Some sharp estimates for the second continuity modulus for periodic functions and for functions extended from A segment

Let C be the space of 2π-periodic continuons real functions with the uniform norm, let Hn be the set of trigonometric polynomials of order not more than n, let ω2(f) be the second continuity modulus for a function f ∈ C, and let Tn(f) be the best approximation polynomial of order n for f ∈ C. Set A0(f) = 1/2π ∫-ππ f; /; U : C - C; C(U,h) = sup f∈C ||U(f) - ||/ω2(f, h), In this paper, for h sufficiently large we find the values C(U,h) for some positive operators U. For example, C(A0, h) and C(T0, h) are found. For n = 1, 2, 3 we find the values C(U, π/n + 1) for some linear positive operators U : C → Hn. We establish relations between C(T0, h) and exact constants in the inequality ω2(f, h1) ≤ C(h1, h)ω2(f, h) for some h and h1 such that 0 < h < h1 < π. For a seminorm P invariant with respect to the shift and majorized by the uniform norm, analogs of C(U, h) are estimated from above. We investigate the problem of extension of a function defined on a segment with preservation of the second continuity modulus. The relation (Matrix equation presented) is established. Here the segment X contains I = [0, 1] as a proper subset, and ω2 (f, X, h) is the second continuity modulus for f on X with step h. Bibliography: 5 titles. © 1998 Plenum Publishing Corporation.