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Orthonormal polynomials with exponential-type weights. (English) Zbl 1152.41001
Authors’ abstract: Let =(-,) and let w ρ (x):=|x| ρ exp(-Q(x)), where ρ>-1 2 and Q(x)C 2 : + =[0,) is an even function. In this paper we consider the properties of the orthonormal polynomials with respect to the weight w ρ 2 (x), obtaining bounds on the orthonormal polynomials and spacing on their zeros. Moreover, we estimate A n (x) and B n (x) defined in Section 4, which are used in representing the derivative of the orthonormal polynomials with respect to the weight w ρ 2 (x).
MSC:
41A10Approximation by polynomials
41A27Inverse theorems in approximation theory
References:
[1]Jung, H. S.; Sakai, R.: Inequalities with exponential weights, J. comput. Appl. math. 212, No. 2, 359-373 (2008) · Zbl 1198.41001 · doi:10.1016/j.cam.2006.12.011
[2]Kasuga, T.; Sakai, R.: Orthonormal polynomials for generalized freud-type weights, J. approx. Theory 121, 13-53 (2003) · Zbl 1034.42021 · doi:10.1016/S0021-9045(02)00041-2
[3]Levin, A. L.; Lubinsky, D. S.: Orthogonal polynomials for exponential weights, (2001)
[4]Levin, A. L.; Lubinsky, D. S.: Orthogonal polynomials for exponential weights x2ρe-2Q(x) on [0,d), J. approx. Theory 134, 199-256 (2005) · Zbl 1079.42017 · doi:10.1016/j.jat.2005.02.006
[5]Levin, A. L.; Lubinsky, D. S.: Orthogonal polynomials for exponential weights x2ρe-2Q(x) on [0,d),II, J. approx. Theory 139, 107-143 (2006) · Zbl 1127.42023 · doi:10.1016/j.jat.2005.05.010
[6]G. Szegő, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, vol. 23, American Mathematical Society, Providence, RI, 1975.