V.N. Isakov

At regular interpolation on each individual interval of sampling interpolation the curve is formed by identical rules. Such interpolation is considered examined, at which the conditions of a coordination (diagram of interpolation function are carried out passes through units of interpolation), linearity (at addition of discrete signals them interpolation functions are develop), convergence (at unlimited reduction of a step of sampling the interpolation function comes nearer to a restored signal). For a signal with the limited spectrum the task of interpolation has the unique decision determined by the theorem of Kotelnikov. However restoration of signals on the basis of a Kotelnikov series meets significant difficulties at physical realization. In practice there are more preferably local methods of interpolation, that causes a urgency of consideration of a question about convergence of interpolation process. It is possible to show, that the convergence of interpolation process at restoration of a signal with the limited spectrum is provided, if the method of interpolation carries out exact restoration of unit. The class of functions satisfying to a condition of restoration units, closes concerning Fourier-transformation. To a condition of convergence satisfy, for example, step interpolation, piecewise-linear interpolation, -interpolation, local interpolation with the Lagrange polynomial, local spline-interpolation.