Real orthogonal polynomials in frequency analysis

Bracciali, C.; Li, Xin; Ranga, A.

doi:10.1090/S0025-5718-04-01672-2

Real orthogonal polynomials in frequency analysis
HTML articles powered by AMS MathViewer

by C. F. Bracciali, Xin Li and A. Sri Ranga PDF

Math. Comp. 74 (2005), 341-362 Request permission

Abstract:

We study the use of para-orthogonal polynomials in solving the frequency analysis problem. Through a transformation of Delsarte and Genin, we present an approach for the frequency analysis by using the zeros and Christoffel numbers of polynomials orthogonal on the real line. This leads to a simple and fast algorithm for the estimation of frequencies. We also provide a new method, faster than the Levinson algorithm, for the determination of the reflection coefficients of the corresponding real Szegő polynomials from the given moments.

References

G. S. Ammar, D. Calvetti, and L. Reichel, Continuation methods for the computation of zeros of Szegő polynomials, Linear Algebra Appl. 249 (1996), 125–155. MR 1417413, DOI 10.1016/0024-3795(95)00324-X
Andrea C. Berti and A. Sri Ranga, Companion orthogonal polynomials: some applications, Appl. Numer. Math. 39 (2001), no. 2, 127–149. MR 1862329, DOI 10.1016/S0168-9274(01)00046-0
C.F. Bracciali, A.P. da Silva and A. Sri Ranga, Szegő polynomials: some relations to $L$-orthogonal and orthogonal polynomials, J. Comput. Appl. Math., 153 (2003), 79-88.
R. Bressan, S.F. Menegasso and A. Sri Ranga, Szegő polynomials: quadrature rules on the unit circle and on $[-1, 1]$, Rocky Mountain J. Math., 33 (2003), 567–584.
L. Daruis, O. Njåstad and W. Van Assche, Para-orthogonal polynomials in frequency analysis, Rocky Mountain J. Math., 33 (2003), 629–645.
Philippe Delsarte and Yves V. Genin, The split Levinson algorithm, IEEE Trans. Acoust. Speech Signal Process. 34 (1986), no. 3, 470–478. MR 844658, DOI 10.1109/TASSP.1986.1164830
P. Delsarte and Y. Genin, An introduction to the class of split Levinson algorithms, Numerical linear algebra, digital signal processing and parallel algorithms (Leuven, 1988) NATO Adv. Sci. Inst. Ser. F: Comput. Systems Sci., vol. 70, Springer, Berlin, 1991, pp. 111–130. MR 1150061
Walter Gautschi, Orthogonal polynomials—constructive theory and applications, Proceedings of the international conference on computational and applied mathematics (Leuven, 1984), 1985, pp. 61–76. MR 793944, DOI 10.1016/0377-0427(85)90007-X
William B. Jones, Olav Njåstad, and E. B. Saff, Szegő polynomials associated with Wiener-Levinson filters, J. Comput. Appl. Math. 32 (1990), no. 3, 387–406. MR 1090886, DOI 10.1016/0377-0427(90)90044-Z
William B. Jones, Olav Njåstad, and W. J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989), no. 2, 113–152. MR 976057, DOI 10.1112/blms/21.2.113
William B. Jones, W. J. Thron, Olav Njåstad, and Haakon Waadeland, Szegő polynomials applied to frequency analysis, J. Comput. Appl. Math. 46 (1993), no. 1-2, 217–228. Computational complex analysis. MR 1222483, DOI 10.1016/0377-0427(93)90297-O
W. B. Jones, O. Njåstad, and H. Waadeland, An alternative way of using Szegő polynomials in frequency analysis, Continued fractions and orthogonal functions (Loen, 1992) Lecture Notes in Pure and Appl. Math., vol. 154, Dekker, New York, 1994, pp. 141–152. MR 1263252
William B. Jones and Vigdis Petersen, Continued fractions and Szegő polynomials in frequency analysis and related topics, Proceedings of the International Conference on Rational Approximation, ICRA99 (Antwerp), 2000, pp. 149–174. MR 1783289, DOI 10.1023/A:1006454131615
William B. Jones, W. J. Thron, and Haakon Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc. 261 (1980), no. 2, 503–528. MR 580900, DOI 10.1090/S0002-9947-1980-0580900-4
A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
K. Pan, A refined Wiener-Levinson method in frequency analysis, SIAM J. Math. Anal. 27 (1996), no. 5, 1448–1453. MR 1402449, DOI 10.1137/S0036141094273878
K. Pan and E. B. Saff, Asymptotics for zeros of Szegő polynomials associated with trigonometric polynomial signals, J. Approx. Theory 71 (1992), no. 3, 239–251. MR 1191574, DOI 10.1016/0021-9045(92)90118-8
R. A. Sack and A. F. Donovan, An algorithm for Gaussian quadrature given modified moments, Numer. Math. 18 (1971/72), 465–478. MR 303693, DOI 10.1007/BF01406683
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
Walter Van Assche, Orthogonal polynomials in the complex plane and on the real line, Special functions, $q$-series and related topics (Toronto, ON, 1995) Fields Inst. Commun., vol. 14, Amer. Math. Soc., Providence, RI, 1997, pp. 211–245. MR 1448688
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Alexei Zhedanov, On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval, J. Approx. Theory 94 (1998), no. 1, 73–106. MR 1637803, DOI 10.1006/jath.1998.3179