Approximation by B-spline convolution operators. A probabilistic approach

This paper is concerned with the approximation properties of convolution operators with respect to univariate B-splines. For such operators, we give rates of uniform convergence in terms of the usual second modulus of smoothness at a length which depends on the distances between the knots and their multiplicity. A reasonable balance between the degree of accuracy in the approximation and the degree of differentiability of the approximants is achieved by considering Steklov operators (built up from B-splines with equidistant simple knots), for which strong converse inequalities are given. Applications to simultaneous approximation and divided difference expansions, and to estimate the infinite time ruin probabilities in a context of risk theory are also provided. We use a probabilistic approach in the spirit of Karlin et al. (J. Multivariate Anal. 20 (1986) 69) and Ignatov and Kaishev (Serdica 15 (1989) 91) based on the representation of B-splines as the probability densities of linear combinations of uniform order statistics.