A novel method of computing the volume of hyperspheres in finite and infinite dimensions

TL;DR

This paper introduces a novel approach to calculating the volume of hyperspheres in finite and infinite dimensions using generalized functions and distributional Fourier transforms, providing new insights into high-dimensional geometry.

Contribution

It presents an original method employing tempered distributions and regularization techniques to compute hypersphere volumes in both finite and infinite-dimensional spaces.

Findings

Recalculated hypersphere volumes in finite dimensions using generalized functions.

Extended the method to infinite dimensions, deriving an anti-measure in Hilbert space.

Demonstrated the applicability of distributional Fourier transforms in geometric measure calculations.

Abstract

In the present article, the volume of the hypersphere in n-dimensional euclidean space is recalculated in a rather original way by using the theory of generalized functions (tempered distributions). The calculation is performed by applying the integral representation of the Heaviside unit step function and furthermore by using the distributional Fourier transform of general power-law functions. The same method (added by a regularization procedure of the occurring infinite-dimensional integral via the Riemann-zeta function) is applied in the case of infinite dimensions giving rise to some sort of anti-measure of spheres in the Hilbert space of square-summable complex sequences.