On the Best Constant in the Jackson–Stechkin Inequality

Abstract

In this article, we obtain an upper bound for the exact constant in the Jackson-Stechkin inequality in the space L^p(T), where T is one-dimensional torus, 1 \le p \le \infty. This inequality relates the best approximation of a function by trigonometric polynomials to its generalized modulus of smoothness of order r with non-constant steps depending on the parameters h and a in the space L^p(T).   We consider the modulus of smoothness with a non-constant step depending on both the classical step h and the auxiliary parameter a. We obtain an upper bound for the exact constant in the Jackson-Stechkin inequality that depends only on the smoothness parameter r and is independent of the other parameters a and p. For r=2 and r=3, we derive more precise bounds than in the general case.