New 2D continuous symmetric Christoffel-Darboux formula for Chebyshev orthonormal polynomials of the first kind
Authors: E. Karoussos, Ć. B.Dolićanin, V. D. Pavlović, J. R. Djordjević-Kozarov
Keywords: Christoffel-Darboux formula; Chebyshev polynomials of the first kind; twodimensional functions; classical orthogonal functions
Abstract:
In this paper, anew two-dimensional continuous symmetric Christoffel-Darboux formula for orthonormal classical Chebyshev polynomials of the first kind is proposed. 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] Ciric, D.G., Pavlovic, V.D. (2012) Linear Phase Two-Dimensional FIR Digital Filter Functions Generated by applying Christoffel-Darboux Formula for Orthonormal Polynomials. Elektronika Ir Elektrotechnika, br. 4, str. 39-42
[7] Ćirić, D.G., Pavlović, V.D. (2012) ‘Generalised Christoffel-Darboux formula most directly applied in generating fully symmetric doubly resistively terminated LC lossless ladder filters. International Journal of Electronics, vol. 99, No. 12, pp. 1-17, ( 727352; 2012
[8] 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
[9] Ghosh, S. (2008) Generalized Christoffel-Darboux formula for classical skew-orthogonal polynomials. Journal of Physics A: Mathematical and Theoretical, 41(43): 435204
[10] Ilic, A.D., Pavlovic, V.D. (2011) New class of filter functions generated most directly by Christoffel-Darboux formula for Gegenbauer orthogonal polynomials. International Journal of Electronics, vol. 98, br. 1, str. 61-79
[11] 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
[12] Johnson, D., Johnson, J. (1966) Low-Pass Filters Using Ultraspherical Polynomials. IEEE Transactions on Circuit Theory, 13(4): 364-369
[13] Karoussos, E., Pavlović, V.D., Đorđević-Kozarov, J.R., Dolićanin, Ć.B. (2013) New 2D continuous symmetric Christoffel-Darboux formula for Chebyshev orthonormal polynomials of the second kind. Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics, vol. 5, br. 1, str. 23-33
[14] Lascoux, A., Pragacz, P. (2005) Bezoutians, Euclidean Algorithm, and Orthogonal Polynomials. Annals of Combinatorics, 9(3): 301-319
[15] Lutovac, M., Rabrenović, D. (1991) All-pole filters using ultra spherical polynomials. u: European conf. circuit theory design, ECCTD91, Sep., Copenhagen, str. 203-212
[16] 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
[17] Mitrinović, D., Đoković, D. (1964) Special function. Beograd: Građevinska knjiga
[18] Mitrinović, D.S. (1972) Uvod u specijalne funkcije. Beograd: Građevinska knjiga
[19] Pavlovic, V.D. (2012) New class of filter functions generated directly by the modified Christoffel-Darboux formula for classical orthonormal Jacobi polynomials. Int. J. Circuit Theory and Applications
[20] Pavlović, V.D. (1984) Direct synthesis of filter transfer functions. IEE Proceedings G (Electronic Circuits and Systems), 131(4): 156
[21] Pavlovic, V.D., Ciric, D.G., Doncov, N.S. (2012) Christoffel-Darboux Formula Most Directly Applied in Generating Economical Linear Phase Low-Pass Digital FIR Filter Functions. Elektronika Ir Elektrotechnika, br. 8, str. 109-112
[22] 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
[23] 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
[24] 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
[25] Pavlović, V.D. (1982) Least-square low-pass filters using Chebyshev polynomials. Int. J. Electronics, vol. 53, br. 4, str. 371-379
[26] 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
[27] Pavlović, V.D., Ilić, A.D. (2011) New class of filter functions generated most directly by the Christoffel-Darboux formula for classical orthonormal Jacobi polynomials. International Journal of Electronics, 98(12): 1603-1624
[28] 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
[29] Rakovich, B.D. (1974) Transitional butterworth-legendre filters. Radio & Electron, Eng. vol. 44, str. 673-680
[30] Raković, B.D., Popović, M.V. (1978) Explicit expression for the characteristic function of generalized legendre filters. International Journal of Circuit Theory and Applications, 6(4): 363-373
[31] 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
[32] Raković, B.D. (1983) Predistortion techniques for increasing the element tolerances in equiripple passband niters revisited. International Journal of Electronics, 54(6): 905-912
[33] 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
[34] Sall, R., Entenmann, W. (1979) Handbook of filter design. Berlin: AEG-TELEFUNKEN
[35] Shi, Y.G. (2012) On generalized Christoffel functions. Acta Mathematica Hungarica, 135(3): 213-228
[36] Szego, G. (1939) Orthogonal polynomials. u: American Mathematical Society Colloquium Publications, Providence, RI: American Mathematical Society / AMS, vol. XXIII
[37] Wesles, R. (1962) Numerical methods for scientists and engineers, Bell telephone laboratories. New York: McGraw- Hill, USA
[38] Zverev, A. (1976) Handbook of filter synthesis. New York, itd: Wiley