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Uniform spacing of zeros of orthogonal polynomials. (English) Zbl 1205.42027

If w is a weight function (id est a non-negative integrable function) on [-1,1], then the doubling property means that for some constant L,

2I wL I w

for all intervals I[-1,1], where 2I denotes the “doubled” interval I (twice enlarged from its center). The constant L is referred to as the doubling constant of w. It is shown that for those doubling weights, the zeros of the associated orthogonal polynomials are uniformly spaced, which means that if cosθ m,k with θ m,k [0,π] are the zeros of the m-th orthogonal polynomial associated with w, then θ m,k -θ m,k+1 1 m. It is also shown that for doubling weights, neighbouring Cotes numbers are of the same order. In fact, it is shown that these two properties are actually equivalent to the doubling property of the weight function.

MSC:
42C05General theory of orthogonal functions and polynomials
References:
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[4]Levin, A.L., Lubinsky, D.: Orthogonal polynomials for exponential weights. In: CMS Books in Mathematics. Ouvrages de Mathématiques de la SMC, vol. 4. Springer, New York (2001)
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