Lacunary interpolation by even degree polynomial splines (Q1826053)

Let \(0=x_ 0<x_ 1<...<x_ N=1\) be an arbitrary partition of the interval [0,1], \(h_ i=x_{i+1}-x_ i\), \(\tau_ i=x_ i+\lambda h_ i\), \(i=0,1,...,N-1\), \(0<\lambda <1\). Denote by \(S^{(n)}_{N,2n}\) the class of all spline functions s(x) satisfying: (i) \(s(x)\in C^ n[0,1]\), (ii) \(s(x)\in \pi_{2n}\) in \([x_ i,x_{i+1}]\), for \(i=0,1....,N-1\). The problem is concerned with the following question called Problem A: For a given integer q, \(0\leq q\leq 2n\), determine \(s(x)\in S^{(n)}_{N,2n}\) such that \(s^{(\nu)}(x_ i)=a_{i,\nu}\) \((i=0,1,...,N\); \(\nu =0,1,...,n-2)\); \(s^{(q)}(\tau_ i)=b_{i,q}\) \((i=0,1,...,N-1)\); \(s^{(n-1)}(x_ 0)=a_{0,n-1}\), \(s^{(n-1)}(x_ N)=a_{N,n-1},\) where \(a_{i,\nu}\), \(b_{i,q}\), \(a_{0,n-1}\), \(a_{N,n-1}\) are given real numbers. It is proved that for \(n\geq 1\), \(q\in \{0,2,...,2n\}\) and \(\lambda =1/2\), Problem A has a unique solution \(s(x)\in S^{(n)}_{N,2n}\). In the case of equispaced knots the authors give evaluations for \(\| f^{(n)}-s^{(n)}\|\) and study the convergence of the interpolation method.