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

Giuseppe Mastroianni, Gradimir V. Milovanović, and Incoronata Notarangelo

### Abstract

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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