On Homogeneous Decomposition Spaces and Associated Decompositions of Distribution Spaces

2017/11/06 by Zeineb Al‐Jawahri, Zeineb Al-Jawahri, Morten Nielsen +2
Mathematics · #41A17 #42B35 #42C15 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #math.FA #msc:41A17 #msc:42B35 #msc:42C15

paper · pdf · doi:10.48550/arxiv.1711.01902

27 pages

openalex publication_date 2017/11/06 · arxiv created 2017/12/19 · arxiv updated 2017/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A new construction of decomposition smoothness spaces of homogeneous type is considered. The smoothness spaces are based on structured and flexible decompositions of the frequency space ℝ^d\backslash\0\. We construct simple adapted tight frames for L₂(ℝ^d) that can be used to fully characterise the smoothness norm in terms of a sparseness condition imposed on the frame coefficients. Moreover, it is proved that the frames provide a universal decomposition of tempered distributions with convergence in the tempered distributions modulo polynomials. As an application of the general theory, the notion of homogeneous α-modulation spaces is introduced.

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