Exploring Chebyshev polynomial approximations: Error estimates for functions of bounded variation

S. Akansha; Aditya Subramaniam; S. Akansha; Aditya Subramaniam

doi:10.3934/math.2025398

AIMS Mathematics

2025, Volume 10, Issue 4: 8688-8706. doi: 10.3934/math.2025398

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Research article

Exploring Chebyshev polynomial approximations: Error estimates for functions of bounded variation

S. Akansha ^,,
Aditya Subramaniam

Department of Mathematics Manipal Institute of Technology, Manipal Academy of Higher Education, Karnataka-576104, India

Received: 14 November 2024 Revised: 04 February 2025 Accepted: 12 February 2025 Published: 15 April 2025
MSC : 65D15, 41A21, 41A25

Approximation theory plays a central role in numerical analysis, evolving through a variety of methodologies, with significant contributions from Lebesgue, Weierstrass, Fourier, and Chebyshev approximations. For sufficiently smooth functions, the partial sum of Chebyshev series expansion provides optimal polynomial approximation, making it a preferred choice in many applications. However, existing literature predominantly focuses on Chebyshev interpolation, which requires exact Chebyshev series coefficients. The computation of these exact coefficients is challenging and often impractical for numerical algorithms, limiting their practical utility. Additionally, traditional approaches typically involve polynomials on fixed intervals where the basis functions of the series are defined. In this article, we have generalized Chebyshev polynomial approximation to a broader domain and presented two optimal error estimations for functions of bounded variation, using approximated Chebyshev series coefficients. This aspect is notably absent in current literature. To support our theoretical findings, we conducted numerical experiments and proposed future research directions, particularly in the fields of machine learning and related areas.
- Chebyshev polynomials,
- Chebyshev approximation,
- error quantification,
- functions of bounded variations
Citation: S. Akansha, Aditya Subramaniam. Exploring Chebyshev polynomial approximations: Error estimates for functions of bounded variation[J]. AIMS Mathematics, 2025, 10(4): 8688-8706. doi: 10.3934/math.2025398

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Abstract

Approximation theory plays a central role in numerical analysis, evolving through a variety of methodologies, with significant contributions from Lebesgue, Weierstrass, Fourier, and Chebyshev approximations. For sufficiently smooth functions, the partial sum of Chebyshev series expansion provides optimal polynomial approximation, making it a preferred choice in many applications. However, existing literature predominantly focuses on Chebyshev interpolation, which requires exact Chebyshev series coefficients. The computation of these exact coefficients is challenging and often impractical for numerical algorithms, limiting their practical utility. Additionally, traditional approaches typically involve polynomials on fixed intervals where the basis functions of the series are defined. In this article, we have generalized Chebyshev polynomial approximation to a broader domain and presented two optimal error estimations for functions of bounded variation, using approximated Chebyshev series coefficients. This aspect is notably absent in current literature. To support our theoretical findings, we conducted numerical experiments and proposed future research directions, particularly in the fields of machine learning and related areas.

References

[1]	W. F. Ames, Numerical methods for partial differential equations, Academic press, 2014.
[2]	J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley & Sons, 2016.
[3]	J. C. Mason, D. C. Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, 2003. https://doi.org/10.1201/9781420036114
[4]	L. N. Trefethen, Approximation theory and approximation practice, Philadelphia: Society for Industrial and Applied Mathematics, 2013.
[5]	G. Meinardus, Approximation of functions: Theory and numerical methods, Springer Science & Business Media, 13 (2012).
[6]	H. Attouch, G. Buttazzo, G. Michaille, Variational analysis in Sobolev and BV spaces: Applications to PDEs and optimization, SIAM, 2014.
[7]	R. F, Gariepy, Functions of bounded variation and free discontinuity problems (oxford mathematical monographs) by luigi ambrosio, nicolo fucso and diego pallara: 434 pp., £ 55.00, isbn 0-19-850254-1 (clarendon press, oxford, 2000), Bulletin of the London Mathematical Society, 33 (2001), 492–512. https://doi.org/10.1017/S0024609301309281
[8]	C. M. Dafermos, C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Springer, 3 (2005). https://doi.org/10.1007/978-3-662-49451-6
[9]	B. Bourdin, A. Chambolle, Implementation of an adaptive finite-element approximation of the mumford-shah functional, Numer. Math., 85 (2000), 609–646. https://doi.org/10.1007/PL00005394 doi: 10.1007/PL00005394
[10]	S. Xiang, On the optimal convergence rates of Chebyshev interpolations for functions of limited regularity, Appl. Math. Lett., 84 (2018), 1–7. https://doi.org/10.1016/j.aml.2018.04.009 doi: 10.1016/j.aml.2018.04.009
[11]	F. Dell'Accio, F. Di Tommaso, F. Nudo, Generalizations of the constrained mock-chebyshev least squares in two variables: Tensor product vs total degree polynomial interpolation, Appl. Math. Lett., 125 (2022), 107732. https://doi.org/10.1016/j.aml.2021.107732 doi: 10.1016/j.aml.2021.107732
[12]	F. Dell'Accio, F. Marcellán, F. Nudo, An extension of a mixed interpolation-regression method using zeros of orthogonal polynomials, J. Comput. Appl. Math., 450 (2024), 116010. https://doi.org/10.1016/j.cam.2024.116010 doi: 10.1016/j.cam.2024.116010
[13]	H. Wang, Analysis of error localization of chebyshev spectral approximations, SIAM J. Numer. Anal., 61 (2023), 952–972. https://doi.org/10.1137/22M1481452 doi: 10.1137/22M1481452
[14]	R. Xie, B. Wu, W. Liu, Optimal error estimates for chebyshev approximations of functions with endpoint singularities in fractional spaces, J. Sci. Comput., 96 (2023), 71. https://doi.org/10.1007/s10915-023-02292-5 doi: 10.1007/s10915-023-02292-5
[15]	H. Y. Wang, New error bounds for legendre approximations of differentiable functions, J. Fourier Anal. Appl., 29 (2023), 42. https://doi.org/10.1007/s00041-023-10024-4 doi: 10.1007/s00041-023-10024-4
[16]	W. J. Liu, L. L. Wang, H. Y. Li, Optimal error estimates for chebyshev approximations of functions with limited regularity in fractional sobolev-type spaces, Math. Comput., 88 (2019), 2857–2895.
[17]	M. Hamzehnejad, M. M. Hosseini, A. Salemi, An improved bound on legendre approximation, arXiv Preprint, 2022. https://doi.org/10.48550/arXiv.2207.13477
[18]	X. L. Zhang, J. P. Boyd, Asymptotic coefficients and errors for chebyshev polynomial approximations with weak endpoint singularities: Effects of different bases, Sci. China Math., 66 (2023), 191–220. https://doi.org/10.1007/s11425-021-1974-x doi: 10.1007/s11425-021-1974-x
[19]	X. L. Zhang, The mysteries of the best approximation and chebyshev expansion for the function with logarithmic regularities, arXiv Preprint, 2021. https://doi.org/10.48550/arXiv.2108.03836
[20]	X. L. Zhang, Comparisons of best approximations with chebyshev expansions for functions with logarithmic endpoint singularities, Numer. Algorithms, 2023, 1–25. https://doi.org/10.1007/s11075-023-01538-5 doi: 10.1007/s11075-023-01538-5
[21]	D. Occorsio, W. Themistoclakis, On the filtered polynomial interpolation at chebyshev nodes, Appl. Numer. Math., 166 (2021), 272–287. https://doi.org/10.1016/j.apnum.2021.04.013 doi: 10.1016/j.apnum.2021.04.013
[22]	D. Occorsio, G. Ramella, W. Themistoclakis, Lagrange-chebyshev interpolation for image resizing, Math. Comput. Simulat., 197 (2022), 105–126. https://doi.org/10.1016/j.matcom.2022.01.017 doi: 10.1016/j.matcom.2022.01.017
[23]	M. C. De Bonis, D. Occorsio, W. Themistoclakis, Filtered interpolation for solving prandtl's integro-differential equations, Numer. Algorithms, 88 (2021), 679–709. https://doi.org/10.1007/s11075-020-01053-x doi: 10.1007/s11075-020-01053-x
[24]	D. I. Shuman, P. Vandergheynst, P. Frossard, Chebyshev polynomial approximation for distributed signal processing, In 2011 International Conference on Distributed Computing in Sensor Systems and Workshops (DCOSS), 2011, 1–8. https://doi.org/10.1109/DCOSS.2011.5982158
[25]	T. T. Lee, J. T. Jeng, The chebyshev-polynomials-based unified model neural networks for function approximation, IEEE T. Syst. Man Cy.-B, 28 (1998), 925–935. https://doi.org/10.1109/3477.735405 doi: 10.1109/3477.735405
[26]	M. G. He, Z. W. Wei, J. R. Wen, Convolutional neural networks on graphs with chebyshev approximation, revisited, Adv. Neural Inform. Pro. Syst., 35 (2022), 7264–7276.
[27]	Z. P. Li, J. Wang, Spectral graph neural networks with generalized laguerre approximation, In: ICASSP 2024-2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2024, 7760–7764. https://doi.org/10.1109/ICASSP48485.2024.10445856
[28]	L. Stankovic, D. Mandic, M. Dakovic, M. Brajovic, B. Scalzo, A. G. Constantinides, Graph signal processing–part ii: Processing and analyzing signals on graphs, arXiv Preprint, 2019. https://doi.org/10.48550/arXiv.1909.10325
[29]	C. C. Tseng, S. L. Lee, Minimax design of graph filter using chebyshev polynomial approximation, IEEE T. Circuits-II, 68 (2021), 1630–1634. https://doi.org/10.1109/TCSII.2021.3065977 doi: 10.1109/TCSII.2021.3065977
[30]	K. O. Geddes, Near-minimax polynomial approximation in an elliptical region, SIAM J. Numer. Anal., 15 (1978), 1225–1233. https://doi.org/10.1137/0715083 doi: 10.1137/0715083
[31]	H. Majidian, On the decay rate of chebyshev coefficients, Appl. Numer. Math., 113 (2017), 44–53. https://doi.org/10.1016/j.apnum.2016.11.004 doi: 10.1016/j.apnum.2016.11.004
[32]	W. M. A. Elhameed, O. M. Alqubori, New results of unified chebyshev polynomials, AIMS Math., 9 (2024), 20058–20088. https://doi.org/10.3934/math.2024978 doi: 10.3934/math.2024978
[33]	W. M. A. Elhameed, H. M. Ahmed, Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of chebyshev polynomials, AIMS Math., 9 (2024), 2137–2166. https://doi.org/10.3934/math.2024107 doi: 10.3934/math.2024107
[34]	A. L. Tampos, J. E. C. Lope, Overcoming Gibbs phenomenon in Padé-Chebyshev approximation, Matimyás Mat., 30 (2007), 127–135.
[35]	T. J. Rivlin, The Chebyshev polynomials, New York: John Wiley & Sons, 1974.
[36]	C. Carle, M. Hochbruck, A. Sturm, On leapfrog-chebyshev schemes, SIAM J. Numer. Anal., 58 (2020), 2404–2433. https://doi.org/10.1137/18M1209453 doi: 10.1137/18M1209453
[37]	J. J. Glunt, J. A. Siefert, A. F. Thompson, H. C. Pangborn, Error bounds for compositions of piecewise affine approximations, arXiv Preprint, 2024. https://doi.org/10.1016/j.ifacol.2024.07.423
[38]	J. V. Schaftingen, Approximation in sobolev spaces by piecewise affine interpolation, J. Math. Anal. Appl., 420 (2014), 40–47. https://doi.org/10.1016/j.jmaa.2014.05.036 doi: 10.1016/j.jmaa.2014.05.036
[39]	S. Lang, Real and functional analysis, New York: Springer, 2012.
[40]	T. M. Apostol, Mathematical analysis, 2Eds., Addison-Wesley series in mathematics, Reading: Addison-Wesley, 1974.
[41]	S. Xiang, X. Chen, H. Wang, Error bounds for approximation in Chebyshev points, Numer. Math., 116 (2010), 463–491. https://doi.org/10.1007/s00211-010-0309-4 doi: 10.1007/s00211-010-0309-4
[42]	L. Fox, I. B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, 1968.

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