**AUTHORS:**Yuri K. Dem’yanovich, Olga V. Belyakova, Bich T. N. Le

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The smoothness of functions is quite essential in applications. This smoothness can be used in functional calculations, in the construction of the finite element method, in the approximation of those or other numerical data, etc. The interest in smooth approximate spaces is supported by the desire to have a coincidence of smoothness of exact and approximate solutions. A lot of papers have been devoted to this problem. The continuity of the function at a point means equality of the limits on the right and left; the generalization of this situation is the equality of values of two linear functionals (at the prescribed function) with supports located on opposite sides of the mentioned point. Such generalization allows us to introduce the concept of generalized smoothness, which gives the ability to cover various cases of singular behavior functions at some point. The generalized smoothness is called pseudo-smoothness, although, of course, we can talk about the different types of pseudo-smoothness depending on the selected functionals mentioned above. 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 some cases, there are two interconnected analog signals, one of which represents the values of some function, and the second one represents the values of its derivative. In this case, it is convenient to use the splines of the Hermite type of the first height for processing. In all cases, it is highly desirable that the generalized smoothness of the resulting spline coincides with the generalized smoothness of the original signal. The concepts, which are introduced in this paper, and the theorems, which are proved here, allow us to achieve this result. The paper discusses the existence and uniqueness of spline spaces of the Hermite type of the first height (under condition of fixing the spline grid and the type of generalized smoothness). The purpose of this paper is to discuss generalized smoothness of the Hermite type spline space (not necessarily polynomial). In this paper we use the necessary and sufficient criterion of the generalized smoothness obtained earlier.

**KEYWORDS:**
splines, smoothness, approximate relations, uniqueness of spline spaces

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