Spectral synthesis with the complexity parameter
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912984389910528 |
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| author | Deodhar, S. Iosevich, A. |
| author_facet | Deodhar, S. Iosevich, A. |
| contents | We show that spectral synthesis thresholds are governed by a quantitative spectral complexity parameter, the Fourier Ratio, in addition to the geometric size of the Fourier support. In the Euclidean setting, we prove that if a compactly supported measure has finite $α$-dimensional packing measure and the associated Fourier ratio decays with asymptotic exponent $κ$, then the classical synthesis threshold improves from $\frac{2d}α$ to $\frac{2(d-2κ)}{α-2κ}$. We then establish an analogous result on compact Riemannian manifolds without boundary. In that setting the relevant object is a localized spectral Fourier ratio defined using Laplace--Beltrami spectral projectors. The resulting synthesis threshold is again determined by the decay exponent of this complexity parameter. These results place Euclidean and manifold spectral synthesis into a common framework in which geometric size and spectral complexity jointly govern uniqueness |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_25998 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral synthesis with the complexity parameter Deodhar, S. Iosevich, A. Classical Analysis and ODEs Spectral Theory 42B10 42B37, 58J40, 35A02 We show that spectral synthesis thresholds are governed by a quantitative spectral complexity parameter, the Fourier Ratio, in addition to the geometric size of the Fourier support. In the Euclidean setting, we prove that if a compactly supported measure has finite $α$-dimensional packing measure and the associated Fourier ratio decays with asymptotic exponent $κ$, then the classical synthesis threshold improves from $\frac{2d}α$ to $\frac{2(d-2κ)}{α-2κ}$. We then establish an analogous result on compact Riemannian manifolds without boundary. In that setting the relevant object is a localized spectral Fourier ratio defined using Laplace--Beltrami spectral projectors. The resulting synthesis threshold is again determined by the decay exponent of this complexity parameter. These results place Euclidean and manifold spectral synthesis into a common framework in which geometric size and spectral complexity jointly govern uniqueness |
| title | Spectral synthesis with the complexity parameter |
| topic | Classical Analysis and ODEs Spectral Theory 42B10 42B37, 58J40, 35A02 |
| url | https://arxiv.org/abs/2603.25998 |