A periodic basis system of the smooth approximation space

摘要

The paper is primarily devoted to the problem of smooth interpolation of data in 1D. In addition to the exact interpolation of data at nodes, we are also concerned with the smoothness of the interpolating curve and its derivatives.The interpolating curve is defined as the solution of a variational problem with constraints. The system of functions exp(-ikx), k being an integer, is taken for the basis of the space where we measure the smoothness of the result. It also generates the functions used for the interpolation itself. Choosing different norms when measuring the smoothness, we arrive at different interpolating functions. We also mention the problem of smooth curve fitting (data smoothing).We discuss the proper choice of different norms for this way of approximation and present the results of several 1D numerical examples that show the advantages and drawbacks of smooth interpolation.