Rogosinski-Szego type inequalities for trigonometric sums

JOURNAL OF APPROXIMATION THEORY

卷:190

Article

Kwong, Man Kam¹ 

[1] Hong Kong Polytech Univ, Dept Appl Math, Hong Kong, Hong Kong, Peoples R China

关键词: Trigonometric sums;    Inequalities;    Sharp bounds;   

DOI  :  10.1016/j.jat.2014.04.007

来源: Elsevier

【 摘 要 】

We prove that the inequalities Sigma(n)(k=1) sin(kx)/k+1 >= 1/384 (9 - root 137)root 110 - 6 root 137 = -0.044419686... and Sigma(n)(k=1) sin(kx) + cos(kx)/k +1 >= -1/2 are valid for all real numbers x is an element of [0, pi] and all positive integers n. The constant lower bounds are sharp. Our theorems complement a classical result of Rogosinski and Szego, who proved in 1928 that the inequality Sigma(n)(k=1) cos(kx)/k+1 >= -1/2 holds for all x is an element of [0, pi] and n >= 1. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free