arXiv subjects

Thomas Lamby

Publications and source records attributed to Thomas Lamby.

5 recordsLinked to original sources

arXiv2026

Repeated principal indefinite summation

Under suitable asymptotic and convexity conditions on a function $g\colon\mathbb{R}_+\to\mathbb{R}$, the solution to $\Delta f=g$, where $\Delta$ is the forward difference operator, is unique up to an additive constant and is called the principal indefinite sum of $g$, generalizing the additive form of Bohr-Mollerup's theorem. We consider the map $\Sigma$, which assigns to each admissible function $g$ its principal indefinite sum that vanishes at $1$, and we naturally explore its iterates, which produce repeated principal indefinite sums, in analogy with the concept of repeated indefinite integrals. Explicit formulas and convergence results are established, highlighting connections with classical combinatorial and special functions, including the multiple gamma functions, for which we also provide integral representations.

math.CA↗

arXiv2026

A Functorial Approach to Multi-Space Interpolation with Function Parameters

We introduce an extension of interpolation theory to more than two spaces by employing a functional parameter, while retaining a fully functorial and systematic framework. This approach allows for the construction of generalized intermediate spaces and ensures stability under natural operations such as powers and convex combinations. As a significant application, we demonstrate that the interpolation of multiple generalized Sobolev spaces yields a generalized Besov space. Our framework provides explicit tools for handling multi-parameter interpolation, highlighting both its theoretical robustness and practical relevance.

math.FA↗

arXiv2025

Rademacher's Theorem for Calderon-Zygmund-type Spaces

Rademacher's Theorem can be interpreted as an almost-everywhere \emph{little-$o$ improvement principle}: if a function admits a uniform pointwise first-order Lipschitz control at every point, then this control improves to a vanishing one at almost every point. In the language of Calder\'on--Zygmund pointwise spaces, this means that \[ f \in T^\infty_1(x) \quad \forall x \in \mathbb{R}^d \qquad \Longrightarrow \qquad f \in t^\infty_1(x) \quad \text{for a.e. } x \in \mathbb{R}^d. \] The purpose of this paper is to establish an analogous almost-everywhere improvement principle in a refined $L^p$ setting. We consider pointwise Calder\'on-Zygmund spaces $T^p_{\phi}(x)$ defined via polynomial approximation in $L^p$ with a function parameter $\phi$, allowing for fractional regularity indices and logarithmic corrections through Boyd functions. We prove that, under natural assumptions on $\phi$, the uniform membership \[ f \in T^p_{\phi}(x) \quad \forall x \in E \] on a measurable set $E \subset \mathbb{R}^d$ implies an almost-everywhere improvement to a vanishing approximation rate, namely \[ f \in t^p_{\phi,n+1}(x) \quad \text{for a.e. } x \in E, \] where $n < \underline{b}(\phi) \leq \overline{b}(\phi) < n+1$. The proof combines measurability arguments, a generalized Whitney extension theorem, and fine properties of Sobolev spaces. We also show that this result is essentially sharp: in general, one cannot expect almost-everywhere membership in $t^p_{\phi,n}(x)$ for fractional indices, and explicit counterexamples are provided.

math.FA↗

arXiv2025

Newton series representation of completely monotone functions

We prove that every completely monotone function defined on a right-unbounded open interval admits a Newton series expansion at every point of that interval. This result can be viewed as an analog of Bernstein's little theorem for absolutely monotone functions. As an application, we use it to study principal indefinite sums, which are constructed via a broad generalization of Bohr-Mollerup's theorem.

math.CA↗

arXiv2025

The Thomae Function: Fractal Insights

This article examines the Thomae function, a paradigmatic example of a function that is continuous on the irrationals and discontinuous elsewhere. Defined for a parameter $θ>0$, it exhibits a rich self-similar structure and intriguing regularity properties. After revisiting its fundamental characteristics, we analyze its Hölder continuity, emphasizing the interplay between its discrete spikes and its behavior on dense subsets of the real line. This study provides a refined perspective on the irregularity of the Thomae function, using classical analytical tools to elucidate its fractal nature.

math.GM↗