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A new generalization of a problem of F. Lukács

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A new generalization of a problem of F. Lukács COMPOSITIO MATHEMATICA LOUISBRICKMAN Anew generalization of a problem of F. Lukács Compositio Mathematica, tome 14 (1959-1960), p. 195-227. <http://www.numdam.org/item?id=CM_1959-1960__14__195_0> © Foundation Compositio Mathematica, 1959-1960, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ A New Generalization of a Problem of F. Lukács by Louis Brickman Acknowledgement The author wishes to acknowledge his indebtedness to Professor Isaac J. Schoenberg for sùggesting the topic, informing him of its history, and guiding him in its investigation. Introduction Let ~(x) be a real valued non-decreasing function with infinitely many points of increase in the finite or infinite interval (a, b ), and let the moments exist. Then there exists a set of polynomials {~v(x)}~0 uniquely determined by the following conditions: (a) ~v(x) is a polynomial of precise degree v in which the co- efficient of xv is positive. (b) The system {~v(x)} is orthonormal, i.e., The natural number n being given, we denote by 03A0n the class of real polynomials f(x) of degree at most n satisfying the following two conditions: This class has received much attention in connection with the problem of determining where z is a real number usually, but not always, assumed to be in (a, b ). F. Lukàcs determined in 1918 the value of Mn(+1) for all n for the special case (a, b ) = ( -1, +1) and ~(x) = x.2) For this pur- 1) G. Szegô, Orthogonal Polynomials, pp. 24-25. 2) F. L

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