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Abstract view

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

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.