On the (0.4) lacunary interpolation problem and some related questions

We consider at first a (0,4) lacunary interpolation problem, and we solve it in an appropriate class of “deficient” splines. Under suitable assumptions, we show existence and uniqueness of the solution. We provide the convergence analysis, showing that the method is of order “almost” five. Some additional considerations are made for problems of similar nature, of arbitrary order and with arbitrary intermediate lacunary conditions. This is a natural method to couple with finite difference schemes for the reconstruction of the solution of boundary value problems. It may constitute a viable alternative to continuous Runge-Kutta methods.