Hardy-Littlewood theorem for series with general monotone coefficients

Abstract

In this work we study trigonometric series with general monotone coefficients. Also, we consider Lqϕ ( Lq ) space. In particular, when ϕ ( t ) ≡ 1 the space Lqϕ ( Lq ) coincides with Lq . Well known the theorem of Hardy and Littlewood about trigonometric series with monotone coefficients. Also known various generalizations of this theorem. In 1982 this theorem was generalized by M.F. Timan for the spaces Lqϕ ( Lq ). And in 2007 S.Tikhonov proved Hardy-Littlewood theorem for trigonometric series with general monotone coefficients. In this work we have generalized Hardy-Littlewood theorem for Fourier series of functions f ∈ Lqϕ ( Lq ) with general monotone coefficients. Also, obtained upper - bound estimate of best approximation of functions f ∈ Lq through its Fourier’s coefficients which are general monotone.

Keywords

• trigonometric series
• Hardy-Littlewood theorem
• general monotone sequences
• convergence
• Fourier’s coefficients