Abstract: The wavelets and scaling functions based on Chebyshev polynomials and their zeros are introduced. The constructed system of functions is proved to be orthogonal. Using this system, an orthonormal basis in the space of square-integrable functions is built. Approximative properties of partial sums of corresponding wavelet series are investigated.
Keywords: polynomial wavelets, Chebyshev polynomials of second kind, orthogonality, Christoffel--Darboux formula, function approximation, wavelet series
For citation: Sultanakhmedov M. S. Approximative Properties of the Chebyshev Wavelet Series of the Second Kind. Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 17, no. 3, pp.56-64.
DOI 10.23671/VNC.2017.3.7265
1. Chui C. K., Mhaskar H. N. On Trigonometric wavelets.
Constructive Approximation, 1993, vol. 9, pp. 167-190.
2. Kilgore T., Prestin J. Polynomial wavelets on an interval.
Constructive Approximation, 1996, vol. 12 (1), pp. 1-18.
3. Davis P. J. Interpolation and Approximation, N. Y., Dover Publ.
Inc., 1973.
4. Fischer B. and Prestin J. Wavelet based on orthogonal
polynomials. Math. Comp., 1997, vol. 66, pp. 1593-1618.
5. Fischer B., Themistoclakis W. Orthogonal polynomial wavelets.
Numerical Algorithms, 2002, vol. 30, pp. 37-58.
6. Capobiancho M. R., Themistoclakis W. Interpolating polynomial
wavelet on [-1,1]. Advanced Comput. Math., 2005, vol. 23, pp.
353-374.
7. Dao-Qing Dai, Wei Lin. Orthonormal polynomial wavelets on the
interval. Proc. Amer. Math. Soc., 2005, vol. 134 (5), pp. 1383-1390.
8. Mohd F., Mohd I. Orthogonal functions based on Chebyshev
polynomials. Matematika, 2011, vol. 27, no. 1, pp. 97-107.
9. Sege G. Ortogonal'nye Mnogochleny, M., Fizmatlit, 1962, 500 p.
(Russian).
10. Jahnin 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.
Uspehi Mat. Nauk [Russian Mathematical Surveys], 1958, vol. 13, no.
6 (84), pp. 207-211 (Russian).
11. Jahnin 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. [Russian Mathematics
(Izvestiya VUZ. Matematika)], 1963, no. 1, pp. 172-178 (Russian).
12. Bernshtejn S. N. O Mnogochlenah, Ortogonal'nyh na Konechnom
Intervale, Har'kov, Gos. Nauch.-Teh. Izd-vo Ukrainy, 1937, 128 p.
