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

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Let C be the space of 2π-periodic continuous 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\(A_0 \left( f \right) = \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi f \); U:C→C;\(C\left( {U,h} \right) = \mathop {\sup }\limits_{f \in C} ^{\frac{{\left\| {U\left( f \right) - f} \right\|}}{{\omega _2 \left( {f,h} \right)}}} \). 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\(C\left( {U,\frac{\pi }{{n + l}}} \right)\) 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 $$\mathop {\sup }\limits_{f \in C\left( I \right)} \mathop {\inf }\limits_{\mathop {g:X \to \mathbb{R},}\limits_{f = g\left| {_I } \right.} } \frac{{\omega _2 \left( {g,X,h} \right)}}{{\omega _2 \left( {f,I,h} \right)}} = \frac{3}{2}$$ 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.