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The Scientific World Journal
Volume 2014, Article ID 782652, 13 pages
http://dx.doi.org/10.1155/2014/782652
Research Article

On Fourier Series of Fuzzy-Valued Functions

1Department of Mathematics, Faculty of Science, Bozok University, Yozgat, Turkey
2Department of Mathematics, Faculty of Science, Gazi University, Ankara, Turkey
3Department of Mathematics, Faculty of Arts and Sciences, Fatih University, 34500 İstanbul, Turkey

Received 13 November 2013; Accepted 30 December 2013; Published 10 April 2014

Academic Editors: A. Bellouquid, T. Calvo, and E. Momoniat

Copyright © 2014 Uğur Kadak and Feyzi Başar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Fourier analysis is a powerful tool for many problems, and especially for solving various differential equations of interest in science and engineering. In the present paper since the utilization of Zadeh’s Extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct a fuzzy-valued function on a closed interval via related membership function. We derive uniform convergence of a fuzzy-valued function sequences and series with level sets. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, Fourier series of periodic fuzzy-valued functions is defined and its complex form is given via sine and cosine fuzzy coefficients with an illustrative example. Finally, by using the Dirichlet kernel and its properties, we especially examine the convergence of Fourier series of fuzzy-valued functions at each point of discontinuity, where one-sided limits exist.