SOME PROPERTIES OF MINIMAL SPLINES

S.G. MIKHLIN was the first to construct systematically coordinate func tions on an equidistant grid solving a system of approximate equations (called ''fundamental relations'', see [5]; GOEL discussed some speci al cases earlier in 1969; see also [1, 4, 6]). Further, the idea was d eveloped in the case of irregular grids (which may have finite accumul ation points, see [1]). This paper is devoted to the investigation of A-minimal splines, introduced by the author; they include polynomial m inimal splines which have been discussed earlier. Using the idea menti oned above, we give necessary and sufficient conditions for existence, uniqueness and g-continuity of these splines. The application of thes e results to polynomial splines of m-th degree on an equidistant grid leads us, in particular, to necessary and sufficient conditions for th e continuity of their i-th derivative (i = 1,..., m). These conditions do not exclude discontinuities of other derivatives (e.g. of order le ss than i). This allows us to give a certain classification of minimal spline spaces. It turns out that the spline classes are in one-to-one -correspondence with certain planes contained in a hyperplane.