de la Vallée-Poussin means of Fourier series for the quadratic spectrum and for spectra with power-like density

A new method is proposed and elaborated for investigating complex or real trigonometric series with various spectra. It is based on new multiplicative inequalities which give a lower bound for the integral norm of the de la Vallée-Poussin means and are themselves based on results establishing corresponding analogues of the Littlewood–Paley theorem in the BMO, Hardy, and Lorentz spaces. For spectra with power-like density a description of the class of absolute values of coefficients such that the corresponding complex or real trigonometric series are Fourier series is found which depends on the arithmetic characteristics of the spectrum and is sharp in limiting cases. Furthermore, for the quadratic spectrum some results of Hardy and Littlewood on elliptic theta functions are generalized and refined. For the quadratic spectrum and power-like spectra with non-integer exponents new lower bounds are found for the integral norms of exponential sums.
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