  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 $L^1_E\subset
L^1(\mathbb{T})$. 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 $d$.
