\mathcal{I}_\mathcal{p}\)-closed sets and other classes of generalized closed sets in ideal topological spaces, such as (\mathcal{I}_\mathcal{g}\)- closed sets, \(\alpha\)\(\mathcal{I}_\mathcal{g}\)-closed sets, and (\mathcal{I}_\mathcal{r}\)\mathcal{g}\)-closed sets. Then, we focus on the topological implications of \(\mathcal{I}_\mathcal{p}\)-closed sets and investigate how they relate to the concepts of \(\mathcal{I}_\mathcal{p}\)-continuous map, (\mathcal{I}_\mathcal{p}\)-irresolute map, and a strongly \(\mathcal{I}_\mathcal{p}\)-continuous map. First and foremost, we define the \(\mathcal{I}_\mathcal{p}\)-continuous map, investigate the behavior of \(\mathcal{I}_\mathcal{p}\)- continuous map with respect to \(\mathcal{I}_\mathcal{p}\)-closed sets, and derive several important properties of \(\mathcal{I}_\mathcal{p}\)-continuous map. Further, we studied their relationships with other classes of continuous maps in ideal topological spaces. Nevertheless, we defined the definitions of \(\mathcal{I}_\mathcal{p}\)-irresolute maps and strongly \(\mathcal{I}_\mathcal{p}\)-continuous maps in ideal topological spaces. We explored the connections with the notions of \(\mathcal{I}_\mathcal{p}\)-continuous map, (\mathcal{I}_\mathcal{p}\)-irresolute map, and a strongly \(\mathcal{I}_\mathcal{p}\)-continuous map. Our results provide new insights into the study of ideal topological spaces.

Boundary conjugation problem for piecewise analytic functions in Besov spaces

The main aim of this work is to show the solvability of the homogeneous and inhomogeneous boundary conjugation problems for piecewise analytic functions in the Besov space, which is embedded into the class of continuous functions.

On the Convergence of Random Fourier-Jacobi Series of Continuous functions

The interest in orthogonal polynomials and random Fourier series in numerous branches of science and a few studies on random Fourier series in orthogonal polynomials inspired us to focus on random Fourier series in Jacobi polynomials. In the present note, an attempt has been made to investigate the stochastic convergence of some random Jacobi series. We looked into the random series $\sum_{n=0}^\infty d_n r_n(\omega)\varphi_n(y)$ in orthogonal polynomials $\varphi_n(y)$ with random variables $r_n(\omega).$ The random coefficients $r_n(\omega)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $\varphi_n(y)$ are chosen to be the Jacobi polynomials and their variants depending on the random variables associated with the kind of stochastic process. The convergence of random series is established for different parameters $\gamma,\delta$ of the Jacobi polynomials with corresponding choice of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class of continuous functions. The sum functions of the random Fourier-Jacobi series associated with continuous stochastic processes are observed to be the stochastic integrals. The continuity properties of the sum functions are also discussed.

A first augmented Lagrangian method for binary QCQP problems based on a class of continuous functions

A first augmented Lagrangian method for binary QCQP problems based on a class of continuous functions

О некотором классе уравнений Вольтерра I рода в задаче идентификации линейной нестационарной динамической системы

This paper proposes an approach to the identification of a nonstationary linear dynamic system. Its input-output mathematical model is presented as a Volterra equation of the first kind. The problem of nonparametric identification of Volterra kernels is solved on the basis of an active experiment using test piecewise linear signals (that have a rising front). The problem statement is based on the conditions for modeling the dynamics of technical devices in the energy and power industry. The choice of an admissible family of input signals is driven by the complexity of generating piecewise-constant type signals for real energy objects. The original problem is reduced to solving Volterra integral equations of the first kind with two variable integration limits. A formula for the inversion of the integral equations under study is constructed. Sufficient conditions are obtained for the solvability of the corresponding equations with respect to Volterra kernels in the class of continuous functions.

A Fixed Point Approach to Lattice Fuzzy Set via F-Contraction

In this work, we design and demonstrate the occurrence of L-fuzzy common fixed points of L-fuzzy mappings (L-FM) meeting contractive criteria in the framework of complete b-metric spaces (b-MS) employing F-contractions and a certain class of continuous functions. We also conduct a case study to determine the implementation of our derived principles. A few other concepts which are the direct consequences of our findings are explained in this paper.

Limit Theorems for Iterates of the Szász–Mirakyan Operator in Probabilistic View

The Szász–Mirakyan operator is known as a positive linear operator which uniformly approximates a certain class of continuous functions on the half line. The purpose of the present paper is to find out limiting behaviors of the iterates of the Szász–Mirakyan operator in a probabilistic point of view. We show that the iterates of the Szász–Mirakyan operator uniformly converge to a continuous semigroup generated by a second-order degenerate differential operator. A probabilistic interpretation of the convergence in terms of a discrete Markov chain constructed from the iterates and a limiting diffusion process on the half line is captured as well.

Universal prediction algorithm in maneuvering target tracking

The paper considers the methods to increase the accuracy of trajectory processing of the results of radar measurements in automated air traffic management systems (AATMS). The recursive filters traditionally used in the secondary processing path of radar information of modern AATMS systems are briefly considered: alpha-beta filter, linear and extended Kalman filters, multi-model filter. Along with multi-model Kalman filters it is proposed to use a universal filter based on the universal prediction theorem, the accuracy of which is asymptotically no worse than any other filter in the class of continuous functions. The algorithm for the operation of a universal alpha-beta filter was developed. As a result of modeling, it was obtained that the mean square of the tracking error of the trajectory of a maneuvering target for the universal filter during the change of the motion model turned out to be lower than for traditional filters.

Approximation of Real Functions by a Generalization of Ismail–May Operator

In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus of continuity and the second-order modulus of Ditzian and Totik. Then, we characterize the bounded functions that can be approximated uniformly by these new operators. In the last section, we obtain the most important results of the paper. We give the complete asymptotic expansion for the operators and we deduce a Voronovskaya-type theorem, results that hold true for smooth functions with exponential growth.

A global solvability of a two‐dimensional kernel determination problem for a viscoelasticity equation

Under study is the two‐dimensional inverse problem of determining the convolutional kernel of the integral term in a viscoelasticity equation. The direct problem is represented by a generalized initial‐boundary value problem for this equation with zero initial data and the Neumann boundary condition in the form of the Dirac delta function. For solving the inverse problem, the traces of the solution to the direct problem on the domain boundary are given as an additional condition. The main result is the theorem of global unique solvability of the inverse problem in the class of functions twice continuously differentiable in the time variable and analytic in the space variable. We apply the methods of scales of Banach spaces of real analytic functions of variable and weight norms in the class of continuous functions.

Cauchy-subregular functions vis-à-vis different types of continuity

A function between two metric spaces is said to be Cauchy-regular if it takes Cauchy sequences to Cauchy sequences. This well-studied class of functions lies strictly in between the class of continuous functions and that of uniformly continuous functions. In this paper, we exhibit another class of functions between metric spaces which is a weaker form of Cauchy-regular functions, called Cauchy-subregular functions. We call a function f:(X,d)→(Y,ρ) to be Cauchy-subregular if for every Cauchy sequence (xn) in (X,d), (f(xn)) has a Cauchy subsequence. We investigate how much it differs from Cauchy-regularity, while using this weaker notion profitably to characterize complete metric spaces. We also compare Cauchy-subregular functions to various other types of continuity such as locally Lipschitz functions and subcontinuous functions. Furthermore, we study relation between Cauchy-subregular functions and the functions that preserve cofinally Cauchy sequences called cofinally Cauchy regular functions.

On the generalized parabolic Hardy-Hénon equation: Existence, blow-up, self-similarity and large-time asymptotic behavior

This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in $C_{b}([0,\infty);L^{q_c,\infty}(\mathbb R^{n}))$. As a direct consequence, global existence for data in strong critical Lebesgue $L^{q_c}(\mathbb R^{n})$ follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time behavior of globally defined solutions. The zero solution $u\equiv 0$ is shown to be asymptotically stable in $L^{q_c}(\mathbb R^{n})$ -- it is the only self-similar solution which is initially small in $L^{q_c}(\mathbb R^{n})$. Moreover, blow-up results are obtained under mild assumptions on the initial data and the corresponding Fujita critical exponent is found.