Regularity of (0,2) interpolation
Given a triangular matrix of nodes (1) \(1\geq x_{1n}> x_{2n}> \cdots> x_{nn}\geq -1\) for \(n=2,3,\dots,\) the problem of (0,2) interpolation on these nodes is called regular if there exist fundamental polynomials \(r(x_{kn})\) and \(\rho_{kn}(x)\in \pi_{2n-1}\) which satisfy \[ \begin{alignedat}{3} r_{kn}(x_{jn}) &= \delta_{kj}, &\qquad r_{kn}^{\prime\prime}(x_{jn}) &=0, &\qquad k,j &= 1,2,\dots,n \tag{2} \\ \rho_{kn}(x_{jn}) &= 0, &\qquad \rho_{kn}^{\prime\prime} (x_{jn}) &= \delta_{kj}, &\qquad k,j &= 1,2,\dots,n \tag{3} \end{alignedat} \] Turán raised the question: Do there exist points (1) for which \(\rho_{kn}(x)\) exist, but not all \(r_{kn}(x)\) of the first kind exist? Here the author answers the above question in the negative. If \(\ell_ k(x)\) is the fundamental polynomial of Lagrange interpolation in (1), set \(B_ k(x)= (x-x_ k) \ell_ k^ 2(x)\). Theorem 1. For a fixed \(n\geq 2\), the following statements are equivalent: (a) The set of nodes (1) is regular for (0,2) interpolation. (b) The matrix \({\mathcal B}_ n:= [B_ k^{\prime\prime} (x_ j)]_{k,j=1}^ n\) is non singular. (c) There is one index \(k\), \(1\leq k\leq n\) such that \(r_ k(x)\in \pi_{2n-1}\) exists uniquely satisfying (2). (d) There is one index \(k\), \(1\leq k\leq n\) such that \(\rho_ k(x)\in \pi_{2n-1}\) exists uniquely. (e) There exists a set of polynomials \(r_ k(x)\in \pi_{2n-1}\) \((k=1,\dots,n)\) satisfying (2). (f) There exists a set of polynomials \(\rho_ k(x)\in \pi_{2n-1}\) \((k=1,\dots,n)\) satisfying (3).
scientific article; zbMATH DE number 540082