Mean convergence of Lagrange interpolation for exponential weights on $[-1,1]$

http://dx.doi.org/10.4153/CJM-1998-062-5
Canad. J. Math. 50(1998), 1273-1297

  Published:1998-12-01
 Printed: Dec 1998

We obtain necessary and sufficient conditions for mean convergence of Lagrange interpolation at zeros of orthogonal polynomials for weights on $[-1,1]$, such as \[ w(x)=\exp \bigl(-(1-x^{2})^{-\alpha }\bigr),\quad \alpha >0 \] or \[ w(x)=\exp \bigl(-\exp _{k}(1-x^{2})^{-\alpha }\bigr),\quad k\geq 1, \ \alpha >0, \] where $\exp_{k}=\exp \Bigl(\exp \bigl(\cdots\exp (\ )\cdots\bigr)\Bigr)$ denotes the $k$-th iterated exponential.