Half-space from $\mathbb Z^n$ forms a Chebyshev subspace in $L_1[0, 1]^n$

B. B. Bednov

I. M. Sechenov First Moscow State Medical University

Abstract: A subspace $Y$ in a Banach space $X$ is called Chebyshev subspace if for every $x \in X$ there exists a unique best approximation element in $Y$. J.L.Doob proved in 1940 that the Hardy space $H_1$ is a Chebyshev subspace in the space $L_1[0, 1]$ of complex-valued functions. So, the Hardy space $H_1$ is isometrically isomorphic to the subspace $Y_{\mathbb N} \subset L_1[0, 1]$ defined as the closure of a linear hull of exponents with spectrum in $\mathbb N$. J.-P. Kahane described in 1974 all sets $M$ in $\mathbb Z$ for which the closure of a linear hull of exponents with spectrum in $M$ forms a Chebyshev subspace in $L_1[0, 1]$. There are infinite arithmetic sequences with odd difference. Two types of such sets are possible up to an integer shift: $(2n + 1)\mathbb N$ and $(2n + 1)\mathbb Z$, $n \in \mathbb N\cup\{0\}$. There are different proofs for the sets $\mathbb N$ and $(2n + 1)\mathbb N,\, n \in \mathbb N$ in Kahane's theorem. In the present paper we attempt to partially generalizing Kahane's result in the case of several variables. Thus, we investigate existence and uniqueness of best approximation element in the closure of a linear hull of exponents with spectrum in the intersection of $\mathbb Z^n$ with a half-space bounded by a hyperplane in the space $L_1[0, 1]^n$ of complex-valued functions of $n$ real variables. The proof follows Kahane's proof for the set $\mathbb N$.

Keywords: complex $L_1$ space, Chebyshev subspace, Kahane's theorem, nicely placed subset, Abelian discrete group.

UDC: 517.982.256 + 515.124.4

MSC: 41A30, 41A50, 41A52, 41A65, 43A20

Received: 10.06.2025
Revised: 01.09.2025
Accepted: 08.09.2025

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