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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2605.04351 |
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| _version_ | 1866911651985358848 |
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| author | Santacana, Andreu Ballus |
| author_facet | Santacana, Andreu Ballus |
| contents | We classify positive linear functionals on $C_c(\mathbb{R}_{>0})$ satisfying scaling covariance of degree $x/2$ and Gaussian normalization to $π^{x/2}$. We prove that the unique such functionals are represented by the Mellin--Gamma measures \[ dμ_x(u) = \frac{π^{x/2}}{Γ(x/2)}\, u^{x/2 - 1}\, du, \quad x > 0. \] The result is a rigidity statement: the Mellin--Gamma structure is forced by the axioms, without assuming analytic continuation, special functions, or a priori formulas. The proof reduces the scaling condition, via a logarithmic change of variables, to translation invariance on $\mathbb{R}$, where Haar measure uniqueness determines the measure up to normalization, which is fixed by the Gaussian integral.
As a consequence, the Euclidean ball volume formula \[ V(x) = \frac{π^{x/2}}{Γ(x/2 + 1)} \] is recovered as the mass of the unit interval. We further analyze the induced dimension-shift structure, identifying two multiplicative cocycles whose ratio is a coboundary given by the dimension function $x$, and give an independent characterization via a shifted Bohr--Mollerup theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04351 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Radial Integration in Continuous Dimension: A Mellin-Gamma Classification of Euclidean Ball Volume Santacana, Andreu Ballus Classical Analysis and ODEs Functional Analysis 28A25, 44A15, 46E27 We classify positive linear functionals on $C_c(\mathbb{R}_{>0})$ satisfying scaling covariance of degree $x/2$ and Gaussian normalization to $π^{x/2}$. We prove that the unique such functionals are represented by the Mellin--Gamma measures \[ dμ_x(u) = \frac{π^{x/2}}{Γ(x/2)}\, u^{x/2 - 1}\, du, \quad x > 0. \] The result is a rigidity statement: the Mellin--Gamma structure is forced by the axioms, without assuming analytic continuation, special functions, or a priori formulas. The proof reduces the scaling condition, via a logarithmic change of variables, to translation invariance on $\mathbb{R}$, where Haar measure uniqueness determines the measure up to normalization, which is fixed by the Gaussian integral. As a consequence, the Euclidean ball volume formula \[ V(x) = \frac{π^{x/2}}{Γ(x/2 + 1)} \] is recovered as the mass of the unit interval. We further analyze the induced dimension-shift structure, identifying two multiplicative cocycles whose ratio is a coboundary given by the dimension function $x$, and give an independent characterization via a shifted Bohr--Mollerup theorem. |
| title | Radial Integration in Continuous Dimension: A Mellin-Gamma Classification of Euclidean Ball Volume |
| topic | Classical Analysis and ODEs Functional Analysis 28A25, 44A15, 46E27 |
| url | https://arxiv.org/abs/2605.04351 |