An elementary algebraic representation of polynomial spline interpolants for equidistant lattices and its condition (Q1071973)

A customary representation formula for periodic spline interpolants contains redundance which, however, can be eliminated by a transcendent method. We use an elementary identity for the generalized Euler- Frobenius-polynomials, which seems to be unknown until now, in order to derive the theory by purely algebraic arguments. The general cardinal spline interpolation theory can be obtained from the periodic case by a simple approach to the limit. Our representation has minimum condition for odd/even degree if the interpolation points are the lattice (mid- )points. We evaluate the corresponding condition numbers and give an asymptotic representation for them.