Strong Converse Inequalities and Qantitative Voronovskaya-Type Theorems for Trigonometric Fej`er Sums

Authors : Jorge BUSTAMANTE, Lázaro FLORES DE JESÚS

Pages : 53-63

Doi:10.33205/cma.653843

View : 13 | Download : 7

Publication Date : 2020-06-01

Article Type : Research Paper

Abstract :Let $\sigma_n$ denotes the classical Fej\`er operator for trigonometric expansions. For a fixed even integer $r$, we characterize the rate of convergence of the iterative operators $insert ignore into journalissuearticles values(I-\sigma_n);^rinsert ignore into journalissuearticles values(f);$ in terms of the modulus of continuity of order $r$ insert ignore into journalissuearticles values(with specific constants); in all $\mathbb{L}^p$ spaces $1\leq p \leq \infty$. In particular, the constants depend not on $p$. Moreover, we present a quantitative version of the Voronovskaya-type theorems for the operators $insert ignore into journalissuearticles values(I-\sigma_n);^rinsert ignore into journalissuearticles values(f);$.
Keywords : Fej`er operators, iterative combinations, rate of convergence, direct and converse results