2021/07/22 by Cyril Belardinelli, Belardinelli, Cyril
Mathematics · Physics and Astronomy · #01A90 #46F10 #Advanced Mathematical Theories and Applications #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical and Theoretical Analysis #Scientific Research and Discoveries
paper · pdf · doi:10.48550/arxiv.2107.14032
openalex publication_date 2021/07/22 · openalex created_date 2021/08/02 · openalex updated_date 2026/07/28
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.