On Fourier coefficients of lacunary systems

Abstract

We prove that the Zygmund space L(lnL)1/2 is the largest among symmetric spaces X in which any uniformly bounded orthonormal system of functions contains a sequence such that the corresponding space of Fourier coefficients F(X) coincides with l2. Moreover, we obtain a description of spaces of Fourier coefficients corresponding to appropriate subsequences of arbitrary uniformly bounded orthonormal systems in symmetric spaces located between the spaces L(lnL)1/2 and L1.