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An extension of Vietoris’s inequalities. (English) Zbl 1138.42002
For which positive nonincreasing sequences (a k ) 0kn is the cosine polynomial k=0 n a k coskx positive on [0,π[? L. Vietoris [Österr. Akad. Wiss., Math.-naturw. Kl., S.-Ber., Abt. II 167, 125–135 (1958; Zbl 0088.27402); Österr. Akad. Wiss., Math.-naturw. Kl. Anz. 192–193 (1959; Zbl 0090.04301)] gave a general sufficient condition: a 2k (1-1/2k)a 2k-1 . It has been noted that this condition may be improved: in particular, G. Brown and Q. H. Yin, [Acta Sci. Math. 67, 221–247 (2001; Zbl 0980.42003)] proposed to determine the optimal constant α 0 in place of the 1/2 above. The author proves that α 0 is the root of the equation 0 3π/2 t -α costdt=0 in the interval ]0·308443,0·308444[: the condition a 2k (1-α 0 /k)a 2k-1 is sufficient for k=0 n a k coskx to be positive, whereas this sum admits no uniform lower bound if a 2k =a 2k+1 =k-α k and α<α 0 . Vietoris’ theorem occurs in the proof of other inequalities, so that the following consequences are derived. (1) An improvement of a result of R. A. Askey and J. Steinig [Trans. Am. Math. Soc. 187, 295–307 (1974; Zbl 0244.42002)]. Let b k be a positive nonincreasing sequence such that b k (1-α 0 /k)b k-1 . The n real and simple zeros s 1 <<s n of k=0 n b k cos(n-k)x in ]0,π[ satisfy (k-1/2)π/(n+1/4)<s k <kπ/(n+1/4). (2) Sufficient conditions for sums of Gegenbauer polynomials and Jacobi polynomials to be positive.

42A05Trigonometric polynomials, inequalities, extremal problems
42A32Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
26D05Inequalities for trigonometric functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
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