Arithmetic from Weighted Operators on Gaussian Spaces
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2025
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| _version_ | 1866901470665768960 |
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| author | Cabrera Vera, Kittim Lorena |
| author_facet | Cabrera Vera, Kittim Lorena |
| contents | <p> </p> <p>This manuscript develops a functional-analytic reconstruction of classical arithmetic from operator actions on a Gaussian-weighted Hilbert space. We introduce a weighted space <span><span>L2(R,w)L^{2}(\mathbb{R}, w)</span><span><span><span><span>L</span><span><span><span><span><span><span>2</span></span></span></span></span></span></span><span>(</span><span>R</span><span>,</span><span>w</span><span>)</span></span></span></span> with <span><span>w(t)=φ−t2w(t)=\varphi^{-t^{2}}</span><span><span><span>w</span><span>(</span><span>t</span><span>)</span><span>=</span></span><span><span><span>φ</span><span><span><span><span><span><span>−<span>t</span><span>2</span></span></span></span></span></span></span></span></span></span></span>, define a dense Sobolev domain <span><span>Hw1(R)H^{1}_{w}(\mathbb{R})</span><span><span><span><span>H</span><span><span><span><span><span><span><span>w</span></span></span><span><span>1</span></span></span><span></span></span></span></span></span><span>(</span><span>R</span><span>)</span></span></span></span>, and equip it with a commuting operator triple <span><span>S,E,RS, E, R</span><span><span><span>S</span><span>,</span><span>E</span><span>,</span><span>R</span></span></span></span> satisfying natural normalization and boundary conditions. A continuous boundary projection <span><span>Π(f)=f(0)\Pi(f)=f(0)</span><span><span><span>Π</span><span>(</span><span>f</span><span>)</span><span>=</span></span><span><span>f</span><span>(</span><span>0</span><span>)</span></span></span></span>, justified via Sobolev embedding, allows classical addition, multiplication, negation, limits, and analytic continuation to emerge as boundary values of operator expressions. Gaussian convolution is shown to be bounded, entire-preserving, and stable under the operator algebra, and symmetric functional equations <span><span>F(s)=F(1−s)F(s)=F(1-s)</span><span><span><span>F</span><span>(</span><span>s</span><span>)</span><span>=</span></span><span><span>F</span><span>(</span><span>1</span><span>−</span></span><span><span>s</span><span>)</span></span></span></span> are preserved. The resulting structure is internally consistent, compatible with standard functional analysis, and provides a rigorous analytic framework suitable for subsequent work on Dirichlet series, zeta-functions, and related problems in analytic number theory.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17782277 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Arithmetic from Weighted Operators on Gaussian Spaces Cabrera Vera, Kittim Lorena functional analysis operator theory weighted sobolev spaces gaussian convolution <p> </p> <p>This manuscript develops a functional-analytic reconstruction of classical arithmetic from operator actions on a Gaussian-weighted Hilbert space. We introduce a weighted space <span><span>L2(R,w)L^{2}(\mathbb{R}, w)</span><span><span><span><span>L</span><span><span><span><span><span><span>2</span></span></span></span></span></span></span><span>(</span><span>R</span><span>,</span><span>w</span><span>)</span></span></span></span> with <span><span>w(t)=φ−t2w(t)=\varphi^{-t^{2}}</span><span><span><span>w</span><span>(</span><span>t</span><span>)</span><span>=</span></span><span><span><span>φ</span><span><span><span><span><span><span>−<span>t</span><span>2</span></span></span></span></span></span></span></span></span></span></span>, define a dense Sobolev domain <span><span>Hw1(R)H^{1}_{w}(\mathbb{R})</span><span><span><span><span>H</span><span><span><span><span><span><span><span>w</span></span></span><span><span>1</span></span></span><span></span></span></span></span></span><span>(</span><span>R</span><span>)</span></span></span></span>, and equip it with a commuting operator triple <span><span>S,E,RS, E, R</span><span><span><span>S</span><span>,</span><span>E</span><span>,</span><span>R</span></span></span></span> satisfying natural normalization and boundary conditions. A continuous boundary projection <span><span>Π(f)=f(0)\Pi(f)=f(0)</span><span><span><span>Π</span><span>(</span><span>f</span><span>)</span><span>=</span></span><span><span>f</span><span>(</span><span>0</span><span>)</span></span></span></span>, justified via Sobolev embedding, allows classical addition, multiplication, negation, limits, and analytic continuation to emerge as boundary values of operator expressions. Gaussian convolution is shown to be bounded, entire-preserving, and stable under the operator algebra, and symmetric functional equations <span><span>F(s)=F(1−s)F(s)=F(1-s)</span><span><span><span>F</span><span>(</span><span>s</span><span>)</span><span>=</span></span><span><span>F</span><span>(</span><span>1</span><span>−</span></span><span><span>s</span><span>)</span></span></span></span> are preserved. The resulting structure is internally consistent, compatible with standard functional analysis, and provides a rigorous analytic framework suitable for subsequent work on Dirichlet series, zeta-functions, and related problems in analytic number theory.</p> |
| title | Arithmetic from Weighted Operators on Gaussian Spaces |
| topic | functional analysis operator theory weighted sobolev spaces gaussian convolution |
| url | https://doi.org/10.5281/zenodo.17782277 |