Smoothness and Asymptotic Properties of Functions with General Monotone Fourier Coefficients

Abstract

In this paper we study trigonometric series with general monotone coefficients, i.e., satisfying

$$\begin{aligned} \sum \limits _{k=n}^{2n} |a_k - a_{k+1}| \le C \sum \limits _{k=[{n}/{\gamma }]}^{[\gamma n]} \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$

for some $C \ge 1$ and $\gamma >1$. We first prove the Lebesgue-type inequalities for such series

$$\begin{aligned} n|a_n|\le C \omega (f,1/n). \end{aligned}$$

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

$$\begin{aligned} |a_n | \le C \sum \limits _{k=[{n}/{\gamma }]}^{\infty } \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$

for some $C \ge 1$ and $\gamma >1$. Sharpness of the obtained results is given. Finally, we study the asymptotic results of Salem–Hardy type.