We investigate the convergence and the rate of convergence in , 1 < p < ∞, of a bivariate interpolating (with respect to a sparse grid) trigonometric polynomial in the framework of Sobolev spaces of dominating mixed smoothness.
The direct and inverse problems of approximation theory in the subspace of weighted generalized grand Lebesgue spaces of 2π-periodic functions with the weights satisfying Muckenhoupt's condition are investigated. Appropriate direct and inverse theorems are proved. As a corollary some results on constructive characterization problems in generalized Lipschitz classes are presented.
It is not the purpose of this paper to construct approximations but to establish a class of almost periodic functions which can be approximated, with an arbitrarily prescribed accuracy, by continuous periodic functions uniformly on .
In this paper we study the approximation of continuous functions F, defined on a compact Hausdorff space S, whose values F(t), for each t in S, are convex subsets of a normed space E. Both quantitative estimates (in the Hausdorff semimetric) and Bohman-Korovkin type approximation theorems for sequences of monotone operators are obtained.
We show the general and precise conditions on the functions and modulus of continuity as well as on the entries of matrices generating the summability means and give the rates of approximation of functions from the generalized integral Lipschitz classes by double matrix means of their Fourier series. Consequently, we give some results on norm approximation. Thus we essentially extend and improve our earlier results [Acta Comment. Univ. Tartu. Math. 13 (2009), 11-24] and the result of S. Lal [Appl....