A formula for inserting point masses. (English) Zbl 1174.42032
Summary: Let d\(\mu\) be a probability measure on the unit circle and d\(\nu\) be the measure formed by adding a pure point to d\(\mu\). We give a formula for the Verblunsky coefficients of d\(\nu\), based on a result of B. Simon [Orthogonal polynomials on the unit circle. II: Spectral theory (2005; Zbl 1082.42021)].
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 30E10 | Approximation in the complex plane |
| 05E35 | Orthogonal polynomials (combinatorics) (MSC2000) |
Citations:
Zbl 1082.42021References:
| [1] | Simon, B., Orthogonal polynomials on the unit circle, Part 1: Classical theory, (AMS Colloquium Series (2005), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1082.42020 |
| [2] | Simon, B., Orthogonal polynomials on the unit circle, Part 2: Spectral theory, (AMS Colloquium Series (2005), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1082.42021 |
| [3] | Szegő, G., (Orthogonal Polynomials. Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23 (1939), American Mathematical Society: American Mathematical Society Providence, RI), 1967 · Zbl 0023.21505 |
| [4] | Geronimus, Ya. L., Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Trans., 1954, 104, p. 79 (1954) · Zbl 0058.19302 |
| [5] | Nevai, P., Orthogonal polynomials, Mem. Amer. Math. Soc., 18, 213, p. 185 (1979) · Zbl 0405.33009 |
| [6] | Cachafeiro, A.; Marcellán, F., Modifications of Toeplitz matrices: Jump functions, Rocky Mountain J. Math., 23, 521-531 (1993) · Zbl 0804.42012 |
| [7] | M.-W.L. Wong, Generalized bounded variations and inserting point masses, Constr. Approx. (in press); M.-W.L. Wong, Generalized bounded variations and inserting point masses, Constr. Approx. (in press) · Zbl 1283.42041 |
| [8] | M.-W.L. Wong, Asymptotics of polynomials and point perturbation in a gap, preprint; M.-W.L. Wong, Asymptotics of polynomials and point perturbation in a gap, preprint |
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