Approximation by \(B\)-spline convolution operators. A probabilistic approach
The authors are concerned with convolution operators with respect to univariate \(B\)-splines. One of the main features of this paper is that a \(B\)-spline convolution operator is represented as expectation with respect to the linear combination of uniform order statistics. This probabilistic approach has several interesting advantages, i.e.: a) It shows that the \(B\)-spline convolution operator belongs to the general class of Bernstein type operators thus allowing for a unified treatment of their approximation properties. b) It makes easier the computation of moments. c) The proofs are more simple. d) Opens the door to applications of \(B\)-splines in statistics and applied probability. Concerning the contents of the paper, at first the probabilistic representation of univariate \(B\)-splines is recalled. Then, the authors consider \(B\)-spline convolution operators and show a direct inequality in terms of the second modulus of smoothness. For Steklov operators built up from \(B\)-splines with equidistant simple knots strong direct and converse inequalities in the sup-norm are given. Finally, applications to the approximation of functions and to the time ruin probabilities in a classical risk model are presented.
scientific article; zbMATH DE number 2137443
scientific article