polynomials orthogonal on the unit circle
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1: 18.33 Polynomials Orthogonal on the Unit Circle
§18.33 Polynomials Orthogonal on the Unit Circle
►§18.33(i) Definition
… ►§18.33(iii) Connection with OP’s on the Line
… ►§18.33(v) Biorthogonal Polynomials on the Unit Circle
… ►Recurrence Relations
…2: 18.1 Notation
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3: Bibliography Z
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On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval.
J. Approx. Theory 94 (1), pp. 73–106.
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4: Bibliography S
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Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory.
American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.
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Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory.
American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.
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5: Errata
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►In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem.
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6: 18.34 Bessel Polynomials
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§18.34(ii) Orthogonality
… ►Hence the full system of polynomials cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if : … ►Orthogonality of the full system on the unit circle can be given with a much simpler weight function: …See Ismail (2009, (4.10.9)) for orthogonality on the unit circle for general values of . … ►In this limit the finite system of Jacobi polynomials which is orthogonal on (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on (see (18.34.5_5)). …7: Bibliography
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Some orthogonal
-polynomials.
Math. Nachr. 30, pp. 47–61.
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Characterization theorems for orthogonal polynomials.
In Orthogonal Polynomials (Columbus, OH, 1989),
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 1–24.
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Classical Orthogonal Polynomials.
In Orthogonal Polynomials and Applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux (Eds.),
Lecture Notes in Math., Vol. 1171, pp. 36–62.
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Recurrence relations, continued fractions, and orthogonal polynomials.
Mem. Amer. Math. Soc. 49 (300), pp. iv+108.
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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.
Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
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8: 10.59 Integrals
9: 18.35 Pollaczek Polynomials
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18.35.7
, .
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10: 3.5 Quadrature
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►For further extensions, applications, and computation of orthogonal polynomials and Gauss-type formulas, see Gautschi (1994, 1996, 2004).
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►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3.
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►Below we give for the classical orthogonal polynomials the recurrence coefficients and in (3.5.30).
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►Complex orthogonal polynomials
of degree , in that satisfy the orthogonality condition
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