On the rate of convergence and the Voronovskaya theorem for the Poisson integrals for Hermite and Laguerre expansions
Let \(L^{(\alpha)}_n\) denote the \(n\)th Laguerre polynomial with parameter \(\alpha> -1\). The authors study properties of the Poisson integral \(A(f)\) defined for \(p\geq 1\) and \(f\in L^p([0,\infty), \omega_\alpha)\) by \[ A(f)(r, x):= \int^\infty_0 K_\alpha(r, x,y) f(y) \omega_\alpha(y) dy\qquad (0< \tau< 1, x> 0), \] where \(\omega_\alpha(y)= y^\alpha\exp(- y)\) is the corresponding weight and \[ K_\alpha(r,x,y)= \sum^\infty_{n=0} {r^n n!\over \Gamma(n+\alpha+1)} L^{(\alpha)}_n(x) L^{(\alpha)}_n(y). \] Continuing earlier work of \textit{B. Muckenhoupt} [Trans. Am. Math. Soc. 139, 231-242 (1969; Zbl 0175.12602)], they prove results on the rate of convergence of \(A(f)\) to \(f\) for \(r\to 1^-\) in terms of the modulus of continuity of \(f\), and on the asymptotic behaviour of \[ (A(f)(r, x)- f(x))/(1- r) \] for \(r\to 1^-\) and fixed \(x\) (so-called Voronovskaya-type-theorem). Similar results are provided for the Poisson integral \(B(f)\), defined with the Hermite weight \(\omega(y)= \exp(- y^2)\) and the Hermite polynomials \(H_n\) by \[ B(f)(r,x):= \int^\infty_{-\infty} P(r,x,y) f(y) \omega(y) dy\qquad (0< r<1, x\in\mathbb{R}) \] for \(f\in L^p(\mathbb{R}, \omega)\), where \[ P(r,x,y)= \sum^\infty_{n=0} {r^n H_n(x) H_n(y)\over \sqrt{\pi} 2^n n!}. \]
scientific article; zbMATH DE number 1800260