Fourier series of functions with infinite discontinuities
DOI:
https://doi.org/10.5644/SJM.10.1.13Keywords:
total H$_{1}$- integrability, Fourier seriesAbstract
Using the total H$_{1}$-integrability concept we shall show that functions, which take on infinite values in the interval $\left( -\pi ,\pi \right)$ at only finitely many places, can be expanded into a \textit{Fourier} series over this interval.
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References
R. G. Bartle, A Modern Theory of Integration, Graduate Studies in Math., Vol. 32, AMS, Providence, 2001.
I. J. L. Garces and P. Y. Lee, Convergence theorems for the H$_{1}$-integral, Taiwanese J. Math., 4 (3) (2000), 439-445.
R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Graduate Studies in Math., Vol. 4, AMS, Providence, 1994.
J. Marcinkiewicz and A. Zygmund, Two theorems on trigonometrical series, Rec. Math., [Mat. Sbornik] N.S., 2 (44) 4, (1937), 733--737.
E. J. McShane, Partial orderings and Moore-Smith limits, Am. Math. Mon., 59 (1952), 1--11.
B. Sarić, The Fourier series of one class of functions with discontinuities, Doctoral dissertation defended on 20th of October 2009 at the University of Novi Sad, Faculty of Science, Department of Mathematics and Informathics.
B. Sarić, Cauchy's residue theorem for a class of real valued functions, Czech. Math. J., 60 (4) (2010), 1043--1048.
B. Sarić, On totalization of the Henstock – Kurzweil integral in the multidimensional space, Czech. Math. J., 61, (4) (2011), 1017--1022.
B. Sarić, On totalization of the H$_{1}$-integral, Taiw. J. Math., {15} (4) (2011), 1691--1700.
V. Sinha and I. K. Rana, On the continuity of associated interval functions, Real Anal. Exch., 29 (2) (2003/2004), 979--981.
A. Zygmund, Trigonometric series, University Press, Cambridge, 2003.