New 2D continuous symmetric Christoffel-Darboux formula for Chebyshev orthonormal polynomials of the second kind
Authors: E. Karoussos, V. D. Pavlovic, J. R. Djordjević-Kozarov, Ć. B. Dolićanin
Keywords: Christoffel-Darboux formula; Chebyshev polynomials of the second kind; two-dimensional functions; classical orthogonal functions
Abstract:
In this paper, we propose a new two-dimensional continuous symmetric Christoffel-Darboux formula for orthonormal classical Chebyshev polynomials of the second kind. This continuous two-dimensional function of two real variables is most directly applied to approximation problems and synthesis of filter functions. The examples of the proposed two-dimensional Christoffel-Darboux formula are illustrated.
References:
[1] Abramowitz, M., Stegun, I.A. (1964) Handbook of Mathematical Functions. New York: Dover Publications, Inc
[2] Andrews, L.C. (1998) Special Functions for Engineers and Applied Mathematicians. New York: MacMillan Publishing Company
[3] Angot, A. (1957) Complements de mathematiques, A lusage des ingenieurs de lElektrotechnique et des telecomunicationss. Paris
[4] Ayers, P.W. (2003) Generalized Christoffel?Darboux formulae and the frontier Kohn?Sham molecular orbitals. Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta), 110(4): 267-275
[5] Beccari, C. (1979) The use of the shifted jacobi polynomials in the synthesis of lowpass filters. International Journal of Circuit Theory and Applications, 7(3): 289-295
[6] Ghosh, S. (2006) Generalized Christoffel-Darboux formula for skew-orthogonal polynomials and random matrix theory. Journal of Physics A: Mathematical and General, 39(28): 8775-8782
[7] Ghosh, S. (2008) Generalized Christoffel-Darboux formula for classical skew-orthogonal polynomials. Journal of Physics A: Mathematical and Theoretical, 41(43): 435204
[8] Jafarizadeh, M.A., Sufiani, R., Jafarizadeh, S. (2009) Recursive calculation of effective resistances in distance-regular networks based on Bose-Mesner algebra and Christoffel-Darboux identity. Journal of Mathematical Physics, 50(2): 023302
[9] Johnson, D., Johnson, J. (1966) Low-Pass Filters Using Ultraspherical Polynomials. IEEE Transactions on Circuit Theory, 13(4): 364-369
[10] Lascoux, A., Pragacz, P. (2005) Bezoutians, Euclidean Algorithm, and Orthogonal Polynomials. Annals of Combinatorics, 9(3): 301-319
[11] Lutovac, M., Rabrenović, D. (1991) All-pole filters using ultra spherical polynomials. u: European conf. circuit theory design, ECCTD91, Sep., Copenhagen, str. 203-212
[12] Milovanović, G.V., Pavlović, V.D. (1981) Uslovna srednje-kvadratna aproksimacija prenosne funkcije sa ebiševljevom težinom. u: Numeričke metode u tehnici (III Znanstveni skup), Stubike toplice, str. 241-247
[13] Mitrinović, D., Đoković, D. (1964) Special function. Beograd: Građevinska knjiga
[14] Mitrinović, D.S. (1972) Uvod u specijalne funkcije. Beograd: Građevinska knjiga
[15] Pavlovic, V.D. (1982) Least-square low-pass filters using Chebyshev polynomials. International Journal of Electronics, 53(4): 371-379
[16] Pavlović, V.D. (1984) Direct synthesis of filter transfer functions. IEE Proceedings G (Electronic Circuits and Systems), 131(4): 156
[17] Pavlović, V. (1988) Filter transfer function synthesis by Gegenbauer generating function. u: YU Proceedings of the XXVII Conference of ETAN, Part III, June, Sarajevo, str. 157-164, 6-10
[18] Pavlović, V.D. (2004) Filter transfer function synthesis by Hermite generating function. Journal of Electrotechn. Math, Kosovska Mitrovica, vol. 9, br. 1, str. 35-41
[19] Pavlović, V.D. (2005) Synthesis of filter function using gene ratings functions of classical orthogonal polynomials. Journal of Technical Sciences and Mathematics, Kosovska Mitrovica, vol. 10, br. 1, str. 35-46
[20] Pavlović, V.D., Popović, M.V. (1987) An iterative method for loss LC ladder filter synthesis. u: Proc. Int. Symp. Network Theory, Paris, str. 185-190
[21] Rakovich, B.D., Pavlovic, V.D. (1987) Method of designing doubly terminated lossy ladder filters with increased element tolerances. IEE Proceedings G (Electronic Circuits and Systems), 134(6): 285
[22] Rakovich, B.D., Popovich, M.V. (1978) Explicit expression for the characteristic function of generalized legendre filters. International Journal of Circuit Theory and Applications, 6(4): 363-373
[23] Rakovich, B.D. (1974) Transitional butterworth-legendre filters. Radio & Electron, Eng. vol. 44, str. 673-680
[24] Raković, B.D. (1972) Characteristic functions for least mean square approximation for all pole filters. Publ. of Electrical Engineering Faculty, vol. 107-108, pp 23-26
[25] Raković, B.D. (1983) Predistortion techniques for increasing the element tolerances in equiripple passband niters revisited. International Journal of Electronics, 54(6): 905-912
[26] Raković, B.D. (1974) Designing monotonic low-pass filters-comparison of some methods and criteria. International Journal of Circuit Theory and Applications, 2(3): 215-221
[27] Sall, R., Entenmann, W. (1979) Handbook of filter design. Berlin: AEG-TELEFUNKEN
[28] Shi, Y.G. (2012) On generalized Christoffel functions. Acta Mathematica Hungarica, 135(3): 213-228
[29] Szego, G. (1939) Orthogonal polynomials. u: American Mathematical Society Colloquium Publications, Providence, RI: American Mathematical Society / AMS, vol. XXIII
[30] Wesles, R. (1962) Numerical methods for scientists and engineers, Bell telephone laboratories. New York: McGraw- Hill, USA
[31] Zverev, A. (1976) Handbook of filter synthesis. New York, itd: Wiley