Hermite interpolation and mean convergence of its derivatives
Let \(\{p_ m(w)\}^ \infty_{m=0}\) be a system of orthogonal polynomials corresponding to a weight function \(w(x)=g(x)(1-x)^ \alpha(1+x)^ \beta\), \(x\in[-1,1]\) where \(0\leq g\in C^ 1\) and \(g'\in\text{Lip}_ M 1\) and let \(\{x_{i,m}(w)\}^ m_{i=1}\) be their zeros labelled in increasing order. Consider also the matrices of additional knots \(Y=\{y_{j,m}: j=1,\dots,s, m\in N\}\) and \(Z=\{z_{i,m}: i=1,\dots,r, m\in N\}\), \(r,s\in N\), satisfying the conditions: \[ -1\leq y_{1,m}<\cdots< y_{s,m}< x_{1,m}<\cdots < x_{m,m}<z_{1,m}<\cdots < z_{r,m}\leq 1; \] \[ x_{i,m}-y_{s,m}\sim m^{-2}\sim z_{i,m}-x_{m,m}; y_{j,m}-y_{j-1,m}\sim m^{-2}, j=2,\dots,s; \] \[ z_{i,m}-z_{i-1,m}\sim m^{-2},\quad i=2,\dots,r, \] uniformly with respect to \(m\in N\). Then there exists a Hermite-type interpolation polynomial \(H_{m,r,s}(w;f)\), verifying the conditions: \[ H^{(i)}_{m,r,s}(w;f;x_{k,m})=f^{(i)}(x_{k,m}),\quad i=0,1, k=1,\dots,m; \] \[ H_{m,r,s}(w;f;y_{j,m})=f(y_{j,m}),\quad j=1,\dots,s; H_{m,r,s}(w;f;z_{i,m})=f(z_{i,m}),\quad i=1,\dots,r. \] Weighted \(L^ p\)-convergence of the derivatives of this Hermite interpolation polynomials on the zeros of Jacobi polynomials plus the additional points in \(Y\) and \(Z\) is investigated.
scientific article; zbMATH DE number 61967