SIAM J. MATH. ANAL. Vol. 10, No. 5, September 1979 (C) 1979 Society for Industrial and Applied Mathematics 0036-1410/79/1005-0018 $01.00/0 THE UNSETTLED PROBLEM OF M. G. KREIN ON NONNEGATIVE POLYNOMIALS IN T-SYSTEMS* R. K. S. RATHORE- AND P. N. AGRAWALt 2m Abstract. The unsettled problem of M. G. Krein was as follows’ If {ui}i--0 (m 1, 2,. .) is a T-system on [a, b] and T ={tl, tz," , tk}c[a, b] is a set of distinct points containing only one of the end points a, b and k =< m, then does there always exist a nonnegative linear combination of ui’s vanishing precisely on T in [a, b ]? In this note we settle this problem by furnishing a counterexample for which such a linear combination does not exist. In this note we furnish a counterexample for the unsettled problem of M. G. Krein on the existence of nonnegative polynomials in T-systems {ui}/o (m 1, 2,. .) on an interval [a, b], vanishing precisely on a given set T {tl, t2, , tk} containing only one of the end points a or b. The problem arose in the following manner. M. G. Krein [2] (see Karlin and Studden [1, Thm. 5.1, p. 28]) proved the following result. THEOREM 1. Let {Ui}= 0 be a T-system on [a, b]. Let T---{tl, t:z," ", tk}c[a, b]. Define 2, ti (a, b), w(ti) 1, ti=a orb. Then, (a) /f E k i--1 w(ti) <-n, there exists a nontrivial, nonnegative polynomial u(t)= in=o [3iui(t) vanishing precisely on T and at no other point of [a, b ]. The only exception is that if n 2m and exactly one of the end points a or b is in T then u(t) may vanish at the other end point as well. (b) If any one of the following further conditions holds then without exception the polynomial may be constructed to vanish precisely at tl, t2, , tk (i) {Ui}’- (i.e. the setoffunctions Uo, Ul, u2, un-1 excluding un) isa T-system. (ii) {ui} 7=o is a T-system on an interval [a’, b’] containing [a, b where a’ < a < b < b " (iii) {ui}=o is an ET-system of order 2 on [a, b]. Regarding the exceptional case in the above theorem Karlin and Studden 1, p. 30] remark that "it is worthwhile to notice the slightly weaker conclusion in the case where only one of the end points a or b is contained in T (and n 2m). Whether or not this exceptional case can be eliminated from the theorem has not been settled." In the sequel we show that this exceptional case cannot be eliminated from the theorem. For this purpose we employ an interesting example of a T-system due to Zielke [3]. PROPOSITION 1. The set {Pi}=o of functions defined by po(t) 1, p(t)=(1-t)t, pi(t) (1 t)ti-2(t2-1), =2, , n, is a T-system on [-1, 1]/f n 2m (m 1, 2,...). To render the paper self contained we first give an elementary new proof of the above proposition. * Received by the editors December 9, 1977. t Department of Mathematics, Indian Institute of Technology, Kanpur, India. 1092 Downloaded 01/20/14 to 202.3.77.183. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php