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Summability of Jacobi series


Author: Richard Askey
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
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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 [Enhancements On Off] (What's this?)

  • [1] R. Askey, Mean convergence of orthogonal series and Lagrange interpolation, Acta Math. 23 (1972), 71-85. MR 0322400 (48:762)
  • [2] -, Nonnegative sums of Jacobi polynomials, Tôhoku Math. J. (2) 24 (1972), 109-119.
  • [3] R. Askey and J. Fitch, Positivity of the Cotes numbers for some ultraspherical abscissas, SIAM J. Numer. Anal. 5 (1968), 199-201. MR 37 #3750. MR 0228166 (37:3750)
  • [4] -, Integral representations for Jacobi polynomials and some applications, J. Math. Anal. Appl. 26 (1969), 411-437. MR 38 #6128. MR 0237847 (38:6128)
  • [5] R. Askey and H. Pollard, Some absolutely monotonic and completely monotonic functions, SIAM J. Math. Anal. 4 (1973). MR 0340935 (49:5685)
  • [6] R. Askey and S. Wainger, A convolution structure for Jacobi series, Amer. J. Math. 91 (1969), 463-485. MR 41 #8728. MR 0264132 (41:8728)
  • [7] W. N. Bailey, The generating function of Jacobi polynomials, J. London Math. Soc. 13 (1938), 8-12.
  • [8] -, Hypergeometric functions, Cambridge Univ. Press, Cambridge, 1935.
  • [9] R. G. Cooke, A monotonic property of Bessel functions, J. London Math. Soc. 12 (1937), 180-185.
  • [10] G. K. Eagleson, A characterization theorem for positive definite sequences on the Krawtchouk polynomials, Austral. J. Statist. 11 (1969), 29-38. MR 0328162 (48:6504)
  • [11] L. Fejér, Sur les fonctions bornées et intégrables, C. R. Acad. Sci. Paris 131 (1900), 984-987; Gesammelte Arbeiten I, 37-41.
  • [12] -, Einige Sätze, die sich aug das Vorzeichen einer ganzen rationalen Funktion beziehen usw., Monatsh. Math. 35 (1928), 305-344; Gesammelte Arbeiten II, 202-237.
  • [13] -, Neue Eigenschaften der Mittelwerte bei den Fourierreihen, J. London Math. Soc. 8 (1933), 53-62; Gesammelte Arbeiten II, 493-501.
  • [14] E. Feldheim, On the positivity of certain sums of ultraspherical polynomials, J. Analyse Math. 11 (1963), 275-284. MR 28 #1333. MR 0158107 (28:1333)
  • [15] G. Gasper, Positivity and convolution structure for Jacobi series, Ann. of Math. (2) 93 (1971), 112-118. MR 0284628 (44:1852)
  • [16] -, Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math. (2) 95 (1972), 261-280. MR 0310536 (46:9634)
  • [17] E. Kogbetliantz, Recherches sur la sommabilité des séries ultersphériques par la méthode des moyennes arithmétiques, J. Math. Pures Appl. (9) 3 (1924), 107-187.
  • [18] T. Koornwinder, Jacobi polynomials. II. An analytic proof of the product formula, SIAM J. Math. Anal. 4 (1973). MR 0385198 (52:6063)
  • [19] L. Lorch, M. E. Muldoon and P. Szego, Higher monotonicity properties of certain Sturm-Liouville functions. III, Canad. J. Math. 22 (1970), 1238-1265. MR 43 #603 MR 0274845 (43:603)
  • [20] E. Makai, On a monotonic property of certain Sturm-Liouville functions, Acta Math. Acad. Sci. Hungar. 3 (1952), 165-172. MR 14, 872. MR 0054103 (14:872e)
  • [21] E. D. Rainville, Special functions, Macmillan, New York, 1960. MR 21 #6447. MR 0107725 (21:6447)
  • [22] E. M. Stein, Interpolation in polynomial classes and Markoff's inequality, Duke Math. J. 24 (1957), 467-476. MR 19, 956. MR 0091368 (19:956b)
  • [23] J. Steinig, On a monotonicity property of Bessel functions, Math. Z. 122 (1971), 363-365. MR 0447654 (56:5964)
  • [24] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1967.
  • [25] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Univ. Press, Cambridge, 1944. MR 6, 64. MR 0010746 (6:64a)
  • [26] A. Zygmund, A property of the zeros of Legendre polynomials, Trans. Amer. Math. Soc. 54 (1943), 39-56. MR 5, 180. MR 0009657 (5:180a)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0315351-7
Keywords: Positive Cesàro sum, Jacobi polynomial, Bessel function, Poisson kernel, delayed means, Lagrange interpolation
Article copyright: © Copyright 1973 American Mathematical Society

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