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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.17019 |
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| _version_ | 1866913668520738816 |
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| author | Lacunza, Diego Castelli Long, Carlos A. Sing |
| author_facet | Lacunza, Diego Castelli Long, Carlos A. Sing |
| contents | Resolving the details of an object from coarse-scale measurements is a classical problem in applied mathematics. This problem is usually formulated as extrapolating the Fourier transform of the object from a bounded region to the entire space, that is, in terms of performing extrapolation in frequency. This problem is ill-posed unless one assumes that the object has some additional structure. When the object is compactly supported, then it is well-known that its Fourier transform can be extended to the entire space. However, it is also well-known that this problem is severely ill-conditioned.
In this work, we assume that the object is known to belong to a collection of compactly supported functions and, instead performing extrapolation in frequency to the entire space, we study the problem of extrapolating to a larger bounded set using dilations in frequency and a single Fourier multiplier. This is reminiscent of the refinement equation in multiresolution analysis. Under suitable conditions, we prove the existence of a worst-case optimal multiplier over the entire collection, and we show that all such multipliers share the same canonical structure. When the collection is finite, we show that any worst-case optimal multiplier can be represented in terms of an Hermitian matrix. This allows us to introduce a fixed-point iteration to find the optimal multiplier. This leads us to introduce a family of multipliers, which we call $Σ$-multipliers, that can be used to perform extrapolation in frequency. We establish connections between $Σ$-multipliers and multiresolution analysis. We conclude with some numerical experiments illustrating the practical consequences of our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17019 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adaptive multipliers for extrapolation in frequency Lacunza, Diego Castelli Long, Carlos A. Sing Functional Analysis Numerical Analysis Signal Processing Resolving the details of an object from coarse-scale measurements is a classical problem in applied mathematics. This problem is usually formulated as extrapolating the Fourier transform of the object from a bounded region to the entire space, that is, in terms of performing extrapolation in frequency. This problem is ill-posed unless one assumes that the object has some additional structure. When the object is compactly supported, then it is well-known that its Fourier transform can be extended to the entire space. However, it is also well-known that this problem is severely ill-conditioned. In this work, we assume that the object is known to belong to a collection of compactly supported functions and, instead performing extrapolation in frequency to the entire space, we study the problem of extrapolating to a larger bounded set using dilations in frequency and a single Fourier multiplier. This is reminiscent of the refinement equation in multiresolution analysis. Under suitable conditions, we prove the existence of a worst-case optimal multiplier over the entire collection, and we show that all such multipliers share the same canonical structure. When the collection is finite, we show that any worst-case optimal multiplier can be represented in terms of an Hermitian matrix. This allows us to introduce a fixed-point iteration to find the optimal multiplier. This leads us to introduce a family of multipliers, which we call $Σ$-multipliers, that can be used to perform extrapolation in frequency. We establish connections between $Σ$-multipliers and multiresolution analysis. We conclude with some numerical experiments illustrating the practical consequences of our results. |
| title | Adaptive multipliers for extrapolation in frequency |
| topic | Functional Analysis Numerical Analysis Signal Processing |
| url | https://arxiv.org/abs/2501.17019 |