Schoenberg's Theorem via the law of large numbers

2005/04/29 by Davar Khoshnevisan, Khoshnevisan, Davar
Computer Science · Mathematics · Physics and Astronomy · #43A35 #60F-xx #Advanced Mathematical Theories and Applications #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Music Technology and Sound Studies #Probability (math.PR) #math.CA #math.PR #msc:43A35 #msc:60F-xx

paper · pdf · doi:10.48550/arxiv.math/0504603

Some errors and misprints corrected; new references have been added

openalex publication_date 2005/04/29 · arxiv created 2005/05/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A classical theorem of S. Bochner states that a function f:Rⁿ → C is the Fourier transform of a finite Borel measure if and only if f is positive definite. In 1938, I. Schoenberg found a beautiful complement to Bochner's theorem. We present a non-technical derivation of of Schoenberg's theorem that relies chiefly on the de Finneti theorem and the law of large numbers of classical probability theory.

Citations

• Positive definite distributions and subspaces of L_-p with applications to stable processes

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