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Real orthogonal polynomials in frequency analysis
Author(s):
C.
F.
Bracciali;
Xin
Li;
A.
Sri
Ranga.
Journal:
Math. Comp.
74
(2005),
341-362.
MSC (2000):
Primary 42C05, 94A11, 94A12
Posted:
May 25, 2004
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Additional information
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 Szego polynomials from the given moments.
References:
-
- 1.
- G.S. Ammar, D. Calvetti and L. Reichel, Continuation methods for the computation of zeros of Szego polynomials, Linear Algebra and its Appl., 249 (1996), 125-155.MR 97j:65081
- 2.
- A.C. Berti and A. Sri Ranga, Companion orthogonal polynomials: some applications, Appl. Numer. Maths., 39 (2001), 127-149.MR 2002k:33006
- 3.
- C.F. Bracciali, A.P. da Silva and A. Sri Ranga, Szego polynomials: some relations to
 -orthogonal and orthogonal polynomials, J. Comput. Appl. Math., 153 (2003), 79-88. - 4.
- R. Bressan, S.F. Menegasso and A. Sri Ranga, Szego polynomials: quadrature rules on the unit circle and on
![$[-1, 1]$](/mcom/2005-74-249/S0025-5718-04-01672-2/gif-references0/img2.gif) , Rocky Mountain J. Math., 33 (2003), 567-584. - 5.
- L. Daruis, O. Njåstad and W. Van Assche, Para-orthogonal polynomials in frequency analysis, Rocky Mountain J. Math., 33 (2003), 629-645.
- 6.
- P. Delsarte and Y. Genin, The split Levinson algorithm, IEEE Trans. Acoust. Speech Signal Process, 34 (1986), 470-478.MR 87f:94007
- 7.
- P. Delsarte and Y. Genin, An introduction to the class of split Levinson algorithms, in Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms (G.H. Golub and P. Van Dooren, eds.), NATO ASI Series F, Vol. 70, pp. 111-130, Springer-Verlag, 1991.MR 92k:65076
- 8.
- W. Gautschi, Orthogonal polynomials--Constructive theory and applications, J. Comput. Appl. Math., 12/13 (1985), 61-76.MR 87a:65045
- 9.
- W.B. Jones, O. Njåstad and E.B. Saff, Szego polynomials associated with Wiener-Levinson filters, J. Comput. Appl. Math., 32 (1990), 387-407.MR 92e:94001
- 10.
- W.B. Jones, O. Njåstad and W.J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc., 21 (1989), 113-152. MR 90e:42027
- 11.
- W.B. Jones, O. Njåstad, W.J. Thron and H. Waadeland, Szego polynomials applied to frequency analysis, J. Comput. Appl. Math., 46 (1993), 217-228.MR 94g:94003
- 12.
- W.B. Jones, O. Njåstad and H. Waadeland, An alternative way of using Szego polynomials in frequency analysis, in Continued Fractions and Orthogonal Functions (S.C. Cooper and W.J. Thron, eds.), Lecture Notes in Pure and Applied Mathematics, Vol. 154, pp. 141-152, Marcel Dekker, 1994. MR 95h:94001
- 13.
- W.B. Jones and V. Peterson, Continued fractions and Szego polynomials in frequency analysis and related topics, Acta Appl. Math., 61 (2000), 149-174. MR 2001f:65019
- 14.
- W.B. Jones, W.J. Thron and H. Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc., 261 (1980), 503-528. MR 81j:30055
- 15.
- N. Levinson, The Wiener RMS (root mean square) error criterion in filter design and prediction, J. Math. Phys. Mass. Inst. Techn., 25 (1947), 261-278.MR 8:391e
- 16.
- K. Pan, A refined Wiener-Levinson method in frequency analysis, SIAM J. Math. Anal., 27 (1996), 1448-1453. MR 97h:94002
- 17.
- K. Pan and E.B. Saff, Asymptotics for zeros of Szego polynomials associated with trigonometric polynomial signals, J. Approx. Theory, 71 (1992), 239-251. MR 94d:41013
- 18.
- R.A. Sack and A.F. Donovan, An algorithm for Gaussian quadrature given modified moments, Numer. Math., 18 (1971/72), 465-478. MR 46:2829
- 19.
- G. Szego, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975. MR 51:8724
- 20.
- W. Van Assche, Orthogonal polynomials in the complex plane and on the real line, in Fields Institute Communications 14: Special functions,
 -series and related topics (M.E.H. Ismail et al., eds.), Amer. Math. Soc. (1997), 211-245. MR 98i:33014 - 21.
- N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series, The Technology Press of the Massachusetts Institute of Technology and John Wiley and Sons, 1949. MR 11:118j
- 22.
- A. 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), 73-106. MR 2000a:42040
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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
DOI:
10.1090/S0025-5718-04-01672-2
PII:
S 0025-5718(04)01672-2
Keywords:
Frequency analysis problem,
frequency estimation,
orthogonal polynomials,
Szeg\H{o} polynomials,
para-orthogonal polynomials,
quadrature
Received by editor(s):
March 8, 2003
Received by editor(s) in revised form:
August 14, 2003
Posted:
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
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Copyright
2004,
American Mathematical Society
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