On the error of optimal interpolation by linear shape-preserving algorithms

We consider the problem of optimal linear interpolation by algorithms positive on a cone describing the properties of the shape of the functions being approximated. We show that such linear shape-preserving methods have a negative property connected with the inability to identically approximate algebraic polynomials of at least the given degree. We also show that the estimation of the error of the problem of linear shape-preserving interpolation can be reduced to the problem of conic optimization. This makes it possible to use the duality principle for obtaining an estimate of the error of the shape-preserving interpolation.