Two-Sided Estimates for Some Functionals in Terms of the Best Approximations

Abstract

Let C be the space of continuous 2π-periodic functions. For some integrals of the form

\( \underset{0}{\overset{\pi }{\int }}{\omega}_r\left(f,t\right)\Phi (t) dt, \)

where ω_r(f, t) is the modulus of continuity of order r of a function f in C, two-sided bounds in terms of the best approximations by trigonometric polynomials are established.