Authors : Pratham Ankurkumar Dungrani

Volume/Issue : Volume 10 - 2025, Issue 12 - December

Google Scholar : https://tinyurl.com/5fvj2yst

Scribd : https://tinyurl.com/4kbnkvus

DOI : https://doi.org/10.38124/ijisrt/25dec1396

Note : A published paper may take 4-5 working days from the publication date to appear in PlumX Metrics, Semantic Scholar, and ResearchGate.

Abstract : Fourier series are fundamental analytical tools for representing periodic functions as infinite sums of sine and cosine components. While their convergence properties for smooth functions are well established, many practical signals, engineering systems, and physical models naturally give rise to piecewise smooth functions—functions that remain smooth over subintervals but exhibit isolated discontinuities in the function or its derivatives. Such functions display a rich and nontrivial convergence behavior, characterized by nonuniform convergence rates, localized oscillations near discontinuities, overshoot phenomena, and slow decay of Fourier coefficients. This paper presents a comprehensive investigation of the convergence behavior of Fourier series for piecewise smooth functions. The study integrates theoretical analysis, convergence criteria, error characterization, and numerical demonstrations to examine how Fourier series converge pointwise, uniformly, and in the mean-square sense under varying degrees of regularity. Particular emphasis is placed on the Gibbs phenomenon, the role of jump discontinuities, endpoint smoothness, coefficient decay rates, and the relationship between differentiability and convergence efficiency. Analytical results and graphical evaluations demonstrate that convergence rates depend critically on function smoothness. For piecewise smooth functions, Fourier coefficients decay proportionally to 1/n, while continuously differentiable functions exhibit a faster 1/n2 decay, and analytic functions display exponential decay. In the presence of finite jump discontinuities, partial sums converge globally in the L2 sense but fail to converge uniformly, producing a persistent overshoot of approximately 8.94% near discontinuities. Numerical experiments further reveal that although partial sums exhibit oscillatory behavior near jump points, alternative summation techniques such as Fejér averaging and spectral smoothing can significantly suppress oscillations and improve convergence. The results presented reinforce fundamental principles of Fourier analysis, clarify the intrinsic limitations of classical Fourier approximations for non-smooth functions, and provide practical insights relevant to signal processing, spectral methods for partial differential equations, and engineering system modeling.

References :

1. Zygmund, A. (2002). Trigonometric series (Vols. 1–2). Cambridge University Press.
2. Tolstov, G. P. (1976). Fourier series. Dover Publications.
3. Katznelson, Y. (2004). An introduction to harmonic analysis (3rd ed.). Cambridge University Press.
4. Stein, E. M., & Shakarchi, R. (2003). Fourier analysis: An introduction. Princeton University Press.
5. Rudin, W. (1976). Principles of mathematical analysis (3rd ed.). McGraw-Hill.
6. Hardy, G. H. (1949). Divergent series. Oxford University Press.
7. Carleson, L. (1966). On convergence and growth of partial sums of Fourier series. Acta Mathematica, 116, 135–157. https://doi.org/10.1007/BF02392815
8. Hunt, R. A. (1968). On the convergence of Fourier series. In Proceedings of the Conference on Orthogonal Expansions and Their Continuous Analogues (pp. 235–255).
9. Timan, A. F. (1963). Theory of approximation of functions of a real variable. Pergamon Press.
10. Butzer, P. L., & Nessel, R. J. (1971). Fourier analysis and approximation. Academic Press.
11. Gibbs, J. W. (1899). Fourier’s series. Nature, 59, 200. https://doi.org/10.1038/059200b0
12. Fejér, L. (1900). Über die Konvergenz der Fourierreihen. Mathematische Annalen, 58, 51–69. https://doi.org/10.1007/BF01448709
13. Jerri, A. J. (1998). The Gibbs phenomenon in Fourier analysis, splines, and wavelet approximations. Springer. https://doi.org/10.1007/978-1-4757-2847-7
14. Gottlieb, D., & Shu, C.-W. (1997). On the Gibbs phenomenon and its resolution. SIAM Review, 39(4), 644–668. https://doi.org/10.1137/S0036144596301390
15. Tadmor, E. (2007). Filters, mollifiers and the computation of the Gibbs phenomenon. Acta Numerica, 16, 305–378. https://doi.org/10.1017/S0962492906000182
16. Weisz, F. (2018). Convergence and summability of Fourier transforms and Hardy spaces. Birkhäuser.
17. Okuyama, Y. (1984). Absolute summability of Fourier series and orthogonal series. Springer.
18. Arias-de-Reyna, J. (2002). Pointwise convergence of Fourier series. Springer.
19. Weisz, F. (2021). Lebesgue points and summability of higher-dimensional Fourier series. Birkhäuser.
20. García, A., Hernández, E., & Sánchez, J. L. (2019). Gibbs phenomenon for subdivision schemes. Journal of Inequalities and Applications, 2019, Article 48. https://doi.org/10.1186/s13660-019-1998-6
21. Boyd, J. P. (2001). Chebyshev and Fourier spectral methods (2nd ed.). Dover Publications.
22. Trefethen, L. N. (2000). Spectral methods in MATLAB. SIAM.
23. Canuto, C., Hussaini, M. Y., Quarteroni, A., & Zang, T. A. (2006). Spectral methods: Fundamentals in single domains. Springer.
24. Gottlieb, D., & Orszag, S. A. (1977). Numerical analysis of spectral methods. SIAM.
25. Shen, J., Tang, T., & Wang, L. L. (2011). Spectral methods: Algorithms, analysis and applications. Springer.
26. Fornberg, B. (1996). A practical guide to pseudospectral methods. Cambridge University Press.
27. LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems. Cambridge University Press.
28. Quarteroni, A., Sacco, R., & Saleri, F. (2007). Numerical mathematics (2nd ed.). Springer.
29. Weisstein, E. W. (n.d.). Gibbs phenomenon. MathWorld—Wolfram Research. https://mathworld.wolfram.com/GibbsPhenomenon.html
30. Encyclopedia of Mathematics. (n.d.). Gibbs phenomenon. https://encyclopediaofmath.org/wiki/Gibbs_phenomenon
31. Feldman, J. (2017). Gibbs phenomenon [PDF]. University of British Columbia. https://personal.math.ubc.ca/~feldman/m321/gibbs.pdf
32. Hagiwara, S. (2018). Convergence of Fourier series [PDF]. University of Chicago. https://math.uchicago.edu/~may/REU2018/REUPapers/Hagiwara.pdf
33. Adcock, B., & Huybrechs, D. (2011). On the resolution power of Fourier extensions for oscillatory functions. arXiv. https://arxiv.org/abs/1105.3426
34. Lacey, M. (2003). Carleson’s theorem: Proof, complements, variations. arXiv. https://arxiv.org/abs/math/0307008
35. Dipierro, S., Pfefferlé, D., & Valdinoci, E. (2025). A primer on Fourier series. arXiv. https://arxiv.org/abs/2510.18203
36. Rapakaa, N. R., & Riahi, M. K. (2025). Generalized Fourier series eliminating Gibbs oscillations. arXiv. https://arxiv.org/abs/2510.14731
37. Harris, F. J. (1978). On the use of windows for harmonic analysis. Proceedings of the IEEE, 66(1), 51–83. https://doi.org/10.1109/PROC.1978.10837
38. Oppenheim, A. V., & Schafer, R. W. (2010). Signals and systems (2nd ed.). Pearson.
39. Papoulis, A. (1987). The Fourier integral and its applications. McGraw-Hill.
40. Proakis, J. G., & Manolakis, D. G. (2007). Digital signal processing. Pearson.
41. Stoica, P., & Moses, R. L. (2005). Spectral analysis of signals. Pearson.
42. Marple, S. L. (1987). Digital spectral analysis. Prentice Hall.
43. Szegő, G. (1975). Orthogonal polynomials (4th ed.). American Mathematical Society.
44. Rivlin, T. J. (1974). The Chebyshev polynomials. Wiley.
45. DeVore, R. A., & Lorentz, G. G. (1993). Constructive approximation. Springer.
46. Cheney, E. W. (1982). Introduction to approximation theory. American Mathematical Society.
47. Evans, L. C. (2010). Partial differential equations (2nd ed.). American Mathematical Society.
48. Strauss, W. A. (2007). Partial differential equations: An introduction. Wiley.
49. Kreyszig, E. (2011). Advanced engineering mathematics (10th ed.). Wiley.
50. Morse, P. M., & Feshbach, H. (1953). Methods of theoretical physics. McGraw-Hill.
51. Unser, M. (2000). Sampling—50 years after Shannon. Proceedings of the IEEE. https://doi.org/10.1109/5.843002
52. Daubechies, I. (1992). Ten lectures on wavelets. SIAM.
53. Meyer, Y. (1993). Wavelets and operators. Cambridge University Press.
54. Cohen, A. (2003). Numerical analysis of wavelet methods. Elsevier.
55. Hansen, P. C. (2010). Discrete inverse problems. SIAM.
56. Boyd, J. P. (2014). Weakly nonlocal solitons and Gibbs phenomenon. Wave Motion, 51, 124–139. https://doi.org/10.1016/j.wavemoti.2013.08.007
57. MathWorks. (n.d.). Fourier series and Gibbs phenomenon. https://www.mathworks.com/help/matlab/ref/fourier.html
58. MIT OpenCourseWare. (n.d.). Signals and systems. https://ocw.mit.edu/courses/6-003-signals-and-systems-fall-2011/
59. KTH Royal Institute of Technology. (n.d.). Fourier series and spectral methods. https://kth-nek5000.github.io/kthNekBook/_md/spectral/fourier.html
60. Harvey Mudd College. (n.d.). Fourier series and Gibbs overshoot. https://saeta.physics.hmc.edu/p064/FO-Gibbs.html