Turán type problems on mean convergence. I: Lagrange type interpolations
The paper is concerned with the convergence, with respect to weighted \(L^ p\)-norms, of Hermite interpolation polynomials for functions in \(C= C[- 1,1]\). For \(0< p< \infty\) and \(u: [-1,1]\to [0, \infty)\) and such that \[ 0< \int^ 1_{-1} u(x) dx< \infty\quad\text{put}\quad \| g\|_{p,u}= \left( \int^ 1_{-1} | g(x)|^ p u(x)d(x)\right)^{1/p} \] and \(\| g\|_{\infty, u}=\text{ess}\sup| g(x)|\), for any measurable function \(g: [-1, 1]\to \overline R\). Denote by \(d\alpha\) the measure generated by a nondecreasing function \(\alpha\) on \([-1, 1]\). For \(\alpha\) absolutely continuous \(w(x)= \alpha'(x)\) a.e. on \([-1, 1]\) is the weight function associated to \(\alpha\). Let \(p_ n(d\alpha, x)\) be the orthogonal polynomial of degree \(n\) and \(x_{kn}\), \(k= 1,\dots,n\) its roots. For \(n,m\in \mathbb{N}\) denote by \(H_{n,m}(f, d\alpha, x)\) the Hermite interpolation polynomial of a function \(f\in C\) on the knots \(x_{kn}\), \(1\leq k\leq n\). The convergence results are totally different for the cases \(m\) odd and \(m\) even both for the uniform convergence as well as for the \(L^ p\)- convergence. For \(m\) even one can find good knot matrices for which \(H_{n,m}(f)\) converges to \(f\), for every \(f\in C\) which is impossible for \(m\) odd. In this part of the paper \(m\in \{1,3,5,\dots\}\) is a fixed odd number. One of the results of the paper (Theorem 2.1) asserts that if \(\text{supp}(d\alpha)= [-1, 1]\) (the support of the measure \(d\alpha\)), \(\alpha'(x)> 0\) a.e. on \([-1, 1]\), \(0< p_ 0\leq \infty\) and \(\|(\alpha'/\sqrt{1- x^ 2})^{m/2}\|_{p, u}= \infty\), for all \(p_ 0< p\leq \infty\) then there exists an \(f\in C\) such that \[ \varlimsup_{n\to\infty} \| H_{n,m} (f, d\alpha)\|_{p, u}= \infty,\qquad\text{for all}\quad p_ 0< p\leq \infty. \] A similar result is obtained for the weight \(w_ \delta(x)= \exp(-(1- x^ 2)^{- \delta})\), where \(\delta> 0\) is fixed. In this case there exists \(f\in C\) such that \[ \varlimsup_{n\to \infty} \int^ 1_{-1} | H_{n,m} (f, w, x)- f(x)|^ p w_ \delta(x)= \infty, \] for every \(p\), \(2/m< p\leq \infty\). The value \(2/m\) is critical in the sense that for \(p= 2/m\), \(\lim_{n\to \infty} \int^ 1_{-1} | H_{n,m} (f, v, x)- f(x)|^ p dx= 0\), for all \(f\in C\), where \(v(x)= (1- x)^ \alpha (1+ x)^ \beta\), \(\alpha,\beta> -1/2- 1/m\). The paper contains also a survey of other convergence and divergence results for Hermite interpolation.
convergence of interpolation polynomials
scientific article; zbMATH DE number 638242
scientific article