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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.09099 |
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| _version_ | 1866909303018881024 |
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| author | Priyanka, Kumari Selvan, A. Antony |
| author_facet | Priyanka, Kumari Selvan, A. Antony |
| contents | For several shift-invariant spaces, there exists a real number $a\in\mathbb{R}$ such that the set $a+\mathbb{Z}$ is a complete interpolation set. In this paper, we characterize the complete interpolation property of the set $(a+\mathbb{N}_0)\cup(α+a+\mathbb{N}^{-})$ for shift-invariant spaces using Toeplitz operators. Using this characterization, we determine all $α$ for which the sample set $\mathbb{N}_0\cupα+\mathbb{N}^{-}$ forms a complete interpolation set for transversal-invariant spaces. We introduce a new recurrence relation for exponential splines, examines the zeros of these splines, and explores the zero-free region of the doubly infinite Lerch zeta function. Consequently, we demonstrate that $\left\langle\frac{m}{2}\right\rangle+\mathbb{N}_0\cupα+\left\langle\frac{m}{2}\right\rangle+\mathbb{N}^{-}$ is a complete interpolation set for a shift-invariant spline space of order $m\geq 2$ if and only if $|α|<1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09099 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction of irregular complete interpolation sets for shift-invariant spaces Priyanka, Kumari Selvan, A. Antony Functional Analysis For several shift-invariant spaces, there exists a real number $a\in\mathbb{R}$ such that the set $a+\mathbb{Z}$ is a complete interpolation set. In this paper, we characterize the complete interpolation property of the set $(a+\mathbb{N}_0)\cup(α+a+\mathbb{N}^{-})$ for shift-invariant spaces using Toeplitz operators. Using this characterization, we determine all $α$ for which the sample set $\mathbb{N}_0\cupα+\mathbb{N}^{-}$ forms a complete interpolation set for transversal-invariant spaces. We introduce a new recurrence relation for exponential splines, examines the zeros of these splines, and explores the zero-free region of the doubly infinite Lerch zeta function. Consequently, we demonstrate that $\left\langle\frac{m}{2}\right\rangle+\mathbb{N}_0\cupα+\left\langle\frac{m}{2}\right\rangle+\mathbb{N}^{-}$ is a complete interpolation set for a shift-invariant spline space of order $m\geq 2$ if and only if $|α|<1/2$. |
| title | Construction of irregular complete interpolation sets for shift-invariant spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2408.09099 |