约束最佳一致逼近理论

Theory of Uniformly Best Approximation Subject to Constraints

本文根据工程中的需要,提出一新型最佳一致逼近问题:函数族u(α,x)在随参数α而变化的定义域X(α)上取值、参数α在点集A(A中的点α对应的X(α)非空)中取值时的最佳一致逼近问题.文中给出了判别最佳逼近函数的Kolmogorov型定理、一阶和二阶必要条件和充分条件、以及一阶二阶局部唯一性定理.

A new type of the uniformly best approximation problem is proposed to meet the requirement in the engineering field. This problem consists in a function family u(a,x) being defined in the domain of definiton X(a) which depends on the parameter a and the parameter o belongs to the point set A (the set X(a) corresponding to the parameter a in A is nonempty). The sufficient condition and necessary condition of the Kolmogorov type, the first-order sufficient condition and necessary condition and the second -order sufficient condition and necessary condition for a function to be best approximation are derived. The first-order and second-order sufficient conditions are also the local uniqueness conditions, and the first-order sufficient condition can be taken as a generalization of the Haar uniqueness theorem in the problem of the uniformly best approximation proposed.