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- W2950691892 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 $alext$ and is a Banach space under the Alexiewicz norm, $|f|_T =sup_{|I|leq 2pi}|int_I f|$, the supremum being taken over intervals of length not exceeding $2pi$. 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 $fhat(n)=o(n)$ as $|n|toinfty$. The convolution is defined for $finalext$ and $g$ a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate $|fast g|_inftyleq |f|_T |g|_bv$. For $gin L^1(T)$, $|fast g|_Tleq |f|_T |g|_1$. As well, $widehat{fast g}(n)=fhatn hat{g}(n)$. There are versions of the Salem-Zygmund-Rudin-Cohen factorization theorem, Fej'er'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 $D_n$ be the Dirichlet kernel and let $fin L^1(T)$. Then $|D_nast f-f|_Tto 0$ as $ntoinfty$. Fourier coefficients of functions of bounded variation are characterized. An appendix contains a type of Fubini theorem." @default.
- W2950691892 created "2019-06-27" @default.
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- W2950691892 date "2011-05-27" @default.
- W2950691892 modified "2023-09-27" @default.
- W2950691892 title "Fourier series with the continuous primitive integral" @default.
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