Necessary and sufficient conditions for interpolation with functions having monotone r-th derivatives (Q912323)

The author gives necessary and sufficient conditions for the existence of a function \(x\in m^ r(I)\) satisfying \(x(t_ i)=c_ i\), \(i=1,2,...,n\) for arbitrary given n tuples \(t^ n\) and \(c^ n(c_ 1,...,c_ n)\in R^ n\). This result is connected then with many problems in the theory of approximations, recovery or smoothing by splines. The main motivation was to apply the solution to some recovery and smoothing problems connected with the uniform approximation of functions of the class \(m^ r(I)\) and of the closely related classes \(W^{r+1}_{m,M}(I)\) of functions, whose \((r+1)\)-st derivatives belong on an I to a given interval [m,M].