On the equivalence between the sets of the trigonometric polynomials

Abstract

In this paper we construct an injection from the linear space of trigonometric polynomials defined on T d with bounded degrees with respect to each variable to a suitable linear subspace L 1 E ⊂ L 1 (T). We give such a quantitative condition on L 1 E that this injection is a isomorphism of a Banach spaces equipped with L 1 norm and the norm of the isomorphism is independent on the dimension d.

Independent sums of $H^1_n(\mathbb{T})$ and $H^1_n(δ)$

“…The main part of the following corollary is somewhat trivial. We state it separately for its similarity with Theorem 2.9 and Theorem 5.8 (compare results [13], [8]) and because we find the fact that the assumption d k N k+a can be weakened in neither Theorem 2.3 nor Corollary 2.7 worth a proof.…”