NDLI: Fourier Series with the Continuous Primitive Integral

Content Provider	SpringerLink
Author	Talvila, Erik
Copyright Year	2011
Abstract	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 ${\mathcal{A}}_{c}(\mathbb{T})$ and is a Banach space under the Alexiewicz norm, $\\|f\\|_{\mathbb{T}}=\sup_{\|I\|\leq2\pi}\|\int_{I} f\|$ , 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 L 1 Fourier series continue to hold for this larger space, with the L 1 norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form $\hat{f}(n)=o(n)$ as \|n\|→∞. The convolution is defined for $f\in{\mathcal{A}}_{c}(\mathbb{T})$ and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate $\\|f\ast g\\|_{\infty}\leq\\|f\\|_{\mathbb{T}} \\|g\\|_{\mathcal{BV}}$ . For $g\in L^{1}(\mathbb{T})$ , $\\|f\ast g\\|_{\mathbb{T}}\leq\\|f\\|_{\mathbb {T}} \\|g\\|_{1}$ . As well, $\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n)$ . There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The trigonometric polynomials are dense in ${\mathcal{A}}_{c}(\mathbb{T})$ . The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D n be the Dirichlet kernel and let $f\in L^{1}(\mathbb{T})$ . Then $\\|D_{n}\ast f-f\\|_{\mathbb{T}}\to0$ as n→∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem.
Ending Page	44
Page Count	18
Starting Page	27
File Format	PDF
ISSN	10695869
e-ISSN	15315851
Journal	Journal of Fourier Analysis and Applications
Issue Number	1
Volume Number	18
Language	English
Publisher	SP Birkhäuser Verlag Boston
Publisher Date	2011-07-01
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Schwartz distribution Signal, Image and Speech Processing Generalized function Fourier series Continuous primitive integral Henstock–Kurzweil integral Distributional integral Abstract Harmonic Analysis Fourier coefficients, Fourier series of functions with special properties, special Fourier series Mathematical Methods in Physics Convolution Fourier Analysis Operations with distributions Approximations and Expansions Denjoy and Perron integrals, other special integrals Partial Differential Equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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