On the equivalence between the sets of the trigonometric polynomials

TL;DR

This paper establishes a dimension-independent isomorphism between spaces of multivariate trigonometric polynomials and certain subspaces of univariate $L^1$, providing a new perspective on their structural equivalence.

Contribution

It constructs a dimension-independent isomorphism between multivariate trigonometric polynomial spaces and univariate $L^1$ subspaces, with a quantitative condition on the subspace.

Findings

The injection is an isomorphism with a norm independent of dimension.

Provides a quantitative condition for the subspace $L^1_E$.

Establishes a structural equivalence between multivariate and univariate polynomial spaces.

Abstract

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 $d$.