Notes on interpolation. III (convergence)

Notes on the convergence of (0, 2) and (0, 1, 3) interpolations

“…In [6], G. FREUD improved the convergence theorem from [3] as follows: [-1, 1] which satisfies the condition The object of this part is to give another and improved form of Theorem 2. 1.…”

“…be an infinite point-system. As in [1], by (0, 2)-interpolating polynomials we mean those polynomials R,(x) of degree =< 2n-1 whose values and second derivatives at the Xk,'S are prescribed: In [2] and [3] P. TURAN and his collaborators proved the following theorems: Let ,us choose as fundamental points Xk, of the (0, 2) interpolation the zeros Xk. '(k = 1, 2, ..., n; n = 1, 2, ...) of the polynomials…”

Divergence of trigonometric lacunary interpolation

“…In their papers [1], [2] and [3] J. Balfizs, J. Surfinyi and P. Turfin investigated the so called (0, 2) algebraic interpolatory polynomials.…”

A problem of P. Turán (on the mean convergence of lacunary interpolation

“…:stands for the formula (6.1.1) from [3]. Denoting _R,(f;x)= ~f(xk,,)ra,,(x ) k=l (n=2, 4, 6 .... ) for a continuous f(x), a theorem proved in [3] gave some conditions regarding convergence of _R,(f; x) to f(x).…”

“…--1=<x~1 It was proved in [3] that for any 0<~< 1 we can find a continuous function fELip c~ such that /~n ( (In this paper we consider only weight functions p (x), which are nonnegative, summable and are zero only on a set of measure zero.) As we know, the Faber's theorem implies that there exists a continuous function f3 such that L n (fz; x) does not converge uniformly to f3(x) ( [10]).…”