Smoothness and Asymptotic Properties of Functions with General Monotone Fourier Coefficients

• Published in: The Journal of fourier analysis and applications 2018-08, Vol.24 (4), p.1072-1097
• Main Authors: Dyachenko, M. I., Tikhonov, S. Yu
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Approximations and Expansions; Asymptotic properties; Coefficients; Economic models; Fourier Analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Sharpness; Signal,Image and Speech Processing; Smoothness
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-017-9553-7

In this paper we study trigonometric series with general monotone coefficients, i.e., satisfying ∑ k = n 2 n | a k - a k + 1 | ≤ C ∑ k = [ n / γ ] [ γ n ] | a k | k , n ∈ N , for some C ≥ 1 and γ > 1 . We first prove the Lebesgue-type inequalities for such series n | a n | ≤ C ω ( f , 1 / n ) . Moreover, we obtain necessary and sufficient conditions for the sum of such series to belong to the generalized Lipschitz, Nikolskii, and Zygmund spaces. We also prove similar results for trigonometric series with weak monotone coefficients, i.e., satisfying | a n | ≤ C ∑ k = [ n / γ ] ∞ | a k | k , n ∈ N , for some C ≥ 1 and γ > 1 . Sharpness of the obtained results is given. Finally, we study the asymptotic results of Salem–Hardy type.