On the equivalence between the sets of the trigonometric polynomials

In this paper we construct an injection from the linear space of trigonometric polynomials defined on ${\mathbb{T}}^d$ with bounded degrees with respect to each variable to a suitable linear subspace $L^1_E\subset L^1($\mathbb{T}$)$\mathrm{.}Wegivesuchaquantitativeconditionon$L^1_E$ 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 $d$.