Volume 50, pp. 36-51, 2018.

Polynomial approximation with Pollaczeck-Laguerre weights on the real semiaxis. A survey

Giuseppe Mastroianni, Gradimir V. Milovanović, and Incoronata Notarangelo


This paper summarizes recent results on weighted polynomial approximations for functions defined on the real semiaxis. The function may grow exponentially both at $0$ and at $+\infty$. We discuss orthogonal polynomials, polynomial inequalities, function spaces with new moduli of smoothness, estimates for the best approximation, Gaussian rules, and Lagrange interpolation with respect to the weight $w(x)=x^\gamma\mathrm{e}^{-x^{-\alpha}-x^\beta}$ ($\alpha>0$, $\beta>1$, $\gamma\geq 0$).

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Key words

orthogonal polynomials, weighted polynomial approximation, polynomial inequalities, Gaussian quadrature rules, Lagrange interpolation, Pollaczeck-Laguerre exponential weights

AMS subject classifications

41A05, 41A10, 41A17, 41A25, 65D05, 65D32

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