Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.08422

The Mehler-Fock transform of arbitrary order

Sudhanshu Ranjan·Akhilesh Prasad
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we construct the Mehler-Fock transform of arbitrary order MF(mu,k) by removing the restriction Re(mu) > -1/2 with the help of a positive integer k such that Re(mu+k) > -1/2 and obtain some transformation formulae based on the differential operator M(mu)N(mu). Furthermore, we define the Mehler-Fock translation and convolution operators of arbitrary order and demonstrate that the factorization property for the convolution holds in the test function space H(mu).

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sudhanshu Ranjan, Akhilesh Prasad. 2026-10-06. The Mehler-Fock transform of arbitrary order. https://arxiv.org/abs/2610.08422

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Operations on $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathrm{Alt}^{\bullet}(U,M)$. Stokes theorem for locally convex linear space valued forms. Divergence theorem for locally convex linear space valued vector fields

We extend to $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathfrak{T}_{\bullet}^{\bullet}(U,M)$ the usual operations defined on $\mathfrak{T}_{\bullet}^{\bullet}(U,M)$, then by employing the properties of the projective tensor product of locally convex spaces, we generalize to $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathrm{Alt}^{\bullet}(U,M)$ the usual wedge product, insertion operator and exterior differential defined on $\mathrm{Alt}^{\bullet}(U,M)$ and establish their properties. Here $M$ is a smooth finite dimensional manifold, $U$ an open submanifold of $M$, $\mathcal{A}(U)$ the ring of smooth maps on $U$, $\mathfrak{T}_{\bullet}^{\bullet}(U,M)$ the $\mathcal{A}(U)$-module of smooth tensor fields of $M$ defined on $U$, $\mathrm{Alt}^{\bullet}(U,M)$ the $\mathcal{A}(U)$-module of smooth alternating tensor fields of $M$ defined on $U$, while $\mathscr{M}$ is a $\mathcal{A}(U)$-module. $\mathscr{M}$ depending by the operation might be either general, or one of the following function spaces: $\mathscr{L}_{c}^{1}(U,G,λ)$, $\mathscr{L}_{loc}^{1}(U,G,λ)$, $\mathcal{B}^{p}(U,G)$ with $G$ a Hausdorff locally convex space. This framenwork permits to construct the divergence of a $G$-valued $p$-times continuously differentiable vector field, to define the weak integral of $G$-valued compactly supported scalarly integrable maximal forms, to associate a $G$-valued measure with any locally integrable $G$-valued form, to establish a Stokes type theorem for $G$-valued compactly supported $p$-times continuously differentiable forms and to obtain a divergence type theorem for $G$-valued $p$-times continously differentiable vector fields.

math.FA↗
arXiv2026

The Grothendieck Constant is Strictly Larger than Davie-Reeds' Bound

The Grothendieck constant $K_{G}$ is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of $K_{G}$ is unknown. The best known lower bound on $K_{G}$ was obtained independently by Davie and Reeds in the 1980s. In this paper we show that their bound is not optimal. We prove that $K_{G} \ge K_{DR} + 10^{-12}$, where $K_{DR}$ denotes the Davie-Reeds lower bound. Our argument is based on a perturbative analysis of the Davie-Reeds operator. We show that every near-extremizer for the Davie-Reeds problem has $Ω(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.

math.FA↗
arXiv2026

Higher-order differentiability of Korevaar--Schoen energy forms and energy measures

In this paper, we investigate the differentiability of Korevaar--Schoen $p$-energy forms and the associated $p$-energy measures. We obtain higher-order derivatives by virtue of explicit realizations of Korevaar--Schoen $p$-energy forms and the associated $p$-energy measures as subsequential pointwise limits of certain double integrals.

math.FA↗