On summability of distributions and spectral geometry

TL;DR

This paper explores the asymptotic behaviors of distributions and spectral densities, demonstrating how Cesaro methods facilitate calculations in spectral geometry and validating a bosonic action functional.

Contribution

It introduces a novel asymptotic analysis framework for spectral densities using Cesaro methods, linking distribution summability to spectral geometry applications.

Findings

Cesaro and parametric behaviors are equivalent modulo moment asymptotic expansion.

Cesaro developments enable efficient calculation of spectral expansion coefficients.

The bosonic action functional by Chamseddine and Connes is validated as a Cesaro asymptotic development.

Abstract

Modulo the moment asymptotic expansion, the Cesaro and parametric behaviours of distributions at infinity are equivalent. On the strength of this result, we construct the asymptotic analysis for spectral densities, arising from elliptic pseudodifferential operators. We show how Cesaro developments lead to efficient calculations of the expansion coefficients of counting number functionals and Green functions. The bosonic action functional proposed by Chamseddine and Connes can more generally be validated as a Cesaro asymptotic development.