Estimates for Sard's best formulas for linear functionals on CS[ a, b

Abstract

Let J:C s[a, b]→ R be a bounded linear functional on the space of s times continuously differentiable functions ( s⩾0), and let Y=( y i, n ) be a triangular matrix of nodes satisfying a= y 0, n < y 1, n ⋯ < y n, n = b. Then one may approximate J by linear functionals Q n of the form Q n[ƒ]=Σ n−m 2 i=m 1a i,nf(y n,n) where m 1, m 2 ϵ {0, 1}. Among these, we consider the best formulas Q B n in the sense of Sard. For certain classes of nodes (which include, e.g., equidistant nodes, and the nodes of the Gauss quadrature formulas), and for arbitrary J, we give estimates for the weights a B i,n of Q B n , for the corresponding Peano kernels, and for the approximation error, including the error for interpolation by natural splines. By choosing special J's, estimates for best formulas for numerical integration, interpolation and differentiation are obtained, and also exponential decay of the fundamental natural splines is proved.