Upper bounds of best one-sided approximations of the classes WrLΨ in the metric of L

Abstract

With a view to estimating LI f-] 1[,+, we consider the expression max [r (~)1~ (?o/y) '.(~>'(2g~Yt can easily be shown that the function +h(r) de~q~/hr+ ' r>O, .fi>O,O<q<l, at the point r* = --hll/ln q, so that: +n (r) < al?h (r*) ----q-t+'ln~ (15/(_ In q))+h+. Note. Let Xz(t) be the characteristic function of the cubeK~={yllyiI~ e/2, i = i,..., m}.We consider coordinate Systems of the type ~h)(x)= ~ (x/h-s), where s is an integer-valued vector in R m. If, in the last expression, we put ~ = ~i*%1 or ~ ~ %1*%1*%1, then, with m = i, we obtain two examples of coordinate systems which satisfy the conditions of Theorem 2. With m > i, the required coordinate systems cannot be obtained by taking ~ equal to a multiple convolution of the function X~ with itself.For m > i, the conditions of Theorem 2 are satisfied by coordinate systems with a function ~ of the type ~ = %1*%E* 9 * ~s, where k f is any integer, and ~ is a sufficiently small number 0 < s < co(k, m). UPPER BOUNDS OF BEST ONE-SIDED APPROXIMATIONS OF THE CLASSES

Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities

“…(10), (11), and Theorems A and B, respectively. Only the statements on their sharpness need to be clarified.…”

“…The best one-sided approximations of the classes W p r by splines were established by Doronin and Ligun in [11]. They proved that, for n, m, r ∈ N , m ≥ r, one has…”