Positivity 5: 1–11, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands. A Continual Analogue of a Theorem by M. Fekete and G. Pólya OLGA M. KATKOVA and ANNA M. VISHNYAKOVA Department of Mathematics Kharkov State University, Svobody Sq., 4, 310077, Kharkov, Ukraine. E-mail: firstname.lastname@example.org; email@example.com (Received 29 December 1998; accepted 7 November 1999) Mathematics Subject Classiﬁcations (2000):: 44A35, 42A85, 60E05, 44A10 1. Introduction and Statement of Results In 1912 M. Fekete and G. Pólya  proved the following theorem. THEOREM A 1. Let f.u/ D p C p uC:::C p u be a polynomial of degree n 0 1 n with real coefﬁcients such that f.u/ >0forany u > 0: (1) Then there exists a positive number such that for any > the entire function " " exp.u/f.u/ has nonnegative Taylor coefﬁcients. We are going to give a continual analogue of this theorem. For a function f VT0IC1/ ! R we denote by L.x;f / its Laplace transform: L.x;f / VD exp.xt/f.t/dt . The main result of this paper is the following: THEOREM 1. Let p VT0IC1/ ! R be a function satisfying the following conditions: 1. p 2 CT0IC1/, 2. L.x;jpj/< 1 for
Positivity – Springer Journals
Published: Oct 3, 2004
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