Borislav R. Draganov

Exact estimates of the rate of approximation of convolution operators

The paper presents a method for establishing direct and strong converse inequalities in terms of K-functionals for convolution operators acting in homogeneous Banach spaces of multivariate functions. The method is based on the behaviour of the Fourier transform of the kernel of the convolution operator.

Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums

We establish matching direct and two-term strong converse estimates of the rate of weighted simultaneous approximation by the Bernstein operator and its iterated Boolean sums for smooth functions in L p -norm, 1 < p ? ∞ . We consider Jacobi weights. The characterization is stated in terms of appropriate moduli of smoothness or K -functionals. Also, analogous results concerning the generalized Kantorovich operators are derived.

A generalized modulus of smoothness

When using an approximation process it is important to have a practical and computable measure of its error. For such a measure one can use the so-called modulus of smoothness. Loosely speaking, it describes structural properties of the function and, in particular, its smoothness. Then error estimates by means of an appropriate modulus of smoothness state that the smoother a function is, the faster it is approximated. Let us recall the definition of the classical unweighted fixed-step modulus of smoothness for functions on a finite interval. As usual, Lp[a, b], 1 ≤ p ≤ ∞, are the Lebesgue spaces of real/complex-valued functions on the interval [a, b] with their standard norm, which we shall denote by ‖ · ‖p. In what follows we can consider C[a, b], the space of continuous real/complex-valued functions on [a, b], in place of L∞[a, b]. The finite difference of order r ∈ N and step h > 0 of the function f ∈ Lp[a, b] is defined by

Simultaneous approximation by Bernstein polynomials with integer coefficients

Abstract We prove that several forms of the Bernstein polynomials with integer coefficients possess the property of simultaneous approximation, that is, they approximate not only the function but also its derivatives. We establish direct estimates of the error of that approximation in uniform norm by means of moduli of smoothness. Moreover, we show that the sufficient conditions under which those estimates hold are also necessary.