Nikol’skii’s inequality for different metrics and properties of the sequence of norms of the Fourier sums of a function in the Lorentz space

Abstract

Let (X, Y) be a pair of normed spaces such that X ⊂ Y ⊂ L 1[0, 1] n and {e k } k be an expanding sequence of finite sets in ℤ n with respect to a scalar or vector parameter k, k ∈ ℕ or k ∈ ℕ n . The properties of the sequence of norms $$\{ \left\| {S_{e_k } (f)} \right\|x\} _k $$ of the Fourier sums of a fixed function f ∈ Y are studied. As the spaces X and Y, the Lebesgue spaces L p [0, 1], the Lorentz spaces L p,q [0, 1], L p,q [0, 1] n , and the anisotropic Lorentz spaces L p,q*[0, 1] n are considered. In the one-dimensional case, the sequence {e k } k consists of segments, and in the multidimensional case, it is a sequence of hyperbolic crosses or parallelepipeds in ℤ n . For trigonometric polynomials with the spectrum given by step hyperbolic crosses and parallelepipeds, various types of inequalities for different metrics in the Lorentz spaces L p,q [0, 1] n and L p,q*[0, 1] n are obtained.