Absolute convergence factors of Lipschitz class functions for general Fourier series

The main aim of this paper is to investigate the sequences of positive numbers, for which multiplication with Fourier coefficients of functions f ∈ Lip ⁡ 1 {f\in\operatorname{Lip}1} class provides absolute convergence of Fourier series. In particular, we found special conditions for the functions of orthonormal system (ONS), for which the above sequences are absolute convergence factors of Fourier series of functions of Lip ⁡ 1 {\operatorname{Lip}1} class. It is established that the resulting conditions are best possible in certain sense.

Summation-Integral Type Operators Based on Lupas-Jain Functions

We introduce a genuine summation-integral type operators based on Lupaş-Jain type base functions related to the unbounded sequences. We investigated their degree of approximation in terms of modulus of continuity and ????-functional for the functions from bounded and continuous functions space. Furthermore, we give some theorems for the local approximation properties of functions belonging to Lipschitz class. Also, we give Voronovskaja theorem for these operators.

Dunkl-type generalization of the second kind beta operators via $(p,q)$-calculus

The main purpose of this research article is to construct a Dunkl extension of $(p,q)$ ( p , q ) -variant of Szász–Beta operators of the second kind by applying a new parameter. We obtain Korovkin-type approximation theorems, local approximations, and weighted approximations. Further, we study the rate of convergence by using the modulus of continuity, Lipschitz class and Peetre’s K-functionals.

Legendre wavelet approximation of functions having derivatives of Lipschitz class

In this paper, Legendre Wavelet approximation of functions f having first derivative f' and second derivative f'' of Lip? class, 0 < ? ? 1, have been determined. These wavelet estimators are sharper, better and best possible in Wavelet Analysis. It is observed that the LegendreWavelet estimator of f whose f'' ? Lip? is sharper than the estimator of f having f ' ?Lip? class.

Trigonometric approximation of functions $f(x,y)$ of generalized Lipschitz class by double Hausdorff matrix summability method

In this paper, we establish a new estimate for the degree of approximation of functions $f(x,y)$ f ( x , y ) belonging to the generalized Lipschitz class $Lip ((\xi _{1}, \xi _{2} );r )$ L i p ( ( ξ 1 , ξ 2 ) ; r ) , $r \geq 1$ r ≥ 1 , by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from $Lip ((\alpha ,\beta );r )$ L i p ( ( α , β ) ; r ) and $Lip(\alpha ,\beta )$ L i p ( α , β ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and $(C, \gamma , \delta )$ ( C , γ , δ ) means.

Asymptotics of approximation of functions by conjugate Poisson integrals

Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha $, i.e. the class of continuous $ 2\pi $-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0<\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha $. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.

Approximation properties of Jain-Appell operators

In this paper, we introduce the Jain-Appell operators by applying Gamma transform to the Jakimovski-Leviatan operators. In their special cases they include not only the Jain-Pethe operators, but also new families of operators, where we call them Appell-Baskakov and Appell-Lupa? operators, since their special cases contain Baskakov and Lupa? operators, respectively. We investigate their weighted approximation properties and compute the error of approximation by using certain Lipschitz class functions. Furthermore, we obtain their A-statistical approximation property.