On uniform spline approximation

Abstract

Let Δ be a partition of [0, 1], m be an integer greater than two, and S be the set of spline functions of order m (degree m − 1) with knots in Δ. It is proved that for functions f( x) continuous on [0, 1], dist (f,S) ⩽ ( m 12 +1 ω(|Δ|) , where ω(·) is the modulus of continuity of f( x), | Δ| is the mesh of Δ, and distance is measured by the sup norm. The proof uses the variation diminishing spline approximation method developed by Schoenberg [13] to get a spline function whose distance from f( x) is bounded by the given expression. Similar bounds on dist(f, S ) have been obtained by de Boor [5], but his coefficients are much larger and are not easily computed for large m. A similar estimate is given for Tchebycheffian spline functions.