On the product of periodic distributions. Product in shift-invariant spaces

2024/03/15 by Aleksandar Aksentijević, Aksentijević, Aleksandar, Suzana Aleksić +3
Mathematics · #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2403.10350

openalex publication_date 2024/03/15 · openalex created_date 2024/03/19 · openalex updated_date 2026/07/28

Abstract

We connect through the Fourier transform shift-invariant Sobolev type spaces V_s⊂ H^s, s∈\mathbb R, and the spaces of periodic distributions and analyze the properties of elements in such spaces with respect to the product. If the series expansions of two periodic distributions have compatible coefficient estimates, then their product is a periodic tempered distribution. We connect product of tempered distributions with the product of shift-invariant elements of V_s. The idea for the analysis of products comes from the Hörmander's description of the Sobolev type wave front in connection with the product of distributions. Coefficient compatibility for the product of f and g in the case of "good" position of their Sobolev type wave fronts is proved in the 2-dimensional case. For larger dimension it is an open problem because of the difficulties on the description of the intersection of cones in dimension d\geqslant3.

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