On example to the Boas theorem

Abstract

In this work we study the relation between summability properties of a sequence tending to zero and integrability properties of the corresponding trigonometric series. More precisely, in this paper is considered the problem of generalizing of Boas' theorem on the Fourier coefficients of monotone functions. According to that theorem the norm of monotone function in the Lorentz space L_p,q[0, 1], 1 < p < ∞ , 1 ≤ q ≤ ∞ is equivalent to the norm of the Fourier coefficients of the function in the discrete Lorentz space l_p′,q . We define the class of α-monotone functions (0 < α ≤ 1) that contains the class of non-increasing absolutely continuous functions. The α-monotone function is defined as function which has non-increasing absolutely continuous right-side fractional Riemann- Liouville integral of order 1 − α . A problem of generalizing of Boas' theorem to the class of α-monotone function is very interesting for us. We give an example of α-monotone function which show impossibility of Boas' theorem in the case p < 1/α. It follows that Boas' theorem for the α-monotone functions should be investigated in the case p ≥ 1/α.