On generalized lacunary series

2022/04/02 by Karagulyan, Grigori A., Karagulyan, Vahe G.
#42A55 #42C05 #42C10 #42C25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2204.00927

Abstract

Given lacunary sequence of integers, n_k, n_k+1/n_k>λ>1, we define a new sequence \m_k\ formed by all possible l-wise sums ± n_k1± n_k2± …± n_kl. We prove if λ>λ_l, then any series ∑_kc_ke^im_kx, (1) with ∑_k|c_k|²2, where S is a finite sum of form (1). For λ≥ 3, we also establish a sharp rate p^l/2 for the growth of the constant C_l,λ,p as p→∞. Such an estimate for the Rademacher chaos sums was proved independently by Bonami and Kiener. In the case of λ≥ 3 we also establish some inverse convergence properties of series (1): 1) if series (1) converges a.e., then ∑_k|c_k|²

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