Some Properties Of B_ψ-splines

摘要

A general approach to the construction of asymptotics of coordinate (not necessarily polynomial) B_ψ-splines of an arbitrary order is proposed. Asymptotic representations for Lagrange type third order B_ψ-splines are obtained. Bibliography: 4 titles. Using spaces of splines of different type, it is possible to develop new methods for processing digital information flows [1]. B_ψ-splines (in general, not polynomial) introduced in [2] yield a lot of versions of such a processing. Asymptotic properties of the coordinate splines play an important role for organizing computations while refining the grid. The asymptotic behavior of the coordinate second order B_ψ-splines was investigated in [3]. However, the methods used in [3] are not extended to splines of higher order. In this paper, we propose another approach: we introduce a function of many variables that is closely connected with the determining vector-valued function ψ and, by using this function, give a new representation of the coordinate B_ψ-splines. Since this function possesses the antisymmetry property under permutation of arguments, it is possible to simplify considerably the study of asymptotic properties of the coordinate splines. In this paper, we deal only with the coordinate third order B_ψ-splines. But the approach proposed could be used for obtaining asymptotic formulas for the coordinate spline of higher order.