Frame Spectral Pairs and Exponential Bases
| Autor: | Azita Mayeli, Christina Frederick |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Lebesgue measure
Applied Mathematics General Mathematics 010102 general mathematics Frame (networking) 010103 numerical & computational mathematics Link (geometry) Discrete set Lambda 01 natural sciences Functional Analysis (math.FA) Exponential function Sampling theory Mathematics - Functional Analysis Combinatorics symbols.namesake Mathematics - Classical Analysis and ODEs Fourier analysis Classical Analysis and ODEs (math.CA) FOS: Mathematics symbols 0101 mathematics Analysis Mathematics |
| Zdroj: | Journal of Fourier Analysis and Applications. 27 |
| ISSN: | 1531-5851 1069-5869 |
| DOI: | 10.1007/s00041-021-09872-9 |
| Popis: | Given a domain $$\varOmega \subset {\mathbb {R}}^d$$ with positive and finite Lebesgue measure and a discrete set $$\varLambda \subset {\mathbb {R}}^d$$ , we say that $$(\varOmega , \varLambda )$$ is a frame spectral pair if the set of exponential functions $${\mathcal {E}}(\varLambda ):=\{e^{2\pi i \lambda \cdot x}: \lambda \in \varLambda \}$$ is a frame for $$L^2(\varOmega )$$ . Special cases of frames include Riesz bases and orthogonal bases. In the finite setting $${\mathbb {Z}}_N^d$$ , $$d, N\ge 1$$ , a frame spectral pair can be similarly defined. In this paper we show how to construct and obtain new classes of frame spectral pairs in $${\mathbb {R}}^d$$ by “adding” a frame spectral pair in $${\mathbb {R}}^{d}$$ to a frame spectral pair in $${\mathbb {Z}}_N^d$$ . Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. We also remark on the link between the spectral property of a domain and sampling theory. |
| Databáze: | OpenAIRE |
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