De la Vallée-Poussin means of Fourier series for quadratic spectrum and power density spectra

摘要

Multiplicative inequalities of the indicated type are established in the most general form involving a lower bound for the integral norm of de la Vallée-Poussin means of Fourier series of functions. It is determined whether a complex or real trigonometric series with given coefficients is a Fourier series. New necessary conditions on the moduli of the coefficients of trigonometric series are established under which series with a quadratic spectrum or other classical power density spectra are Fourier series. In the case of the classical quadratic spectrum, the number of solutions is estimated using Ramanujan's asymptotic formula. Applying Salem's theorem on Fourier series transformations, the important assertion on the quadratic spectrum are obtained.