**AUTHORS:**Yuri K. Dem’yanovich, Tatjana O. Evdokimova, Evelina V. Prozorova

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In many cases the smoothness of splines is important (for qualitative approximation, for the calculation of a number of functionals, etc.). In the case of discontinuity of approximated functions it is difficult to use ordinary splines. It is desirable to have splines with similar properties of the approximated function. The purpose of this paper is to introduce the concept of general smoothness with the aid of linear functionals having a definite location of supports. Splines are often used for processing numerical information flows; a lot of scientific papers are devoted to these investigations. Sometimes spline treatment implies to the filtration of the mentioned flows or to their wavelet decomposition. A discrete flow often appears as a result of analog signal sampling, representing the values of a function, and in this case, the splines of the Lagrange type are used. In all cases, it is highly desirable that the generalized smoothness of the resulting spline coincides with the generalized smoothness of the original signal. Here we formulate the necessary and sufficient conditions for general smoothness of splines, and also a toolkit is being developed to build mentioned splines. The proposed scheme allows us to consider splines generated by functions from different spaces and to apply the obtained result to sources which can appear in physics, chemistry, biology, etc

**KEYWORDS:**
splines, general smoothness, chains of vectors, approximate relations

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