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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Braun, W., Feichtinger, H.G.: Banach spaces of distributions having two module structures. J. Funct. Anal. 51, 174–212 (1983)
Čelidze, V.G., Džvaršeı̌švili, A.G.: The theory of the Denjoy integral and some applications. World Scientific, Singapore (1989) (Trans. P.S. Bullen)
Dales, H.G., Lau, A.T.-M.: The second duals of Beurling algebras. Mem. Amer. Math. Soc. 177(836) (2005)
Edwards, R.E.: Fourier Series: A Modern Introduction, vol. I. Springer, New York (1979)
Edwards, R.E.: Fourier Series: A Modern Introduction, vol. II. Springer, New York (1982)
Friedlander, F.G., Joshi, M.: Introduction to the Theory of Distributions. Cambridge University Press, Cambridge (1999)
Grafakos, L.: Classical Fourier Analysis. Springer, New York (2008)
Grafakos, L.: Modern Fourier Analysis. Springer, New York (2009)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. II. Springer, Berlin (1970)
Jeffery, R.L.: The Theory of Functions of a Real Variable. University of Toronto Press, Toronto (1951)
Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge University Press, Cambridge (2004)
Reiter, H., Stegeman, J.D.: Classical Harmonic Analysis and Locally Compact Groups. Oxford University Press, Oxford (2000)
Rudin, W.: Real and Complex Analysis. McGraw–Hill, New York (1987)
Talvila, E.: Henstock–Kurzweil Fourier transforms. Ill. J. Math. 46, 1207–1226 (2002)
Talvila, E.: The distributional Denjoy integral. Real Anal. Exch. 33, 51–82 (2008)
Talvila, E.: Convolutions with the continuous primitive integral. Abstr. Appl. Anal. 2009, 307404 (2009) 18 pp.
Talvila, E.: The regulated primitive integral. Ill. J. Math. 53, 1187–1219 (2009)
Titchmarsh, E.C.: The order of magnitude of the coefficients in a generalised Fourier series. Proc. Lond. Math. Soc. (2) 22, xxv–xxvi (1923/1924)
Zemanian, A.H.: Distribution Theory and Transform Analysis. Dover, New York (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hans G. Feichtinger.
Supported by the Natural Sciences and Engineering Research Council of Canada.
Rights and permissions
About this article
Cite this article
Talvila, E. Fourier Series with the Continuous Primitive Integral. J Fourier Anal Appl 18, 27–44 (2012). https://doi.org/10.1007/s00041-011-9183-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-011-9183-4
Keywords
- Fourier series
- Convolution
- Distributional integral
- Continuous primitive integral
- Henstock–Kurzweil integral
- Schwartz distribution
- Generalized function