The abstract variational theory of splines in the Hilbert space originated from the well-known paper by M. Atteia (1965) and supported by P.J. Laurent's researches (1968, 1973) is today a well-developed field in the approximation theory. We mean that the forthcoming researches in abstract theory were initiated by the problem of high-quality approximation of the functions at the multi-dimensional scattered meshes. But the efforts in this particular problem have already led to more powerful results both in abstract theory and in practice: new kinds of characterization theorems, convergence and general estimation techniques, finite element approach in the construction of complicated non-polynomial splines, theory of traces of splines onto smooth manifolds (new algorithms for the approximation of complicated surfaces in engineering), general theory of tensor splines, including the famous blending splines, variational theory of the vector and rational splines and so on. We think that the investigations of the Russian mathematicians from the Computing Center of USSR Academy of Sciences in Novosibirsk during recent years and after first fundamental successes of our French collegues, were quite significant in each of these fields. And the last but not least: the powerful software library based on these theoretical grounds was also created in Novosibirsk.
Indeed, this paper is only a short review and is not complete. We inform the reader who has interest in theory of splines and its various applications that the book Bezhaev A.Yu., Vasilenko V.A. Variational Spline Theory (255p.) will be published in special issue 3 (1993) of the Bulletin of Novosibirsk Computing Center, series Numerical Analysis, till summer 1993.