This paper is devoted to the processing of large numerical signals which arise in different technical problems (for example, in positioning systems, satellite maneuvers, in the prediction a lot of phenomenon, and so on). The main tool of the processing is polynomial and nonpolynomial splines of the Hermite type, which are obtained by the approximation relations. These relations allow us to construct splines with approximate properties, which are asymptotically optimal as to N-width of the standard compact sets. The interpolation properties of the mentioned splines are investigated. Such properties give opportunity to obtain the solution of the interpolation Hermite problems without solution of equation systems. The calibration relations on embedded grids are established in the case of deleting the grid knots and in the case of the addition of the last one. A consequence of the obtained results is the embedding of the Hermite spline spaces on the embedded grids. The mentioned embedding allows us to obtain wavelet decomposition of the Hermite spline spaces.