Estimates for functionals with a known, finite set of moments, in terms of moduli of continuity, and behavior of constants, in the Jackson-type inequalities

Abstract

A new technique is developed for estimating functionals by moduli of continuity. The generalized Jackson inequality A σ−0 (f) ≤ 1 2m m m−1 k=0 K 2k (γπ) 2k ν k m + K 2m (γπ) 2m ν m m 2 2m ω 2m f, γπ σ is an example of such an estimate. Here r, m ∈ N, σ, γ > 0, a function f is uniformly continuous and bounded on R, A σ−0 is the best uniform approximation by entire functions of type less than σ, ω 2m is a uniform modulus of continuity of order 2m, K s are the Favard constants, and ν m = 8 2m m (m−1)/2 l=0 2m m−2l−1 (2l + 1) 2 , where x is the entire part of x. Similar inequalities are obtained for best approximations of periodic functions by splines. In some cases, the constants in inequalities are close to optimal.

Constants in Jackson Type Inequalities for the Best Approximation of Periodic Functions Such that Some of Their Fourier Coefficients Vanish

“…Apparently, the bounds obtained in [2] are the best ones at present. Such bounds can be obtained by linear approximation methods, where the sets U n,l are mapped to the space of trigonometric polynomials of degree at most n. Therefore, in addition to the estimate (1), we have…”

“…Regarding bounds for the constant C(r, γ), this question has not been completely investigated yet and we refer to [2] for more information. Apparently, the bounds obtained in [2] are the best ones at present.…”

Improvement of Jackson Type Inequalities for Moduli of Continuity of Order 4, 6, 8

“…For m = 2 and m = 3 and some admissible γ this method yields better constants than the known ones and, in particular, better than the constants obtained by a direct estimation of E n ( Uf) P in terms of the modulus of continuity (cf., for example, [2]). It is easy to see that the inequality τ 2m,P (γ) < 1 fails for m 4 and any γ.…”

“…The following inequality are known [2] E n (f ) P 0, 32441 ω 6 (f, π/(2n)) P , E n (f ) P 0, 16085 ω 8 (f, π/(2n)) P .…”

“…For certain values of order and step of the modulus of continuity this method leads to better constants than the known ones. The review on results concerning constants in Jackson type inequalities can be found in [2].…”