Continuation of Direct Products of Distributions

2000/01/18 by A. Petermann, Arne Petermann, Petermann, A.
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical and Theoretical Analysis #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.math-ph/0001025

3 pages, latex

arxiv created 2000/01/18 · openalex publication_date 2000/01/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If, in some problems, one has to deal with the ``product'' of distributions \rm f_i (also called generalized functions) \rm T = Π^m_i=1 f_i, this product has a priori no definite meaning as a functional (\rm T, ϕ) for \rmϕ∈ S. But if \rm x^κ+1 Π^m_i=1 f_i exists, whatever the associativity is between some powers \rm r_i of \rm x (\rm r_i ∈ \Bbb N, ∑_i r_i≤ κ+1, r_i ≥ 0) and the various \rm f_i, then a continuation of the linear functional \rm T from \rm M onto \rm S^(N) for some \rm N is shown to exist in such a way that \rm x^κ+1 T is defined unambiguously, and \rm ( T, ϕ), ϕ∈ S, significant, though not unique.

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