Hermite and Hermite-Fejér interpolations of higher order. II: Mean convergence (Q1175249)

[For part I see ibid. 54, No. 1/2, 135-152 (1989; Zbl 0703.41004).] The author studies weight \(L^ p\)-convergence of Hermite and Hermite- type interpolatory operators of higher order. Let \(X=\{x_{kn}=\cos \theta_{kn}\}\subset[-1,1]\), \(-1\leq x_{nn}<x_{n- 1,n}<\dots<x_{1n}\leq1\), \(n=1,2,3,\dots\) be an infinite triangular matrix. If \(f\in C\), let \(H_{n,m}(f,X,x)\) denote the unique Hermite Fejer polynomial of degree \(\leq mn-1\) satisfying the following conditions for a fixed m: \[ H_{nm}(x_{kn})=f(x_{kn}),\quad 1\leq k\leq n,\quad H^{(t)}_{n,m}(f_{xn})=0,\quad 1\leq t\leq m-1,\quad 1\leq k\leq n.\tag{1} \] If \(f^{(m-1)}\in C\), the Hermite interpolatory polynomial denoted by \({\mathcal H}_{n,m}(f,X,x)\) is of degree \(\leq mn-1\) and satisfies \[ H^{(t)}_{n,m}(x_{kn})=f^{(t)}(x_{kn}),\quad 1\leq k\leq n,\quad 0\leq t\leq m-1.\tag{2} \] \(\omega=\omega^{(\alpha,\beta)}=(1- x)^{\alpha}(1+x)^{\beta}\), \(\alpha,\beta>-1\), \(| x|\leq 1\) denotes the Jacobi weight function \((\omega\in J\) or \(\omega\in J(\alpha,\beta))\). Let \(\gamma=\min(\alpha,\beta)\), \(\Gamma=\max(\alpha,\beta)\), and let \(A_ m=-1/2-2/m\), \(B_ m=- 1/2+1/m\), \(C_ m=-1/2-1/m\), \(m=1,2,\dots\). Suppose that \(\Gamma- \gamma\leq2/m\) and either \(\gamma\geq C_ m\), or \(A_ m\leq\gamma<C_ m\) hold. The following two theorems are proved: Theorem 1. Let \(m=2,4,\dots\) be fixed, even, \(p>0\), \(u^ p\in J\). Then \[ \lim_{n\to\infty}\|(H_{n,m}(f,\omega)-f)u\|_ p=0\forall f\in C \] if \[ v(x):-{u(x)\sqrt{1-x^ 2}\over(\omega(x)\sqrt{1-x^ 2})^{m/2}}\in L_ p. \] A similar result is proved for the polynomials \({\mathcal H}_{n,m}(f,\omega)\), when \(f^{(m-1)}\in C\). The author points out the advantage of his Lemma 3.2 in proving mean convergence over subintervals also.