## Orthogonal polynomials on several intervals via a polynomial mapping

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- by J. S. Geronimo and W. Van Assche
- Trans. Amer. Math. Soc.
**308**(1988), 559-581 - DOI: https://doi.org/10.1090/S0002-9947-1988-0951620-6
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## Abstract:

Starting from a sequence $\{ {p_n}(x; {\mu _0})\}$ of orthogonal polynomials with an orthogonality measure ${\mu _0}$ supported on ${E_0} \subset [ - 1, 1]$, we construct a new sequence $\{ {p_n}(x; \mu )\}$ of orthogonal polynomials on $E = {T^{ - 1}}({E_0})$ ($T$ is a polynomial of degree $N$) with an orthogonality measure $\mu$ that is related to ${\mu _0}$. If ${E_0} = [ - 1, 1]$, then $E = {T^{ - 1}}([ - 1, 1])$ will in general consist of $N$ intervals. We give explicit formulas relating $\{ {p_n}(x; \mu )\}$ and $\{ {p_n}(x; {\mu _0})\}$ and show how the recurrence coefficients in the three-term recurrence formulas for these orthogonal polynomials are related. If one chooses $T$ to be a Chebyshev polynomial of the first kind, then one gets sieved orthogonal polynomials.## References

- N. I. Ahiezer,
*Orthogonal polynomials on several intervals*, Soviet Math. Dokl.**1**(1960), 989–992. MR**0110916** - Waleed Al-Salam, W. R. Allaway, and Richard Askey,
*Sieved ultraspherical polynomials*, Trans. Amer. Math. Soc.**284**(1984), no. 1, 39–55. MR**742411**, DOI 10.1090/S0002-9947-1984-0742411-6 - A. I. Aptekarev,
*Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains*, Mat. Sb. (N.S.)**125(167)**(1984), no. 2, 231–258 (Russian). MR**764479** - M. F. Barnsley, J. S. Geronimo, and A. N. Harrington,
*Almost periodic Jacobi matrices associated with Julia sets for polynomials*, Comm. Math. Phys.**99**(1985), no. 3, 303–317. MR**795106** - Pierre Barrucand and David Dickinson,
*On cubic transformations of orthogonal polynomials*, Proc. Amer. Math. Soc.**17**(1966), 810–814. MR**197805**, DOI 10.1090/S0002-9939-1966-0197805-1 - J. Bellissard,
*Stability and instability in quantum mechanics*, Schrödinger operators (Como, 1984) Lecture Notes in Math., vol. 1159, Springer, Berlin, 1985, pp. 204–229. MR**824989**, DOI 10.1007/BFb0080334 - D. Bessis, J. S. Geronimo, and P. Moussa,
*Function weighted measures and orthogonal polynomials on Julia sets*, Constr. Approx.**4**(1988), no. 2, 157–173. MR**932652**, DOI 10.1007/BF02075456 - D. Bessis and P. Moussa,
*Orthogonality properties of iterated polynomial mappings*, Comm. Math. Phys.**88**(1983), no. 4, 503–529. MR**702566** - Jairo Charris and Mourad E. H. Ismail,
*On sieved orthogonal polynomials. II. Random walk polynomials*, Canad. J. Math.**38**(1986), no. 2, 397–415. MR**833576**, DOI 10.4153/CJM-1986-020-x - T. S. Chihara,
*An introduction to orthogonal polynomials*, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR**0481884** - Carl de Boor and John R. Rice,
*Extremal polynomials with application to Richardson iteration for indefinite linear systems*, SIAM J. Sci. Statist. Comput.**3**(1982), no. 1, 47–57. MR**651866**, DOI 10.1137/0903004 - J. S. Geronimo and W. Van Assche,
*Orthogonal polynomials with asymptotically periodic recurrence coefficients*, J. Approx. Theory**46**(1986), no. 3, 251–283. MR**840395**, DOI 10.1016/0021-9045(86)90065-1 - J. Geronimus,
*On the character of the solution of the moment-problem in the case of the periodic in the limit associated fraction*, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR]**5**(1941), 203–210 (Russian, with English summary). MR**0005157**
—, - Mourad E. H. Ismail,
*On sieved orthogonal polynomials. I. Symmetric Pollaczek analogues*, SIAM J. Math. Anal.**16**(1985), no. 5, 1093–1113. MR**800799**, DOI 10.1137/0516081 - Jairo Charris and Mourad E. H. Ismail,
*On sieved orthogonal polynomials. II. Random walk polynomials*, Canad. J. Math.**38**(1986), no. 2, 397–415. MR**833576**, DOI 10.4153/CJM-1986-020-x - Jairo Charris and Mourad E. H. Ismail,
*On sieved orthogonal polynomials. II. Random walk polynomials*, Canad. J. Math.**38**(1986), no. 2, 397–415. MR**833576**, DOI 10.4153/CJM-1986-020-x - V. I. Lebedev,
*Iterative methods of solving operator equations with spectrum located on several segments*, Ž. Vyčisl. Mat i Mat. Fiz.**9**(1969), 1247–1252 (Russian). MR**272171** - Attila Máté, Paul Nevai, and Vilmos Totik,
*Asymptotics for orthogonal polynomials defined by a recurrence relation*, Constr. Approx.**1**(1985), no. 3, 231–248. MR**891530**, DOI 10.1007/BF01890033 - Pierre Moussa,
*Itération des polynômes et propriétés d’orthogonalité*, Ann. Inst. H. Poincaré Phys. Théor.**44**(1986), no. 3, 315–325 (French, with English summary). MR**846471** - Paul G. Nevai,
*Orthogonal polynomials*, Mem. Amer. Math. Soc.**18**(1979), no. 213, v+185. MR**519926**, DOI 10.1090/memo/0213 - D. G. Pettifor and D. L. Weaire (eds.),
*The recursion method and its applications*, Springer Series in Solid-State Sciences, vol. 58, Springer-Verlag, Berlin, 1985. MR**798478**, DOI 10.1007/978-3-642-82444-9 - Youcef Saad,
*Iterative solution of indefinite symmetric linear systems by methods using orthogonal polynomials over two disjoint intervals*, SIAM J. Numer. Anal.**20**(1983), no. 4, 784–811. MR**708457**, DOI 10.1137/0720052 - Marshall Harvey Stone,
*Linear transformations in Hilbert space*, American Mathematical Society Colloquium Publications, vol. 15, American Mathematical Society, Providence, RI, 1990. Reprint of the 1932 original. MR**1451877**, DOI 10.1090/coll/015
G. Szegö, - M. Tsuji,
*Potential theory in modern function theory*, Maruzen Co. Ltd., Tokyo, 1959. MR**0114894**
J. C. Wheeler,

*On some finite difference equations and corresponding systems of orthogonal polynomials*, Zap. Mat. Otdel. Fiz.-Mat. Fak. i Kharkov Mak. Obsc. (4)

**25**(1957), 87-100. (Russian)

*Orthogonal polynomials*, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1975.

*Modified moments and continued fraction coefficients for the diatomic linear chain*, J. Chem. Phys.

**80**(1984), 472-476.

## Bibliographic Information

- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**308**(1988), 559-581 - MSC: Primary 42C05; Secondary 30E05, 33A65
- DOI: https://doi.org/10.1090/S0002-9947-1988-0951620-6
- MathSciNet review: 951620