Definability and Strength of Fourier Series Transform Obtained by Employing Parametric Integration
Abstract
Regardless of the term coinage, this article is a story of adding variables that act in a similar way to the “s” in the Laplace transform. This idea is similar to the avoiding of points in improper integration, but instead of taking limits, the endpoints of each maximum integrable interval are made to be variables, but they still preserve the orders. The result is here called a parametric integral. It is applied here to Fourier series transform as an example. Therefore, some functions that formerly had no Fourier series expansion may no longer exist if one employs parametric integration. The resulting transform has advantages in terms of definability and strength. In definability because the transform exists for more functions than even for the well-known Fourier transform. It gives hope for solving more problems that may be of certain practical interests, especially when endpoint variables in the solution are replaced by respective limiting processes. The possibility that a solution still contains non-eliminable variables may lead to interesting non-standard analysis with the parameters as new “numbers”, some of which are given here. The parameters may also serve in determining the partial inverses of the transform.
Keywords: Fourier series expansion, Fourier series transform, half-range expansion, Laplace transform, parametric integration.
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MAZA R. E., & CANOY S. R. Denjoy-type Integrals in Locally Convex Topological Vector Space. European Journal of Pure and Applied Mathematics, 2021, 14(4): 1169-1183. https://doi.org/10.29020/nybg.ejpam.v14i4.4115
THANGE T. G., & GANGANE S. S. Henstock-Kurzweil Integral for Banach Valued Function. Baghdad Science Journal, 2023, 20(1): 388-393. https://doi.org/10.21123/bsj.2023.8421
ROBDERA M. A. A Unified Approach to Integration Theory. International Journal of Applied Physics and Mathematics, 2019, 9(1): 21–28. http://www.ijapm.org/show-79-604-1.html
MUNKRES J. R. Analysis on manifolds, Advanced Book Program. Redwood City, Addison-Wesley Publishing Company, 1991.
SAVAŞ R., & PATTERSON R. F. Gauge Strongly Summability of Measurable Functions. Carpathian Journal of Mathematics, 2021, 37(1): 109-117. https://www.jstor.org/stable/27082087
BARTLE R. G., & SHERBERT D. R. Introduction to Real Analysis, 4th edition. Hoboken, John Wiley & Sons, 1927.
SUMALPONG F. R., & BENITEZ J. V. McShane Integrability Using Variational Measure. European Journal of Pure and Applied Mathematics, 2020, 13(2): 303-313. https://doi.org/10.29020/nybg.ejpam.v13i2.3659
PAŞAOGLU B., & HÜSEYIN T. Uniform Convergence of Generalized Fourier Series of Hahn-Sturm-Liouville Problem. Konuralp Journal of Mathematics, 2021, 9(2): 250–259. https://dergipark.org.tr/tr/pub/konuralpjournalmath/issue/31497/823776
GUO T., ZHANG T., LIM E., LOPEZ-BENITEZ M., MA F., and YU L. A Review of Wavelet Analysis and Its Applications: Challenges and Opportunities. IEEE Access, 2022, 10. https://doi.org/10.1109/ACCESS.2022.3179517
KOSSEK W. Is a Taylor Series also a Generalized Fourier series? College Mathematics Journal, 2018, 49(1): 54-56. https://doi.org/10.1080/07468342.2017.1397449
BENHADID Y. A Wavelets Based Scheme for Solving Schrodinger’s Equation. Journal of Wavelet Theory and Applications, 2021, 15(1): 1–8. http://www.ripublication.com/jwta21/jwtav15n1_01.pdf
KREYZIG E. Advanced Engineering Mathematics, Wiley & Sons, Chicago, 1980.
CONNON D. F. Fourier Series and Periodicity. arXiv, 2014, 150103037. https://doi.org/10.48550/arXiv.1501.03037
RANE V. V. Classical Theory of Fourier Series: Demystified and Generalized. arXiv, 2008, 08072955. https://doi.org/10.48550/arXiv.0807.2955
CHAKRABARTI D., & DAWN A. Power Series as Fourier Series. Rocky Mountain Journal of Mathematics, 2022, 52(5). https://doi.org/10.48550/arXiv.2107.04201
LALITHAMBIGAI K., & GNANACHANDRA P. Topological Structures on Vertex Set of Digraphs, 2023, 20(1). https://doi.org/10.21123/bsj.2023.8432
KREYSZIG E. Introductory Functional Analysis with Applications, Vol. 17. Hoboken, John Wiley & Sons, 1991.
TOVKACH R., KHARKEVYCH Y., and KAL’CHUK I. Application of a Fourier Series for an Analysis of a Network Signals. Proceedings of the 2019 IEEE International Conference on Advanced Trends in Information Theory, 2019, Kyiv, pp. 107–110. https://doi.org/10.1109/ATIT49449.2019.9030488
MARTIROSYAN A. A., MARTIROSYAN K. V., and CHERNYSHEV A. B. Application of Fourier Series in Distributed Control Systems Simulation. Proceedings of the 2019 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering, Saint Petersburg, 2019, pp. 609–613. https://doi.org/10.1109/EIConRus.2019.8656865
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