Operations that are incompatible with certain systems of translates in $L^2(\mathbb{R})$

Abstract

We say that closed subspace $M$ of $L^2(\R)$ admits a \emph{complete set of semi-regular a-translates} if there exist some $a>0$, finitely many functions $g_1,\dots,g_N$, some subsets $J_1,\dots,J_N$ of $\Z$ and some finite subsets $\set{\al_{1j}}_{j=1}^{K_1},\dots,\set{\al_{Nj}}_{j=1}^{K_N}$ of $\R$ such that $$M=\clspan{\bigset{g_i(\cdot-ak), ~g_i(\cdot-\al_{ij})\,|\,k\in J_i,1\leq j\leq K_i\,}}_{i=1}^N.$$ Here $\cdot$ denotes a generic variable. In the first half of this paper, we study whether the properties of being closed under modulation, dilation, reflection or Fourier transform is compatible with the existence of a complete set of semi-regular $a$-translates in closed subspaces of $L^2(\R)$. Specifically, we prove that a closed subspace of $L^2(\R)$ does not admit a complete set of semi-regular $a$-translates if it is closed under modulation or if it is closed under dilation with respect to a scaling factor $b$ satisfying $|b|>1.$ We also show that no infinite-dimensional closed subspace of $L^2(\R)$ can simultaneously be closed under Fourier transform and admit a complete set of semi-regular $a$-translates with $a^2\in \Q$, whereas for any $a>0$, there do exist closed subspaces that are closed under reflection and admit a complete set of semi-regular $a$-translates. In the second half of this paper, we prove that a closed subspace of $L^2(\R)$ does not admit a frame formed by a system of translates if it contains a closed subspace that is closed under modulation and contains a nonzero function in $M^1(\R)$. In addition, we present related results concerning the incompatibility between being closed under Fourier transform and the existence of frames or Schauder bases of translates in closed subspaces of $L^2(\R)$. All results in this half can be extended to $L^2(\R^d)$ for any $d>1.$