\(L^p\)-convergence of Hermite and Hermite-Fejér interpolation (Q390682)

The authors generalize convergence results on \(L^p\)-convergence of Lagrange interpolation (cf. [\textit{P. Nevai}, Trans. Am. Math. Soc. 282, 669--698 (1984; Zbl 0577.41001)]) to the case of Hermite and Hermite-Fejér interpolation.NEWLINENEWLINELet \(w(x)=v^{\alpha,\beta}(x)=(1-x)^{\alpha}(1+x)^{\beta}\), \(\alpha,\beta>-1\), be the Jacobi weight and denote the \(m\) zeros of the \(m\)th Jacobi polynomial by \(x_k\), \(1\leq k\leq m\); furthermore, \(\varphi(x)=\sqrt{1-x^2}\).NEWLINENEWLINENow \(H_{m,r}(w;f)\), \(r\geq 1\), denotes the Hermite interpolation polynomial based on the Jacobi zeros, corresponding to the function \(f\in \text{C}^{r-1}(-1,1)\): NEWLINE\[NEWLINEH^{(i)}_{m,r}(w;f,x_k)=f^{(i)}(x_k),\;1\leq k\leq m,\;0\leq i\leq r-1.NEWLINE\]NEWLINE The main results areNEWLINENEWLINE{ Theorem 1.} Let \(u\in L^p\), \(p\in (1,\infty)\), and \(f\in \text{C}^{r-1}(-1,1)\), \(r\geq 1\); then the inequality NEWLINE\[NEWLINE||H_{m,r}(w;f)u||_p\leq C\left[ ||H_{m,r-1}(w;f)u||_p+ \left(\sum_{k=1}^m\Delta x_k\left|\left(f^{(r-1)}\left({\varphi\over m}\right)^{r-1} u\right) (x_k)\right|^p\right)^{1/p}\right]NEWLINE\]NEWLINE holds with \(C\not= C(m,f)\) if and only if NEWLINE\[NEWLINE{u\over(\sqrt{w\varphi})^r}\in L^p,\;{(\sqrt{w\varphi})^r\over u}\in L^q\;(p^{-1}+q^{-1}=1).\eqno{(*)}NEWLINE\]NEWLINENEWLINENEWLINE{ Theorem 2.} Let \(u\in L^p\), \(p\in (1,\infty)\), and \(f\in\text{C}^{r-1}(-1,1),\;r> 1\); then we have the following equivalence, uniformly in \(m\) and \(f\) NEWLINE\[NEWLINE||H_{m,r}(w;f)u||_p\sim \left(\sum_{k=1}^m\Delta x_k\sum_{i=0}^{r-1}\left|\left(f^{(i)}\left({\varphi\over m}\right)^{i} u\right) (x_k)\right|^p\right)^{1/p}NEWLINE\]NEWLINE if and only if NEWLINE\[NEWLINE{u\over(\sqrt{w\varphi})^r}\in L^p,\;{(\sqrt{w\varphi})^i\over u}\in L^q,~ 1\leq i\leq r\;(p^{-1}+q^{-1}=1).\eqno{(**)}NEWLINE\]NEWLINENEWLINENEWLINEWriting \(H_{m,r}(w;f)=F_{m,r}(w;f)+G_{m,r}(w;f)\), \(r>1\), with \(F_{m,r}\) the Hermite-Fejér interpolation polynomial of higher order, given for \(1\leq k\leq m\) by NEWLINE\[NEWLINEF_{m,r}(w;f,x_k)=f(x_k);\;F_{m,r}(w;f)^{(i)}(x_k)=0\;(1\leq i\leq r-1)NEWLINE\]NEWLINE and NEWLINE\[NEWLINEG_{m,r}(w;f,x_k)=0;\;G_{m,r}(w;f)^{(i)}(x_k)=f^{(i)}(x_k)\;(1\leq i\leq r-1),NEWLINE\]NEWLINE the result isNEWLINENEWLINE{ Theorem 3.} Condition \((*)\) is equivalent to NEWLINE\[NEWLINE||uF_{m,r}(w;f)||_p\leq C||uF_{m,r-1}(w;f)||_p\;(C\not= C(m,f)),NEWLINE\]NEWLINE for any \(f\in\text{C}^0(-1,1)\) and \(p\in (1,\infty)\).NEWLINENEWLINENEWLINEMoreover, under condition \((**)\) and if \(|\Omega_{\varphi}(f,t)_{u,p}|t^{-1-1/p}\in L^1\) NEWLINE\[NEWLINE||[f-F_{m,r}(w;f)]u||_p\leq{C\over m^{1/p}}\int_0^{1/m}\,{|\Omega_{\varphi}(f,t)_{u,p}|\over t^{1+1/p}}dtNEWLINE\]NEWLINE (here \(\Omega^r_{\varphi}(f,t)_{u,p}\) is the main part of the weighted \(L^r\)-th \(\varphi\)-modulus of continuity).