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Orthonormal polynomials with exponential-type weights. (English) Zbl 1152.41001
Authors’ abstract: Let $ℝ=\left(-\infty ,\infty \right)$ and let ${w}_{\rho }\left(x\right):={|x|}^{\rho }exp\left(-Q\left(x\right)\right)$, where $\rho >-\frac{1}{2}$ and $Q\left(x\right)\in {C}^{2}:ℝ\to {ℝ}^{+}=\left[0,\infty \right)$ is an even function. In this paper we consider the properties of the orthonormal polynomials with respect to the weight ${w}_{\rho }^{2}\left(x\right)$, obtaining bounds on the orthonormal polynomials and spacing on their zeros. Moreover, we estimate ${A}_{n}\left(x\right)$ and ${B}_{n}\left(x\right)$ defined in Section 4, which are used in representing the derivative of the orthonormal polynomials with respect to the weight ${w}_{\rho }^{2}\left(x\right)$.
MSC:
 41A10 Approximation by polynomials 41A27 Inverse theorems in approximation theory
References:
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