On Freud’s equations for exponential weights. (English) Zbl 0619.42015
Let \(\{p_ n\}^{\infty}_{n=0}\) be the sequence of orthonormal polynomials associated with the weight exp(-f(x)), \(x\in (- \infty,\infty)\), and let \(a_{n+1}p_{n+1}(x)=(x-b_ n)p_ n(x)-a_ np_{n-1}(x)\) be the corresponding three-term recurrence relation. For the case \(f(x)=| x|^{\alpha},\alpha >1\), Freud formulated a conjecture concerning the asymptotic behavior, as \(n\to \infty\), of the recursion coefficients \(a_ n\) and \(b_ n\). In this important paper the author proves Freud’s conjecture when f(x) is a polynomial of even degree with positive leading coefficient. Extensions of the method used for the proof, which should lead to a proof in the general case, are suggested.
Reviewer: L.Gatteschi
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
References:
| [1] | Aitken, A. C., Determinants and Matrices (1946), Oliver & Boyd: Oliver & Boyd Edinburgh · Zbl 0063.00033 |
| [2] | Akhiezer, N. I., The Classical Moment Problem (1965), Oliver & Boyd: Oliver & Boyd Edinburgh · Zbl 0135.33803 |
| [4] | Bauldry, W. C., Orthogonal Polynomials Associated with Exponential Weights, (Ph.D. dissertation (1985), Ohio State Univ. Columbus: Ohio State Univ. Columbus Ohio) |
| [6] | Berezanskiĭ, Ju. M., Expansions in Eigenfunctions of Selfadjoint Operators, (Transl. Math. Monographs, Vol. 17 (1968), Amer. Math. Soc: Amer. Math. Soc Providence, R.I) · Zbl 0142.37203 |
| [7] | Bessis, D., A new method in the combinatorics of the topological expansion, Commun. Math. Phys., 69, 147-163 (1979) |
| [8] | Bessis, D.; Itzykson, C.; Zuber, J. B., Quantum field theory techniques in graphical enumeration, Advan. Appl. Math., 1, 109-157 (1980) · Zbl 0453.05035 |
| [9] | Bonan, S. S., Applications of G. Freud’s theory, I, (Chui, C. K.; etal., Approximation Theory, IV (1984), Academic Press: Academic Press New York), 347-351 · Zbl 0541.33004 |
| [11] | Bonan, S. S.; Nevai, P., Orthogonal polynomials and their derivatives. I, J. Approx. Theory, 40, 134-147 (1984) · Zbl 0533.42015 |
| [13] | Brualdi, A.; Schneider, H., Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley, Linear Algebra Appl., 52-53, 769-795 (1983) · Zbl 0533.15007 |
| [14] | Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon & Breach: Gordon & Breach New York · Zbl 0389.33008 |
| [15] | Freud, G., Orthogonal Polynomials (1971), Akadémiai Kiadó-Pergamon: Akadémiai Kiadó-Pergamon Budapest · Zbl 0226.33014 |
| [16] | Freud, G., On the coefficients in the recursion formulae of orthogonal polynomials, (Proc. Roy. Irish Acad. Sect. A, 76 (1976)), 1-6, (1) · Zbl 0327.33008 |
| [17] | Freud, G., On the zeros of orthogonal polynomials with respect to measures with non-compact support, Anal. Numér. Théor. Approx., 6, 125-131 (1977) · Zbl 0383.33008 |
| [18] | Freud, G., On the greatest zero of an orthogonal polynomial, J. Approx. Theory, 46, 15-23 (1986) |
| [19] | Goh, W., Asymptotic expansions for Pollaczek polynomials, manuscript (1978), Ohio State Univ: Ohio State Univ Colombus, Ohio |
| [20] | Goncar, A. A.; Rahmanov, E. A., Equilibrium measure and distribution of zeros of extremal polynomials, Mat. Sb. (N.S.), 125, 167, 117-127 (1984) |
| [21] | Grenander, U.; Szegö, G., Toeplitz Forms and Their Applications (1958), Univ. of California Press: Univ. of California Press Berkley and Los Angeles · Zbl 0080.09501 |
| [23] | Knopfmacher, A., Linear Operators and Christoffel Functions associated with Orthogonal Polynomials, (Ph.D. dissertation (1985), Univ. of Witwatersrand: Univ. of Witwatersrand Johannesburg) |
| [24] | Laguerre, E., Sur la réduction en fractions continues d’une fraction qui satisfait a une équation différentielle linéaire du premier ordre dont les coefficients sont rationnels, J. Math. Pure Appl.. (Oeuvres de Laguerre, Vol. II (1972), Chelsea: Chelsea New York), 1, 685-711 (1885), reprinted in · JFM 17.0304.01 |
| [27] | Lew, J. S.; Quarles, D. A., Nonnegative solutions of a nonlinear recurrence, J. Approx. Theory, 38, 357-379 (1983) · Zbl 0518.42029 |
| [28] | Lubinsky, D. S., A weighted polynomial inequality, (Proc. Amer. Math. Soc., 92 (1984)), 263-267 · Zbl 0518.41009 |
| [30] | Lubinsky, D. S., Estimates of Freud-Christoffel functions for some weights with the whole real line as support, J. Approx. Theory, 44, 343-379 (1985) · Zbl 0584.42015 |
| [31] | Lubinsky, D. S., On Nevai’s bounds for orthogonal polynomials associated with exponential weights, J. Approx. Theory, 44, 86-91 (1985) · Zbl 0605.42020 |
| [32] | Lubinsky, D. S., Even entire functions absolutely monotone in [0, ∞) and weights on the whole real line, (Brezinski, C.; etal., Orthogonal Polynomials and Their Applications. Orthogonal Polynomials and Their Applications, Lecture Notes in Mathematics, Vol. 1171 (1985), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0599.41048 |
| [34] | Lubinsky, D. S.; Rabinowitz, P., Rates of convergence of Gaussian quadrature for singular integrands, Math. Comput., 43, 219-242 (1984) · Zbl 0574.41028 |
| [35] | Magnus, Al, Recurrence coefficients for orthogonal polynomials on connected and non-connected sets, (Wuytack, L., Padé Approximation and Its Applications, Antwerp 1979. Padé Approximation and Its Applications, Antwerp 1979, Lecture Notes in Math., Vol. 765 (1979), Springer-Verlag: Springer-Verlag New York), 150-171 · Zbl 0431.41021 |
| [36] | Magnus, Al, Recurrence coefficients in case of Anderson localization, (de Bruin, M. G.; van Rossum, H., Padé Approximation and its Applications, Amsterdam 1980. Padé Approximation and its Applications, Amsterdam 1980, Lecture Notes in Math., Vol. 888 (1981), Springer-Verlag: Springer-Verlag New York), 309-313 · Zbl 0477.40003 |
| [37] | Magnus, Al, Riccati acceleration of Jacobi continued fractions and Laguerre-Hahn orthogonal polynomials, (Werner, H.; Bünger, H. J., Padé Approximation and its Applications Bad Honnef 1983. Padé Approximation and its Applications Bad Honnef 1983, Lecture Notes in Math., Vol. 1071 (1984), Springer-Verlag: Springer-Verlag New York), 213-230 · Zbl 0539.40003 |
| [38] | Magnus, Al, A proof of Freud’s conjecture about orthogonal polynomials related to \(¦x¦^ϱ exp(−x^{2m})\), (Brezinski, C.; etal., Orthogonal Polynomials and Their Applications, Laguerre Symposium. Orthogonal Polynomials and Their Applications, Laguerre Symposium, Bar-le-Duc, 1984. Orthogonal Polynomials and Their Applications, Laguerre Symposium. Orthogonal Polynomials and Their Applications, Laguerre Symposium, Bar-le-Duc, 1984, Lecture Notes in Mathematics, Vol. 1171 (1985), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0606.42019 |
| [39] | Magnus, Al, Asymptotic behaviour of continued fraction coefficients related to singularities of the weight function, (Pettifor, D. G.; Weaire, D. L., The Recursion Method and Its Applications. The Recursion Method and Its Applications, Solid-State Sciences, Vol. 58 (1985), Springer-Verlag: Springer-Verlag New York), 22-45 · Zbl 0625.42011 |
| [40] | Máté, A.; Nevai, P., Asymptotics for solutions of smooth recurrence equations, (Proc. Amer. Math. Soc., 93 (1985)), 423-429 · Zbl 0574.42017 |
| [43] | Máté, A.; Nevai, P.; Zaslavsky, T., Asymptotic Expansion of Ratios of Coefficients of Orthogonal Polynomials with Exponential Weights, Trans. Amer. Math. Soc., 287, 495-505 (1985) · Zbl 0536.42023 |
| [44] | Mhaskar, H. N.; Saff, E. B., Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc., 285, 203-234 (1984) · Zbl 0546.41014 |
| [45] | Mhaskar, H. N.; Saff, E. B., Polynomials with Laguerre weights in \(L_p\), (Graves-Morris, P. R.; Saff, E. B.; Varga, R. S., Rational Approximation and Interpolation. Rational Approximation and Interpolation, Lecture Notes in Math., Vol. 1105 (1984), Springer-Verlag: Springer-Verlag New York), 510-523 · Zbl 0565.41018 |
| [46] | Mhaskar, H. N.; Saff, E. B., Weighted polynomials on finite and infinite intervals: A unified approach, Bull. Amer. Math. Soc., 11, 351-354 (1984) · Zbl 0565.41017 |
| [47] | Mhaskar, H. N.; Saff, E. B., Where does the sup norm of a weighted polynomial live?, Constructive Approx., 1, 71-91 (1985) · Zbl 0582.41009 |
| [48] | Nevai, P., Some properties of orthogonal polynomials corresponding to the weight \((1 + x^{2k})^{α\) · Zbl 0288.42008 |
| [49] | Nevai, P., Orthogonal polynomials on the real line associated with the weight \(¦x¦^α exp(−¦x¦^β)\), I\), Acta Math. Sci. Math. Hungar., 24, 335-342 (1973), [Russian] · Zbl 0293.33010 |
| [50] | Nevai, P., Lagrange interpolation at zeros of orthogonal polynomials, (Lorentz, G. G.; Chui, C. K.; Schumaker, L. L., Approximation Theory, I (1976), Academic Press: Academic Press New York), 163-201 · Zbl 0343.41002 |
| [51] | Nevai, P., Orthogonal Polynomials, Mem. Amer. Math. Soc., 213, 1-185 (1979) · Zbl 0405.33009 |
| [52] | Nevai, P., Orthogonal polynomials associated with exp(−\(x^4)\), (Second Edmonton Conference on Approximation Theory. Second Edmonton Conference on Approximation Theory, Canadian Math. Soc. Conference Proceedings, Vol. 3 (1983)), 263-285 · Zbl 0561.42013 |
| [53] | Nevai, P., Asymptotics for orthogonal polynomials associated with exp(−\(x^4)\), SIAM J. Math. Anal., 15, 1177-1187 (1984) · Zbl 0566.42016 |
| [54] | Nevai, P., Two of my favorite ways of obtaining asymptotics for orthogonal polynomials, (Butzer, P. L.; Stens, R. L.; Sz.-Nagy, B., Functional Analysis and Approximation. Functional Analysis and Approximation, ISNM 65 (1984), Birkhäuser Verlag: Birkhäuser Verlag Basel), 417-436 · Zbl 0628.33009 |
| [55] | Nevai, P., Exact bounds for orthogonal polynomials associated with exponential weights, J. Approx. Theory, 44, 82-85 (1985) · Zbl 0605.42019 |
| [59] | Nevai, P.; Dehesa, J. S., On asymptotic average properties of zeros of orthogonal polynomials, SIAM, J. Math. Anal., 10, 1184-1192 (1979) · Zbl 0434.33010 |
| [61] | Poincaré, H., On linear ordinary differential and difference equations, Amer. J. Math., 7, 203-258 (1885), [French] · JFM 17.0290.01 |
| [62] | Pollaczek, F., On a four parameter family of orthogonal polynomials, Co. Re. Acad. Sci. Paris, 230, 2254-2256 (1950), [French] · Zbl 0038.22403 |
| [63] | Pollaczek, F., On a Generalization of Jacobi Polynomials, Mémorial des Sciences Mathématiques, Vol. 131 (1956), [French] |
| [64] | Rahmanov, E. A., On asymptotic properties of polynomials orthogonal on the real axis, Soviet Math. Dokl., 24, 505-507 (1981) · Zbl 0498.30048 |
| [65] | Rahmanov, E. A., On asymptotic properties of polynomials orthogonal on the real axis, Math. USSR Sb., 47, 155-193 (1984) · Zbl 0522.42018 |
| [66] | Saff, E. B., Incomplete and orthogonal polynomials, (Chui, C. K.; etal., Approximation Theory, IV (1983), Academic Press: Academic Press New York), 219-256 · Zbl 0563.41006 |
| [67] | Sheen, R., Orthogonal Polynomials Associated with \(exp(−x^66)\), (Ph.D. dissertation (1984), Ohio State Univ: Ohio State Univ Columbus, Ohio) |
| [69] | Shohat, J. A., Mechanical quadratures and zeros of Chebyshev polynomials in infinite intervals, Co. Re. Acad. Sci. Paris, 185, 597-598 (1927), [French] · JFM 53.0335.01 |
| [70] | Shohat, J. A., On a wide class of algebraic continued fractions and the corresponding Chebyshev polynomials, Co. Re. Acad. Sci. Paris, 191, 989-990 (1930), [French] · JFM 56.0946.01 |
| [71] | Shohat, J. A., General Theory of Orthogonal Polynomials of Chebyshev, Mémorial des Sciences Mathématiques, Vol. 66, 1.69 (1934), Paris · JFM 60.1037.01 |
| [72] | Shohat, J. A., A differential equation for orthogonal polynomials, Duke Math. J., 5, 401-417 (1939) · JFM 65.0285.03 |
| [73] | Szegö, G., Orthogonal Polynomials, (Amer. Math. Soc. Coll. Publ., Vol. 23 (1975), Amer. Math. Soc: Amer. Math. Soc Providence, R.I) · JFM 65.0278.03 |
| [74] | Szegö, G., (Askey, R., Collected Papers (1982), Birkhäuser-Verlag: Birkhäuser-Verlag Boston) |
| [75] | Ullman, J. L., Orthogonal polynomials associated with an infinite interval, Michigan Math. J., 27, 353-363 (1980) · Zbl 0455.33004 |
| [76] | Ullman, J. L., On orthogonal polynomials associated with an infinite interval, (Cheney, E., Approximation Theory, III (1980), Academic Press: Academic Press New York), 889-895 · Zbl 0455.33004 |
| [80] | Wall, H. S., Analytic Theory of Continued Fractions (1948), Van Nostrand: Van Nostrand Princeton · Zbl 0035.03601 |
| [81] | Zygmund, A., (Trigonometric Series, Vol. 2 (1977), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0367.42001 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
