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Exact bounds for orthogonal polynomials associated with exponential weights. (English) Zbl 0605.42019

The main result is: \(| x| <C_ mn^{1/m}\) implies \(w(x)p^ 2_ n(x)\leq C'n^{-1/m},\) \(n=0,1,2...\); where \(w(x)=\exp (-x^ m)\) for real x, and an even integer m, \(p_ n(x)\) is the normalized orthogonal polynomial of degree n for the weight function w, and C’, \(C_ m\) are certain constants. Specifically, for any C with \(0<C<1\) there exists C’ so that the inequality holds for \(C_ m=C(B(1/2,m/2))^{1/m},\) (the Beta function).
Reviewer: C.F.Dunkl

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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