Fourier Series with the Continuous Primitive Integral

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 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 \(\hat{f}(n)=o(n)\) 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, Fejer’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 Dn 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.