2011/05/27 by Erik Talvila, Talvila, Erik
Mathematics · #26A39 #42A16 #46F10 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #math.CA #msc:26A39 #msc:42A16 #msc:46F10
paper · pdf · doi:10.48550/arxiv.1105.5620
To appear in {\it Journal of Fourier Analysis and Applications}
arxiv created 2011/05/27 · openalex publication_date 2011/05/27 · arxiv updated 2011/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted \alext and is a Banach space under the Alexiewicz norm, ‖f‖_\T =sup|I|≤ 2π|∫I f|, the supremum being taken over intervals of length not exceeding 2π. It contains the periodic functions integrable in the sense of Lebesgue and Henstock-Kurzweil. Many of the properties of L1 Fourier series continue to hold for this larger space, with the L1 norm replaced by the Alexiewicz norm. The Riemann-Lebesgue lemma takes the form \fhat(n)=o(n) as |n|→∞. The convolution is defined for f∈\alext and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate ‖f∗ g‖_∞≤ ‖f‖_\T ‖g‖_\bv. For g∈ L1(\T), ‖f∗ g‖_\T≤ ‖f‖_\T ‖g‖1. As well, \widehatf∗ g(n)=\fhatn g(n). There are versions of the Salem-Zygmund-Rudin-Cohen factorization theorem, Fejér's lemma and the Parseval equality. The trigonometric polynomials are dense in \alext. The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let Dn be the Dirichlet kernel and let f∈ L1(\T). Then ‖Dn∗ f-f‖_\T→ 0 as n→∞. Fourier coefficients of functions of bounded variation are characterized. An appendix contains a type of Fubini theorem.