Relations between L^p- and Pointwise Convergence of Families of Functions Indexed by the Unit Interval

2012/02/26 by Vaios Laschos, Christian Mönch, Laschos, Vaios +1
Mathematics · #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis #math.CA

paper · pdf · doi:10.48550/arxiv.1202.5792

13 pages

openalex publication_date 2012/02/26 · arxiv created 2012/07/15 · arxiv updated 2012/07/17 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

We construct a variety of mappings of the unit interval into L^p([0,1]) to generalize classical examples of L^p-convergence of sequences of functions with simultaneous pointwise divergence. By establishing relations between the regularity of the functions in the image of the mappings and the topology of [0,1], we obtain examples which are L^p-continuous but exhibit discontinuity in a pointwise sense to different degrees. We conclude by proving an Egorov-type theorem, namely that if almost every function in the image is continuous, then we can remove a set of arbitrarily small measure from the index set [0,1] and establish pointwise limits for all functions in the remaining image.

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