Error estimations of Turán formulas with Gori-Micchelli and generalized Chebyshev weight functions

Abstract

S. Li in [Studia Sci. Math. Hungar. 29 (1994) 71-83] proposed a Kronrod type extension to the well-known Turán formula. He showed that such an extension exists for any weight function. For the classical Chebyshev weight function of the first kind, Li found the Kronrod extension of Turán formula that has all its nodes real and belonging to the interval of integration, [−1, 1]. In this paper we show the existence and the uniqueness of the additional two cases-the Kronrod exstensions of corresponding Gauss-Turán quadrature formulas for special case of Gori-Micchelli weight function and for generalized Chebyshev weight function of the second kind, that have all their nodes real and belonging to the integration interval [−1, 1]. Numerical results for the weight coefficients in these cases are presented, while the analytic formulas of the nodes are known.

Digital Media Teaching and Effectiveness Evaluation Integrating Big Data and Artificial Intelligence

“…The activation functions of the hidden layer are radial basis functions, which usually satisfy Micchelli's theorem [ 25 ]. Micchelli's theorem states that as long as each sample point is a mutually distinct point in the space, the resulting interpolation matrix Φ is nonsingular [ 26 ]. Three radial basis functions occupy an important position in the RBF neural network.…”