Lagrange interpolation with exponential weights on \(( -1,1)\)
It is the aim of the authors to study interpolation processes which behave like best approximations in wide subspaces of weighted \(L^p\)-spaces and that are suitable for approximating functions with exponential growth at \(x=\pm 1\). They use weighted \(L^p\)-spaces \(L^p_u\), with \[ u(x)=(1-x^2)^{\mu}e^{-{1\over 2}(1-x^2)^{-\alpha}},\;\alpha>0, \;\mu\geq 0, \;x\in (-1,1), \] and a weight of the form \[ \sigma(x)=(1-x^2)^{\lambda}e^{-(1-x^2)^{-\alpha}},\;\alpha>0, \;\lambda\geq 0, \;x\in (-1,1), \] for the orthonormal system \(\{p_m(\sigma)\}_{m\in\mathbb{N}}\). \vskip0.2cm The main results are as follows. Let \(L_m(w,f)\) be the Lagrange polynomial based on the zeros of \(p_m(w)\) in the space \(C_{\sqrt{w}}\) with \(w(x)=\exp(-(1-x^2)^{-\alpha})\), \(\alpha>0\). The Lebesgue constant is given by \[ ||L_m(w)||_{\infty}=\max_{x\in (-1,1)}\sqrt{w(x)}\sum_{|k|\leq [m/2]}\,{|\ell_k(w,x)|\over\sqrt{w(x)}}, \] and put \({\mathcal L}_m(\sigma,f)=L_m(\sigma,\chi_jf)\), where \(\chi_j\) is the characteristic function of \([-x_j,x_j]\) and \(f\in C_{\mu}\). Then they find amongst other results: a bound on \(||\chi_j{\mathcal L}_m(\sigma,f)u||_{\infty}\) and on \(||f-\chi_j{\mathcal L}_m(\sigma,f)u||_{\infty}\) (Theorem 3.1); for \(u=v^{\mu}\sqrt{w}\), \(\sigma=v^{\lambda}w\), \(1\leq p<\infty\), a bound on \(||\chi_j{\mathcal L}_m(\sigma,f)u||_p\) and on \(||f-\chi_j{\mathcal L}_m(\sigma,f)u||_p\) (Theorem 3.2); for \(u=v^{\mu}\sqrt{w}\), \(\sigma=v^{\lambda}w\), \(1\leq p<4\): a bound on \(||{\mathcal L}_m(\sigma,f)u||_p\) (Theorem 3.5); denoting by \(L_{m,2}^{\ast}(\sigma,f)\) the Lagrange polynomial, where two points are added to the zeros of \(p_m(\sigma)\), \(x_{\pm([m/2]+1)}:=\pm a_m(\sqrt{\sigma})\), a bound on \(||{\mathcal L}_{m,2}(\sigma,f)u||_{\infty}\) and \(||f-{\mathcal L}_{m,2}(\sigma,f)u||_{\infty}\) (Theorem 3.7); for \(u=v^{\mu}\sqrt{w},\sigma=v^{\lambda}w, 1\leq p<\infty\), a bound on \(||{\mathcal L}_m(\sigma,f)u||_p\) (Theorem 3.8); and that the operator \({\mathcal L}_{m,2}(\sigma,f)u\) is uniformly bounded on the Sobolev-type space \(W_r^p(u)\) (Theorem 3.9).
scientific article; zbMATH DE number 6150697