Discrete Modulus of Smoothness of Splines with Equally Spaced Knots

Abstract

We study the behavior of moduli of smoothness of splines s of order r with equally spaced knots {xi}, x,+l xi h. The main results are as follows.(1) For each 0 <_ rn < r, all quantities hJa,_j(s(J),h)p, 0 <_ j <_ rn, are equivalent and can be measured by a discrete norm of the ruth differences of the B-spline coefficients of s, which we call the rnth discrete modulus of smoothness of s.(2) All quantities hJwm_j(s(J), h)p, m >_ r, 0 _< j <_ r-1, are equivalent to cot(s, h)p, which can be measured by the rth discrete modulus of s.(3) When h is replaced by t, 0 _< _< h, in the results above, all the quantities can still be measured by the corresponding discrete modulus multiplied by a power of t/h. The results generalize the notion of a discrete norm of B-spline series in the case of equal spacing.As an application, we use these results to prove that w3 is the best rate of convex approximation by such splines.

On Equivalence of Moduli of Smoothness

“…K Proof of Theorem 2. The proof is almost identical to that of (1.9) in Hu and Yu [9,Theorem 2]. The difference is, of course, that the spline s in [9] has equal spacing.…”

“…The proof is almost identical to that of (1.9) in Hu and Yu [9,Theorem 2]. The difference is, of course, that the spline s in [9] has equal spacing. Note that, however, their proof does not use equal spacing of s, but that of the B-splines N i defined in (2.10 11) of [9], which is introduced by the difference operator 2 m t .…”

“…When t is large, however, there is no equivalence among t j | m& j (S ( j) , t) p unless we allow C to depend on the ratio tÂ$ . See [9] for a counterexample in C. For general case, consider t=1 and…”

“…Yu and Zhou [17] proved part of the inverse in a special case, namely, h| r&1 (S$, h) C r | r (S, h) , (1.2) where S is any spline of order r with equally spaced knots, and h is the mesh size. Hu and Yu [9] proved that for such splines S the whole inverse of (1.1) holds true, thus | m (S, t) p tt| m&1 (S$, t) p tt 2 | m&2 (S", t) p } } } . In this paper we generalize it to splines with arbitrary (fixed) knots, and further to shift-invariant spaces and wavelets under certain conditions.…”

On Positive and Copositive Polynomial and Spline Approximation inLp[−1, 1], 0<p<∞

“…Soon after, Hu and Yu [9] proved an analogue of (2) for such splines (in fact, (2) is derived from this result for splines, with the aid of results in [10]). If p< , the rate drops to | 1 for copositive spline approximation, too.…”