On Fourier Series of Fuzzy-Valued Functions
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...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2014/782652 |
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| _version_ | 1832562374382452736 |
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| author | Uğur Kadak Feyzi Başar |
| author_facet | Uğur Kadak Feyzi Başar |
| author_sort | Uğur Kadak |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-ae14289353324a1d892a22d5aee64ebc |
| institution | Kabale University |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-ae14289353324a1d892a22d5aee64ebc2025-02-03T01:22:47ZengWileyThe Scientific World Journal2356-61401537-744X2014-01-01201410.1155/2014/782652782652On Fourier Series of Fuzzy-Valued FunctionsUğur Kadak0Feyzi Başar1Department of Mathematics, Faculty of Science, Bozok University, Yozgat, TurkeyDepartment of Mathematics, Faculty of Arts and Sciences, Fatih University, 34500 İstanbul, TurkeyFourier 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.http://dx.doi.org/10.1155/2014/782652 |
| spellingShingle | Uğur Kadak Feyzi Başar On Fourier Series of Fuzzy-Valued Functions The Scientific World Journal |
| title | On Fourier Series of Fuzzy-Valued Functions |
| title_full | On Fourier Series of Fuzzy-Valued Functions |
| title_fullStr | On Fourier Series of Fuzzy-Valued Functions |
| title_full_unstemmed | On Fourier Series of Fuzzy-Valued Functions |
| title_short | On Fourier Series of Fuzzy-Valued Functions |
| title_sort | on fourier series of fuzzy valued functions |
| url | http://dx.doi.org/10.1155/2014/782652 |
| work_keys_str_mv | AT ugurkadak onfourierseriesoffuzzyvaluedfunctions AT feyzibasar onfourierseriesoffuzzyvaluedfunctions |