Lacunary \((0;0,1)\) interpolation on the roots of Jacobi polynomials and their derivatives, respectively. I (Existence, explicit formulae, unicity)
The second author in \textit{L.G. Pál} [Analysis Math. 1, 197-205 (1975; Zbl 0333.41001)] introduced a lacunary \((0;0,1)\) interpolation method using the roots of the polynomials \[ \omega_n(x)=(x-x_1)(x-x_2)\dots(x-x_n)\text{ and }\omega_n'(x)=n(x-x_1^*)(x-x_2^*)\dots(x-x_{n-1}^*) \] with respect to the interscaled system of points \(x_1<x_1^*<x_2<\dots<x_k<x_k^*<x_{k+1}<\dots<x_{n-1}<x_{n-1}^*<x_n \), where the nodal points \(x_k\in\mathbb R\), \(k=1,\dots,n\) are given arbitrarily. In this paper, the authors investigate the case when \(\omega_n(x)=P_n^{(\alpha,\beta)}(x)\), the \(n\)-th Jacobi polynomial, for an arbitrary pair of indices \(\alpha\neq\beta\) \((\alpha,\beta>-1)\) or, if \(\alpha=\beta\), then \(n (>2)\) is an odd number. The authors give an algebraic polynomial \(P\) with minimal degree for which the interpolation properties \(P(x_k)=y_k\) \((k=0,1,\dots,n+1) \), \(P'(x_k^*)=y_k'\) \((k=1,2,\dots,n-1) \) are satisfied with respect to the nodal points \( -1=x_{n+1}<x_n<x_{n-1}^*<x_{n-1}<\dots<x_1^*<x_1<x_0=1\), for arbitrary real numbers \( y_k\) and \(y_k'\). Also, a uniqueness property of \(P\) is proved. (Part II, the continuation of the paper will contain convergence and approximation theorems.).
scientific article; zbMATH DE number 1102591