PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 2, February 2007, Pages 427–435 S 0002-9939(06)08694-1 Article electronically published on August 4, 2006 ZERO DISTRIBUTION OF M ¨ UNTZ EXTREMAL POLYNOMIALS IN L p [0, 1] D. S. LUBINSKY AND E. B. SAFF (Communicated by Jonathan M. Borwein) Abstract. Let {λ j } ∞ j=0 be a sequence of distinct positive numbers. Let 1 ≤ p ≤∞ and let T n,p = T n,p {λ 0 ,λ 1 ,λ 2 ,...,λ n } (x) denote the L p extremal M¨ untz polynomial in [0, 1] with exponents λ 0 ,λ 1 ,λ 2 ,...,λ n . We investigate the zero distribution of {T n,p } ∞ n=1 . In particular, we show that if lim n→∞ λ n n = α> 0, then the normalized zero counting measure of T n,p converges weakly as n →∞ to α π t α−1  t α (1 − t α ) dt, while if α = 0 or ∞, the limiting measure is a Dirac delta at 0 or 1, respectively. 1. Introduction and results Let λ 1 ,λ 2 ,... be a sequence of distinct positive numbers. An expression of the form (1.1) n  j=0 c j x λ j is called a M¨ untz polynomial. The name refers, of course, to the famous theorem of M¨ untz that if inf j λ j > 0, these polynomials are dense in L p spaces iff ∞  j=0 1 λ j = ∞. M¨ untz polynomials share many of the properties of ordinary algebraic polynomials. The most fundamental is that a polynomial of the form (1.1) has at most n distinct zeros in (0, ∞), or is identically zero. M¨ untz extremal polynomials are generalizations of classical orthogonal and Che- byshev polynomials. They have been investigated by, amongst others, Borwein and Erdelyi [2] and Milovanovic and his coworkers [3]. Let 1 ≤ p ≤∞. We denote by Received by the editors August 29, 2005. 2000 Mathematics Subject Classification. Primary 41A10, 41A17, 42C99; Secondary 33C45. Key words and phrases. Zero distribution, M¨ untz extremal polynomials, M¨ untz orthogonal polynomials. The research of the first author was supported by NSF grant DMS0400446. The research of the second author was supported by NSF grant DMS0532154. c 2006 American Mathematical Society 427 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use