Orthogonal polynomials on the unit circle

In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle. They were introduced by Szegő (1920, 1921, 1939).

Definition

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Let   be a probability measure on the unit circle   and assume   is nontrivial, i.e., its support is an infinite set. By a combination of the Radon-Nikodym and Lebesgue decomposition theorems, any such measure can be uniquely decomposed into

 ,

where   is singular with respect to   and   with   the absolutely continuous part of  .[1]


The orthogonal polynomials associated with   are defined as

 ,

such that

 .

The Szegő recurrence

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The monic orthogonal Szegő polynomials satisfy a recurrence relation of the form

 
 

for   and initial condition  , with

 

and constants   in the open unit disk   given by

 

called the Verblunsky coefficients. [2] Moreover,

 .

Geronimus' theorem states that the Verblunsky coefficients associated with   are the Schur parameters:[3]

 

Verblunsky's theorem

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Verblunsky's theorem states that for any sequence of numbers   in   there is a unique nontrivial probability measure   on   with  .[4]

Baxter's theorem

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Baxter's theorem states that the Verblunsky coefficients form an absolutely convergent series if and only if the moments of   form an absolutely convergent series and the weight function   is strictly positive everywhere.[5]

Szegő's theorem

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For any nontrivial probability measure   on  , Verblunsky's form of Szegő's theorem states that

 

The left-hand side is independent of   but unlike Szegő's original version, where  , Verblunsky's form does allow  .[6] Subsequently,

 .

One of the consequences is the existence of a mixed spectrum for discretized Schrödinger operators.[7]

Rakhmanov's theorem

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Rakhmanov's theorem states that if the absolutely continuous part   of the measure   is positive almost everywhere then the Verblunsky coefficients   tend to 0.

Examples

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The Rogers–Szegő polynomials are an example of orthogonal polynomials on the unit circle.

See also

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Notes

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  1. ^ Simon 2005a, p. 43.
  2. ^ Simon 2010, p. 44.
  3. ^ Simon 2010, p. 74.
  4. ^ Schmüdgen 2017, p. 265.
  5. ^ Simon 2005a, p. 313.
  6. ^ Simon 2010, p. 29.
  7. ^ Totik 2016, p. 269.

References

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  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials on the unit circle", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
  • Schmüdgen, Konrad (2017). The Moment Problem. Graduate Texts in Mathematics. Vol. 277. Cham: Springer International Publishing. doi:10.1007/978-3-319-64546-9. ISBN 978-3-319-64545-2. ISSN 0072-5285.
  • Simon, Barry (2005). Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications. Vol. 54. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3446-6. MR 2105088.{{cite book}}: CS1 maint: date and year (link)
  • Simon, Barry (2005). Orthogonal polynomials on the unit circle. Part 2. Spectral theory. American Mathematical Society Colloquium Publications. Vol. 54. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3675-0. MR 2105089.{{cite book}}: CS1 maint: date and year (link)
  • Simon, Barry (2010). Szegő's theorem and its descendants: spectral theory for L² perturbations of orthogonal polynomials. Princeton University Press. ISBN 978-0-691-14704-8.
  • Szegő, Gábor (1920), "Beiträge zur Theorie der Toeplitzschen Formen", Mathematische Zeitschrift, 6 (3–4): 167–202, doi:10.1007/BF01199955, ISSN 0025-5874, S2CID 118147030
  • Szegő, Gábor (1921), "Beiträge zur Theorie der Toeplitzschen Formen", Mathematische Zeitschrift, 9 (3–4): 167–190, doi:10.1007/BF01279027, ISSN 0025-5874, S2CID 125157848
  • Szegő, Gábor (1939), Orthogonal Polynomials, Colloquium Publications, vol. XXIII, American Mathematical Society, ISBN 978-0-8218-1023-1, MR 0372517
  • Totik, V. (2016). "Barry Simon and the János Bolyai International Mathematical Prize" (PDF). Acta Mathematica Hungarica. 149 (2). Springer Science and Business Media LLC: 263–273. doi:10.1007/s10474-016-0618-x. ISSN 0236-5294. S2CID 254236846.