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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.02741 |
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| _version_ | 1866917975833968640 |
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| author | Gonçalves, Felipe Vedana, Guilherme |
| author_facet | Gonçalves, Felipe Vedana, Guilherme |
| contents | We completely classify Fourier summation formulas of the form $$
\int_{\mathbb{R}} \widehatφ(t) dμ(t)=\sum_{n=0}^{\infty} a(λ_n)φ(λ_n), $$ that hold for any test function $φ$, where $\widehatφ$ is the Fourier transform of $φ$, $μ$ is a fixed complex measure on $\mathbb{R}$ and $a:\{λ_n\}_{n\geq 0}\to\mathbb{C}$ is a fixed function. We only assume the decay condition $$ \int_{\mathbb{R}} \frac{d |μ|(t)}{(1+t^2)^{c_1}} + \sum_{n\geq 0} |a(λ_n)|e^{-c_2 |λ_n|}<\infty, $$ for some $c_1,c_2>0$. This completes the work initiated by the first author previously, where the condition $c_1\leq 1$ was required. We prove that any such pair $(μ,a)$ can be uniquely associated with a holomorphic map $F(z)$ in the upper-half space that is both almost periodic and belongs to a certain higher index Nevanlinna class. The converse is also true: For any such function $F$ it is possible to generate a Fourier summation pair $(μ,a)$. We provide important examples of such summation formulas not contemplated by the previous results, such as Selberg's trace formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02741 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Complete Classification of Fourier Summation Formulas on the real line Gonçalves, Felipe Vedana, Guilherme Classical Analysis and ODEs Metric Geometry Number Theory 52C23, 30D10 We completely classify Fourier summation formulas of the form $$ \int_{\mathbb{R}} \widehatφ(t) dμ(t)=\sum_{n=0}^{\infty} a(λ_n)φ(λ_n), $$ that hold for any test function $φ$, where $\widehatφ$ is the Fourier transform of $φ$, $μ$ is a fixed complex measure on $\mathbb{R}$ and $a:\{λ_n\}_{n\geq 0}\to\mathbb{C}$ is a fixed function. We only assume the decay condition $$ \int_{\mathbb{R}} \frac{d |μ|(t)}{(1+t^2)^{c_1}} + \sum_{n\geq 0} |a(λ_n)|e^{-c_2 |λ_n|}<\infty, $$ for some $c_1,c_2>0$. This completes the work initiated by the first author previously, where the condition $c_1\leq 1$ was required. We prove that any such pair $(μ,a)$ can be uniquely associated with a holomorphic map $F(z)$ in the upper-half space that is both almost periodic and belongs to a certain higher index Nevanlinna class. The converse is also true: For any such function $F$ it is possible to generate a Fourier summation pair $(μ,a)$. We provide important examples of such summation formulas not contemplated by the previous results, such as Selberg's trace formula. |
| title | A Complete Classification of Fourier Summation Formulas on the real line |
| topic | Classical Analysis and ODEs Metric Geometry Number Theory 52C23, 30D10 |
| url | https://arxiv.org/abs/2504.02741 |