Topological splines in locally convex spaces

In the present paper, we propose a new approximation method in different function spaces. A specific feature of this method is that the choice of the basis approximating elements significantly depends on the topology of the given function space. Basis elements are constructed using the duality theory of locally convex spaces. A method of their exact calculation is presented. The approximating constructions are far-reaching generalizations of the classical Schoenberg splines and, by analogy with the latter, may be called topological splines. In the general case, such a definition of splines is not related to the choice of the grid. In this paper, we give many examples that are useful for practical applications.