Spline approximation in $H_{p}(T)$, p≤ 1

Positive and Copositive Spline Approximation in L P 0; 1]

The order of positive and copositive spline approximation in the L p-norm, 1 p < 1, is studied, the main results are (1) the error of positive approximation by splines is bounded by C! 2 (f; 1=n) p if f has a nonnegative extension; (2) the order deteriorates to ! 1 if f does not have such an extension; (3) the error of copositive spline approximation is bounded by C!(f; 1=n) p ; (4) if f is als...

Convex Polynomial and Spline Approximation in Lp , 0 < p <

We prove that a convex function f 6 L p [ -1 , 1], 0 < p < cx~, can be approximated by convex polynomials with an error not exceeding Cw~ (f, 1/n)p where w~(f, .) is the Ditzian-Totik modulus of smoothness of order three of f . We are thus filling the gap between previously known estimates involving og~(f, I/n)p, and the impossibility of having such estimates involving ~o4. We also give similar...

Perfect spline approximation

Our study of perfect spline approximation reveals: (i) it is closely related to Σ∆ modulation used in one-bit quantization of bandlimited signals. In fact, they share the same recursive formulae, although in different contexts; (ii) the best rate of approximation by perfect splines of order r with equidistant knots of mesh size h is hr−1. This rate is optimal in the sense that a function can be...

Best Quadratic Spline Approximation

We present a method for hierarchical data approximation using quadratic simplicial elements for domain decomposition and field approximation. Higher-order simplicial elements can approximate data better than linear elements. Thus, fewer quadratic elements are required to achieve similar approximation quality. We use quadratic basis functions and compute best quadratic simplicial spline approxim...

Parallel Spline Approximation