## 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

## Additional Information

**C. F. Bracciali**- Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, UNESP- Universidade Estadual Paulista, 15054-000 São José do Rio Preto, São Paulo, Brazil
**Xin Li**- Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
**A. Sri Ranga**- Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, UNESP- Universidade Estadual Paulista, 15054-000 São José do Rio Preto, São Paulo, Brazil
- MR Author ID: 238837
- Received by editor(s): March 8, 2003
- Received by editor(s) in revised form: August 14, 2003
- Published electronically: May 25, 2004
- Additional Notes: This research was started while the second author was visiting the campus of UNESP at São José do Rio Preto, during September/October 2002, with a Fellowship from FAPESP. The first and the third authors’ research is supported by grants from CNPq and FAPESP
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp.
**74**(2005), 341-362 - MSC (2000): Primary 42C05, 94A11, 94A12
- DOI: https://doi.org/10.1090/S0025-5718-04-01672-2
- MathSciNet review: 2085896