Cite this article as:
Sultanakhmedov M. S. Special Wavelets Based on Chebyshev Polynomials of the Second Kind and their Approximative Properties. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2016, vol. 16, iss. 1, pp. 34-41. DOI: https://doi.org/10.18500/1816-9791-2016-16-1-34-41
Special Wavelets Based on Chebyshev Polynomials of the Second Kind and their Approximative Properties
The system of wavelets and scalar functions based on Chebyshev polynomials of the second kind and their zeros is considered. With the help of them we construct a complete orthonormal system of functions. A certain disadvantage is shown in approximation properties of partial sums of the corresponding wavelet series, related to the properties of Chebyshev polynomials themselves and meaning a significant decrease of the rate of their convergence to the original function at the endpoints of orthogonality segment. As an alternative, we propose a modification of Chebyshev wavelet series of the second kind by analogy to the special polynomial series with the property of adhesion. These new special wavelet series is proved to be deprived of the mentioned disadvantage and to have better approximative properties.
1. Meyer Y. Ondelettes et Operateurs, vol. I–III, Paris, Hermann, 1990.
2. Daubechies L. Ten Lectures on Wavelets. CBMSNSF Regional Conference Series in Applied Mathematics Proceedings, vol. 61, Philadelphia, PA, SIAM, 1992, 357 p. DOI: https://doi.org/10.1137/1.9781611970104.
3. Chui C. K. An Introduction to Wavelets. Wavelet Analysis and its Applications, vol. 1, Boston, Academic Press, 1992, 264 p.
4. Chui C. K., Mhaskar H. N. On Trigonometric Wavelets. Constructive Approximation, 1993, vol. 9, iss. 2–3, pp. 167–190. DOI: https://doi.org/10.1007/BF01198002.
5. Kilgore T., Prestin J. Polynomial wavelets on an interval. Constructive Approximation, 1996. Vol. 12, iss. 1, pp. 95–110. DOI: https://doi.org/10.1007/BF02432856.
6. Fischer B., Prestin J. Wavelet based on orthogonal polynomials. Mathematics of computation, 1997, vol. 66, no. 220, pp. 1593–1618. DOI: https://doi.org/10.1090/S0025-5718-97-00876-4.
7. Fischer B., Themistoclakis W. Orthogonal polynomial wavelets. Numerical Algorithms, 2002, vol. 30, iss. 1, pp. 37–58. DOI: https://doi.org/10.1023/A:1015689418605.
8. Capobiancho M. R., Themistoclakis W. Interpolating polynomial wavelet on [−1, 1]. Advanced in Computational Mathematics, 2005, vol. 23, iss. 4, pp. 353–374. DOI: https://doi.org/10.1007/s10444-004-1828-2.
9. Dao-Qing Dai, Wei Lin. Orthonormal polynomial wavelets on the interval. Proc. Amer. Math. Soc., 2005, vol. 134, iss. 5, pp. 1383–1390. DOI: https://doi.org/10.1090/S0002-9939-05-08088-3.
10. Mohd F., Mohd I. Orthogonal Functions Based on Chebyshev Polynomials. Matematika, 2011, vol. 27, no. 1, pp. 97–107.
11. Sultanakhmedov M. S. Approximative properties of the Chebyshev wavelet series of the second kind. Vladikavkazskij matematicheskij zhurnal, 2015, vol. 17, iss. 3, pp. 56–64 (in Russian).
12. Sharapudinov I. I. Limit ultraspherical series and their approximative properties. Math. Notes, 2013, vol. 94, iss. 2, pp. 281–293. DOI: https://doi.org/10.1134/S0001434613070274.
13. Sharapudinov I. I. Some special series in ultraspherical polynomials and their approximative properties. Izv. Math., 2014, vol. 78, no. 5, pp. 1036–1059. DOI: https://doi.org/10.1070/IM2014v078n05ABEH002718.
14. Yakhnin B. M. Lebesgue functions for expansions in series of Jacobi polynomials for the cases α = β = 1/2 , α = β = -1/2 , α = 1/2 , β = -1/2 . Uspekhi Mat. Nauk, 1958, vol. 13, iss. 6(84), pp. 207–211 (in Russian).
15. Yakhnin B. M. Approximation of functions of class Lip α by partial sums of a Fourier series in Chebyshev polynomials of second kind. Izv. Vyssh. Uchebn. Zaved. Mat., 1963, no. 1, pp. 172–178 (in Russian).
16. Szego G. Orthogonal Polynomials. ¨ Colloquium Publications (Amer. Math. Soc.), 1939, vol. 23, 432 p.
17. Timan A. F. Teorija priblizhenija funkcij dejstvitel’nogo peremennogo [Theory of approximation of functions of a real variable]. Moscow, Fizmatgiz, 1960, 626 p. (in Russian).
18. Sharapudinov I. I. Best approximation and the Fourier – Jacobi sums. Math. Notes, 1983, vol. 34, iss. 5, pp. 816–821. DOI: https://doi.org/10.1007/BF01157445.
