A new proof of A. F. Timan's approximation theorem, II

Abstract

2016-12-26T15:05:21

Pointwise estimates for an interpolation process of S. N. Bernstein

“…In an earlier work the first author [13] gave a new proof of the inequality (1.2) for the case r = 1. Here the process is a weakly interpolatory in the sense that it is uniquely determined by the values of the given function at the zeros of TchebychefF polynomial of the first kind.…”

“…Let (1.4) r n (jc) = cosrt0,cos0 = ;c, -I <x< I, be the Tchebycheff polynomial of degree n . We denote by n n ^K (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) n [4] Pointwise estimates for an interpolation process 287…”

“…. , « -2 (see [13]), terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700034248 [12] Pointwise estimates for an interpolation process 295…”

On a new interpolation process of S. N. Bernstein

“…k=l where ma(x) = ~(3#1(x)+ #2(x)), 1 (3) ink(X) = ~(~tk-l(X)-{-2~k(X )-~-~k+l(X)), k----2,3,...,n--1, 1 mn(X ) ----~(~tn_l(X ) -~-3~tn(X)) .…”

“…Let f(x) C C[_I,1] and Tn(x) = cosn0 (x = cos0, 0 _< 0 g 7r) the first kind of Chebyshev polynomial of degree n. Tn(x) has all zeros in (-1, 1) at the points (1) zk = cosOk, Ok -(2k-1)~r 2n , k=l,2,...,n…”