Lagrange interpolation on extended generalized Jacobi nodes

We consider theL ^p -convergence of interpolatory processes for nonsmooth functions. Therefore we use generalizations of the well-known Marcinkiewicz-Zygmund inequality for trigonometric polynomials to the case of algebraic polynomials, extending a result of Y. Xu. Particularly, we obtain the order of convergence for certain Lagrange and quasi-Lagrange interpolatory processes on generalized Jacobi nodes. Our approach enables us also to discuss the influence of additional nodes near the endpoints ±1.     

Convolution functions of several variables with generalized bounded variation

Let L ¹ be the class of all complex-valued functions, with period 2π in each variable, in the space , where $\mathbb{T} = [0,2\pi )$ is the one-dimensional torus. Here, it is observed that L ¹ * E ? E for E = Lip(p; α ₁, α ₂, ..., α _N ) over , for , for , and for in the sense of Vitali as well as Hardy.     

A NOTE ON MEYER—KONIG AND ZHLLER OPERATORS FOR FUNCTIONS OF BOUNDED VARIATION

In the present note we give the correct and improved estimate on the rate of convergence of integrated Meyer-Konig and Zeller operators for function of bounded variation.     

Convergence of two-dimensional trigonometric interpolation polynomials of functions of bounded harmonic variation

The paper considers a question of convergence of partial sums of two-dimensional trigonometric interpolation polynomials. Convergence by Pringsheim for functions of two variables with bounded harmonic variation is established.     

Summability of multiple Fourier series of functions of bounded generalized variation

The paper is devoted to summability problems for multiple Fourier series of functions of bounded generalized variation. We find necessary and sufficient conditions on a sequence Λ = {λ _n } that guarantee the convergence of the Cesàro means of multiple Fourier series of functions of bounded Λ-variation.     

Lagrange interpolation for continuous piecewise smooth functions

This note is devoted to Lagrange interpolation for continuous piecewise smooth functions. A new family of interpolatory functions with explicit approximation error bounds is obtained. We apply the theory to the classical Lagrange interpolation.     

A remainder formula of numerical differentiation for the generalized Lagrange interpolation

Through exploiting the generalized Lagrange interpolation for continuous piecewise smooth functions, a remainder term of numerical differentiation is given. Also, the error estimates for the numerical differentiation formula are presented.     

The exact correction theorem for spaces of functions of generalized bounded variation

Mathematical Notes -     

A NOTE ON INTERPOLATION OF A CERTAIN CLASS OF FUNCTIONS

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On the convergence of multiple Walsh-Fourier series of functions of bounded generalized variation

The convergence of multiple Walsh-Fourier series of functions of bounded generalized variation is investigated. The sufficient and necessary conditions on the sequence ?? = {?? _n } are found for the convergence of multiple Walsh-Fourier series of functions of bounded partial ??-variation.     

Conditions for the existence of stieltjes integral of functions of bounded generalized variation

, , , , - , , , , . . . .