×

zbMATH — the first resource for mathematics

Extensions of Szegö’s theory of orthogonal polynomials. II. (English) Zbl 0635.42023
[For part I see the second author, Lect. Notes Math. 1171, 230-238 (1985; Zbl 0591.42016)]. Let \(d\mu\) be a finite positive measure on the unit circle in the complex plane and \(\{\phi_ n(D\mu)\}\) be a system of orthonormal polynomials on the unit circle with respect to a measure \(d\mu\). If supp(d\(\mu)\) is an infinite set then \(d\mu\) is a distribution. Szegö’s theory of orthogonal polynomials is concerned with the asymptotic behavior of \(\phi_ n(d\mu)\) when log \(\mu\) ’\(\in L^ 1.\)
The authors study the asymptotic behavior of the fraction \(\phi_ n(d\mu_ 1)/\phi_ n(d\mu_ 2)\) outside the unit circle. The consequences for orthogonal polynomials on the real line are also discussed. A few printing mistakes are there in the text.
Reviewer: A.N.Srivastava

MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R. Askey, M. Ismail (1984):Recurrence relations, continued fractions and orthogonal polynomials. Mem. Amer. Math. Soc.,300. · Zbl 0548.33001
[2] G. Freud (1971): Orthogonal Polynomials. Oxford, New York: Pergamon Press. · Zbl 0226.33014
[3] Ya. L. Geronimus (1961): Orthogonal Polynomials. New York: Consultants Bureau. · Zbl 0093.26503
[4] U. Grenander, G. Szegö (1958): Toeplitz Forms and Their Applications. Berkeley: University of California Press.
[5] A. N. Kolmogorov (1941):Stationary sequences in Hilbert spaces (in Russian). Bull. Moscow State University,2:1–40. · Zbl 0063.03291
[6] M. Krein (1945):On a generalization of investigations of G. Szegö, V. Smirnoff and A. Kolmogoroff. Dokl. Adad. Nauk SSSR,46:91–94. · Zbl 0063.03355
[7] A. Máté, P. Nevai (1982):Remarks on B. A. Rahmanov’s paper ”On the asymptotics of the ratio of orthogonal polynomials”. J. Approx. Theory,36:64–72. · Zbl 0509.30029 · doi:10.1016/0021-9045(82)90071-5
[8] A. Máté, P. Nevai (1984):Sublinear perturbations of the differential equation y (n) =0and of the analogous difference equation. J. Differential Equations,53:234–257. · Zbl 0529.34044 · doi:10.1016/0022-0396(84)90041-X
[9] A. Máté, P. Nevai, V. Totik (1984):What is beyond Szegö’s theory of orthogonal polynomials. In: Rational Approximation and Interpolation (P. R. Graves-Morris, E. B. Saff, R. S. Varga, eds.). Lecture Notes in Mathematics, vol. 1105. New York: Springer-Verlag, pp. 502–510. · Zbl 0564.42012
[10] A. Máté, P. Nevai, V. Totik (1985):Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle. Constr. Approx.,1:63–69. · Zbl 0582.42012 · doi:10.1007/BF01890022
[11] A. Máté, P. Nevai, V. Totik (1984):Mean Cesàro summability of orthogonal polynomials. In: Constructive Theory of Function (B. Sendov, P. Petrushev, R. Maleev, S. Tashev, eds.). Sofia: Publishing House of the Bulgarian Academy of Sciences, pp. 588–599.
[12] A. Máté, P. Nevai, V. Totik (1986):Necessary conditions for weighted mean convergence of Fourier series in orthogonal polynomials. J. Approx. Theory,46:314–322. · Zbl 0605.42023 · doi:10.1016/0021-9045(86)90068-7
[13] A. Máté, P. Nevai, V. Totik (to appear):Strong and weak convergence of orthogonal polynomials. Amer. J. Math.
[14] A. Máté, P. Nevai, V. Totik (1986):Oscillatory behavior of orthogonal polynomials. Proc. Amer. Math. Soc.,96:261–268. · Zbl 0585.42024 · doi:10.1090/S0002-9939-1986-0818456-1
[15] A. Máté, P. Nevai, V. Totik (1987):Extensions of Szegö’s theory of orthogonal polynomials, III. Constr. Approx.,3:73–96. · Zbl 0635.42024 · doi:10.1007/BF01890554
[16] P. Nevai (1979):Orthogonal Polynomials. Mem. Amer. Math. Soc.,213:1–185. · Zbl 0405.33009
[17] P. Nevai (1979):Distribution of zeros of orthogonal polynomials. Trans. Amer. Math. Soc.,249:341–361. · Zbl 0413.42016 · doi:10.1090/S0002-9947-1979-0525677-5
[18] P. Nevai (1984):A new class of orthogonal polynomials. Proc. Amer. Math. Soc.,91:409–415. · Zbl 0572.42019 · doi:10.1090/S0002-9939-1984-0744640-X
[19] P. Nevai (1984):Two of my favorite ways of obtaining asymptotics for orthogonal polynomials. In Anniversary Volume on Approximation Theory and Functional Analysis (P. L. Butzer, R. L. Stens, B. Sz.-Nagy, eds.). International Series of Numerical Mathematics 65. Basel: Birkhäuser-Verlag, pp. 417–436.
[20] P. Nevai (1985):Extensions of Szegö’s theory of orthogonal polynomials. In: Orthogonal Polynomials and Their Applications (C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, A. Ronveaux, eds.). Lecture Notes in Mathematics, vol. 1171. Berlin: Springer-Verlag, pp. 230–238. · Zbl 0591.42016
[21] P. Nevai (1986):Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory,48:3–167. · Zbl 0606.42020 · doi:10.1016/0021-9045(86)90016-X
[22] F. Pollaczek (1949):Sur une généralisation des Polynomes de Legendre. C. R. Acad. Sci. Paris,228:1363–1365. · Zbl 0041.03502
[23] F. Pollaczek (1950):Sur une famille de polynomes orthogonaux à quatre paramètres. C. R. Acad. Sci. Paris,230:2254–2256. · Zbl 0038.22403
[24] F. Pollaczek (1956): Sur une Généralisation des Polynomes de Jacobi. Mémorial des Sciences Mathématiques, vol. 131. Paris: Gauthier-Villars.
[25] E. A. Rahmanov (1977):On the asymptotics of the ratio of orthogonal polynomials. Math. USSR-Sb.,32:199–213. · Zbl 0401.30033 · doi:10.1070/SM1977v032n02ABEH002377
[26] E. A. Rahmanov (1983):On the asymptotics of the ratio of orthogonal polynomials, II. Math. USSR-Sb.,46:105–117. · Zbl 0515.30030 · doi:10.1070/SM1983v046n01ABEH002749
[27] W. Rudin (1974): Real and Complex Analysis, 2nd edn. New York: McGraw-Hill. · Zbl 0278.26001
[28] V. I. Smirnov (1932):Sur les formules de Cauchy et de Green et quelques problèmes qui s’y ratachent. Izv. Akad. Nauk SSSR, 337–372. · JFM 58.1076.02
[29] V. I. Smirnov, N. A. Lebedev (1964): Constructive Theory of Functions of Complex Variables (in Russian). Moscow: Nauka.
[30] G. Szegö (1950):On certain special sets of orthogonal polynomials. Proc. Amer. Math. Soc.,1:731–737. · Zbl 0041.39202
[31] G. Szegö (1939, 1975 (4th edn.)): Orthogonal Polynomials. American Mathematical Society Colloquium Publications, vol. 23. Providence, RI: American Mathematical Society. · JFM 65.0278.03
[32] P. Turán (1975):On orthogonal polynomials. Anal. Math.,1:297–311. · Zbl 0331.42006 · doi:10.1007/BF02333179
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.