On estimating the rate of best trigonometric approximation by a modulus of smoothness

Abstract

Best trigonometric approximation in Lp, 1 p ∞, is characterized by a modulus of smoothness, which is equivalent to zero if the function is a trigonometric polynomial of a given degree. The characterization is similar to the one given by the classical modulus of smoothness. The modulus possesses properties similar to those of the classical one.

An Improved Jackson Inequality for the Best Trigonometric Approximation

“…To verify the latter, we take into account that F r f = A r F with F = F r−1 f ; hence B r F r f = = B r A r F = F + η r * F with η r (x) = −1 − 2 cos rx as was established in [11] ((2.9)). Set G = F r f and let G t ∈ W 2r−1 p (T) satisfy (4.6), (4.7) for G in the place of F. Then…”

“…Set F = F r−1 f. Then F r f = F + r 2 a * F. In [11] (3.2) it was proved that (a * g) = g + const for any g ∈ L p (T). Then, using basic properties of the classical modulus, we get…”

“…, 2r − 1, whose coefficients depend on h. Basic properties of these coefficients including their explicit form are given e.g. in [10] (Section 2) (see also [13]). On the other hand, the finite difference ∆ 2r−1 h F r−1 f is the classical symmetric finite difference on the same nodes but of the function F r−1 f.…”

“…where D r is the differential operator whose kernel is T r−1 (see [10] (Theorem 4.2) and [11] (Theorem 2.1)). Theorem 3.1 is proved.…”

Approximation of Unbounded Functions by Trigonometric Polynomials

“…This notion could considered as direct generalization of classical trigonometric approximation in weighted space , ( ) and it is more natural to use it for number of problems of approximation functions ( see, for example [3], [5], [6], [9]). The important problem of approximation theory and theory of Fourier series is the problem of description of best approximation of functions by trigonometric polynomials see [7] and [10].One can consider this problem from the viewpoint of description of Roman Taberski of trigonometric approximation.…”