Summability of Jacobi series
HTML articles powered by AMS MathViewer
- by Richard Askey PDF
- Trans. Amer. Math. Soc. 179 (1973), 71-84 Request permission
Abstract:
The positivity of some Cesàro mean is proven for Jacobi series $\Sigma {a_n}P_n^{(\alpha ,\beta )}(x),\alpha ,\beta \geqq - \tfrac {1}{2}$. This has applications to the mean convergence of Lagrange interpolation at the zeros of Jacobi polynomials. The positivity of the $(C,\alpha + \beta + 2)$ means is conjectured and proven for some $(\alpha ,\beta )$. One consequence of this conjecture would be the complete monotonicity of ${x^{ - c}}{({x^2} + 1)^{ - c}},c \geqq 1$.References
- R. Askey, Mean convergence of orthogonal series and Lagrange interpolation, Acta Math. Acad. Sci. Hungar. 23 (1972), 71–85. MR 322400, DOI 10.1007/BF01889904 —, Nonnegative sums of Jacobi polynomials, Tôhoku Math. J. (2) 24 (1972), 109-119.
- Richard Askey and James Fitch, Positivity of the Cotes numbers for some ultraspherical abscissas, SIAM J. Numer. Anal. 5 (1968), 199–201. MR 228166, DOI 10.1137/0705016
- Richard Askey and James Fitch, Integral representations for Jacobi polynomials and some applications, J. Math. Anal. Appl. 26 (1969), 411–437. MR 237847, DOI 10.1016/0022-247X(69)90165-6
- Richard Askey and Harry Pollard, Some absolutely monotonic and completely monotonic functions, SIAM J. Math. Anal. 5 (1974), 58–63. MR 340935, DOI 10.1137/0505008
- Richard Askey and Stephen Wainger, A convolution structure for Jacobi series, Amer. J. Math. 91 (1969), 463–485. MR 264132, DOI 10.2307/2373520 W. N. Bailey, The generating function of Jacobi polynomials, J. London Math. Soc. 13 (1938), 8-12. —, Hypergeometric functions, Cambridge Univ. Press, Cambridge, 1935. R. G. Cooke, A monotonic property of Bessel functions, J. London Math. Soc. 12 (1937), 180-185.
- G. K. Eagleson, A characterization theorem for positive definite sequences on the Krawtchouk polynomials, Austral. J. Statist. 11 (1969), 29–38. MR 328162, DOI 10.1111/j.1467-842X.1969.tb00004.x L. Fejér, Sur les fonctions bornées et intégrables, C. R. Acad. Sci. Paris 131 (1900), 984-987; Gesammelte Arbeiten I, 37-41. —, Einige Sätze, die sich aug das Vorzeichen einer ganzen rationalen Funktion beziehen usw., Monatsh. Math. 35 (1928), 305-344; Gesammelte Arbeiten II, 202-237. —, Neue Eigenschaften der Mittelwerte bei den Fourierreihen, J. London Math. Soc. 8 (1933), 53-62; Gesammelte Arbeiten II, 493-501.
- Ervin Feldheim, On the positivity of certain sums of ultraspherical polynomials, J. Analyse Math. 11 (1963), 275–284. MR 158107, DOI 10.1007/BF02789988
- George Gasper, Positivity and the convolution structure for Jacobi series, Ann. of Math. (2) 93 (1971), 112–118. MR 284628, DOI 10.2307/1970755
- George Gasper, Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math. (2) 95 (1972), 261–280. MR 310536, DOI 10.2307/1970800 E. Kogbetliantz, Recherches sur la sommabilité des séries ultersphériques par la méthode des moyennes arithmétiques, J. Math. Pures Appl. (9) 3 (1924), 107-187.
- Tom Koornwinder, Jacobi polynomials. II. An analytic proof of the product formula, SIAM J. Math. Anal. 5 (1974), 125–137. MR 385198, DOI 10.1137/0505014
- Lee Lorch, M. E. Muldoon, and Peter Szego, Higher monotonicity properties of certain Sturm-Liouville functions. III, Canadian J. Math. 22 (1970), 1238–1265. MR 274845, DOI 10.4153/CJM-1970-142-1
- E. Makai, On a monotonic property of certain Sturm-Liouville functions, Acta Math. Acad. Sci. Hungar. 3 (1952), 165–172 (English, with Russian summary). MR 54103, DOI 10.1007/BF02022519
- Earl D. Rainville, Special functions, The Macmillan Company, New York, 1960. MR 0107725
- Elias M. Stein, Interpolation in polynomial classes and Markoff’s inequality, Duke Math. J. 24 (1957), 467–476. MR 91368
- John Steinig, On a monotonicity property of Bessel functions, Math. Z. 122 (1971), no. 4, 363–365. MR 447654, DOI 10.1007/BF01110172 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1967.
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
- A. Zygmund, A property of the zeros of Legendre polynomials, Trans. Amer. Math. Soc. 54 (1943), 39–56. MR 9657, DOI 10.1090/S0002-9947-1943-0009657-X
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 179 (1973), 71-84
- MSC: Primary 42A56; Secondary 33A65
- DOI: https://doi.org/10.1090/S0002-9947-1973-0315351-7
- MathSciNet review: 0315351