Smoothness and Asymptotic Properties of Functions with General Monotone Fourier Coefficients

In this paper we study trigonometric series with general monotone coefficients, i.e., satisfying Sigma(2n)(k=n) vertical bar a(k) - a(k+1)vertical bar <= C Sigma([gamma n])(k=[n/gamma]) vertical bar a(k)vertical bar/k, n is an element of N, for some C >= 1 and gamma > 1. We first prove the Lebesgue-type inequalities for such series n vertical bar a(n)vertical bar <= C omega(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 vertical bar a(n)vertical bar <= C Sigma(infinity)(k=[n/gamma]) vertical bar a(k)vertical bar/k, n is an element of N, for some C >= 1 and gamma > 1. Sharpness of the obtained results is given. Finally, we study the asymptotic results of Salem-Hardy type.