Coefficients de Fourier de fonctions à variation bornée
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- by Jie Wu PDF
- Proc. Amer. Math. Soc. 117 (1993), 689-690 Request permission
Abstract:
Let $f:\mathbb {R} \to \mathbb {C}$ be a function of period $2\pi$ and of bounded variation on $[0,2\pi ]$ with the total variation $V(f)$. Its Fourier coefficients are denoted by $\hat f(n)$. In this short note, we give a very simple proof of the known result $|\hat f(n)| \leqslant V(f)/2\pi |n|\;(n \in \mathbb {Z},\;n \ne 0)$.References
- R. E. Edwards, Fourier series. A modern introduction. Vol. 1, 2nd ed., Graduate Texts in Mathematics, vol. 64, Springer-Verlag, New York-Berlin, 1979. MR 545506, DOI 10.1007/978-1-4612-6208-4
- G. H. Hardy and W. W. Rogosinski, Fourier Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 38, Cambridge University Press, 1944. MR 0010206
- Mitchell Taibleson, Fourier coefficients of functions of bounded variation, Proc. Amer. Math. Soc. 18 (1967), 766. MR 212477, DOI 10.1090/S0002-9939-1967-0212477-6
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 117 (1993), 689-690
- MSC: Primary 42A16
- DOI: https://doi.org/10.1090/S0002-9939-1993-1150658-3
- MathSciNet review: 1150658