Abstract
In the present paper we study the orthogonal polynomials with respect to a measure which is the sum of a finite positive Borel measure on [0,2π] and a Bernstein–Szegö measure. We prove that the measure sum belongs to the Szegö class and we obtain several properties about the norms of the orthogonal polynomials, as well as, about the coefficients of the expression which relates the new orthogonal polynomials with the Bernstein–Szegö polynomials. When the Bernstein–Szegö measure corresponds to a polynomial of degree one, we give a nice explicit algebraic expression for the new orthogonal polynomials.
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References
A. Cachafeiro, F. Marcellán, and C. Pérez, Lebesgue perturbation of a quasi-definite hermitian functional. The positive definite case, Linear Algebra Appl. 369 (2003) 235–250.
A. Cachafeiro and C. Pérez, A study of the Laguerre–Hahn affine functionals on the unit circle, J. Comput. Anal. Appl. (2004).
L. Daruis, P. González Vera, and F. Marcellán, Gaussian quadrature formulae on the unit circle, J. Comput. Appl. Math. 140 (2002) 159–183.
S. Elhay, G.H. Golub, and J. Kautsky, Jacobi matrices for sums of weight functions, BIT 32 (1992) 144–166.
P. García-Lázaro and F. Marcellán, Christoffel formulas for n-kernels associated to Jordan arcs, in: Orthogonal Polynomials and Applications, ed. C. Brezinski et al., Lecture Notes in Mathematics, Vol. 1171 (Springer, New York, 1985) pp. 195–203.
Y.L. Geronimus, Polynomials Orthogonal on a Circle and Their Applications, American Mathematical Society Translations Series 1, Vol. 3 (Amer. Math. Soc., Providence, RI, 1962) pp. 1–78.
U. Grenander and G. Szegö, Toeplitz Forms and Their Applications (Univ. of California Press, Berkeley, 1958).
T. Kailath and B. Porat, State-space generators for orthogonal polynomials, in: Prediction Theory and Harmonic Analysis, eds. V. Mandrekar and H. Salehi (North-Holland, Amsterdam, 1983) pp. 131–163.
V.F. Pisarenko, The retrieval of harmonics from a covariance function, Geophys. J. Royal Austral. Soc. 33 (1973) 347–366.
G. Szegö, Orthogonal Polynomials, 4th ed., American Mathematics Society Colloquium Publications, Vol. 23 (Amer. Math. Soc., Providence, RI, 1975).
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Communicated by Jesus Carnicer and Juan Manuel Peña
To Professor Dr. Mariano Gasca with occasion of his 60th anniversary.
The research was supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant number BFM2000-0015, as well as BFM2003-06335-C03-C02.
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Cachafeiro, A., Marcellán, F. & Pérez, C. Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein–Szegö measure. Adv Comput Math 26, 81–104 (2007). https://doi.org/10.1007/s10444-004-7644-x
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DOI: https://doi.org/10.1007/s10444-004-7644-x