Operations that are incompatible with certain systems of translates in $L^2(\mathbb{R})$
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2025
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| _version_ | 1866918129104322560 |
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| author | Yu, Pu-Ting |
| author_facet | Yu, Pu-Ting |
| contents | We say that a closed subspace $M$ of $L^2(\mathbb{R})$ 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 $\mathbb{Z}$ and some finite subsets $\{α_{1j}\}_{j=1}^{K_1},\dots,\{α_{Nj}\}_{j=1}^{K_N}$ of $\mathbb{R}$ such that $$M={\overline{\text{span}}}\{g_i(\cdot-ak), ~g_i(\cdot-α_{ij})\,|\,k\in J_i,1\leq j\leq K_i\,\}_{i=1}^N.$$
Here $\cdot$ denotes a generic variable. In the first half of this paper, we prove that a closed subspace of $L^2(\mathbb{R})$ 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 $b\neq 0$ and $b^{-1}\notin \mathbb{Z}.$ We also show that no infinite-dimensional closed subspace of $L^2(\mathbb{R})$ can simultaneously be closed under Fourier transform and admit a complete set of semi-regular $a$-translates with $a^2\in \mathbb{Q}$, 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, we prove that a closed subspace of $L^2(\mathbb{R})$ 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 $M^1(\mathbb{R})$. We also prove that no closed subspace of $L^2(\mathbb{R})$ can simultaneously be closed under modulation and admit a Schauder basis of translates generated by finitely many functions in $M^1(\mathbb{R}).$ 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 $L^2(\mathbb{R})$. All results in this half can be extended to $L^2(\mathbb{R}^d)$ for any $d>1.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_16529 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Operations that are incompatible with certain systems of translates in $L^2(\mathbb{R})$ Yu, Pu-Ting Functional Analysis Classical Analysis and ODEs We say that a closed subspace $M$ of $L^2(\mathbb{R})$ 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 $\mathbb{Z}$ and some finite subsets $\{α_{1j}\}_{j=1}^{K_1},\dots,\{α_{Nj}\}_{j=1}^{K_N}$ of $\mathbb{R}$ such that $$M={\overline{\text{span}}}\{g_i(\cdot-ak), ~g_i(\cdot-α_{ij})\,|\,k\in J_i,1\leq j\leq K_i\,\}_{i=1}^N.$$ Here $\cdot$ denotes a generic variable. In the first half of this paper, we prove that a closed subspace of $L^2(\mathbb{R})$ 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 $b\neq 0$ and $b^{-1}\notin \mathbb{Z}.$ We also show that no infinite-dimensional closed subspace of $L^2(\mathbb{R})$ can simultaneously be closed under Fourier transform and admit a complete set of semi-regular $a$-translates with $a^2\in \mathbb{Q}$, 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, we prove that a closed subspace of $L^2(\mathbb{R})$ 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 $M^1(\mathbb{R})$. We also prove that no closed subspace of $L^2(\mathbb{R})$ can simultaneously be closed under modulation and admit a Schauder basis of translates generated by finitely many functions in $M^1(\mathbb{R}).$ 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 $L^2(\mathbb{R})$. All results in this half can be extended to $L^2(\mathbb{R}^d)$ for any $d>1.$ |
| title | Operations that are incompatible with certain systems of translates in $L^2(\mathbb{R})$ |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2508.16529 |