Error Bounds for Linear Approximations on the Real Line

Abstract

On the real line, several known error bounds for quasi-interpolants with slowly decaying basis functions do not yield the full order of approximation of the underlying approximation space. On the other hand, the unimprovability of these bounds is only known in some special cases. Developping new Peano kernel techniques for such operators, new pointwise error bounds as well as general unimprovability results are given. MATHEMATICS SUBJECT CLASSIFICATION: 41A80, 65D99 INTRODUCTION AND STATEMENT OF THE MAIN RESULTSWe consider quasi-interpolants of the type with Λ > 0, η € Ν, 0 < ε < 1, C independent of k and t, deg(Qh) > s -1. Here, deg(Q/J > s -1 means that for every ieR and every polynomial ρ of degree < s -1 we have Qh\p](x) = p(x)· The essential requirement in (1) is that all basis functions have a common majorant. An important problem for such approximations is the estimation of the error. In (1), the quasiinterpolants are based on localising basis functions, and their approximation properties depend on the local smoothness of /. Hence, pointwise error bounds are particularly interesting. In this paper, we develop new Peano kernel techniques for quasi-interpolants on the real line, in particular with slowly decaying basis functions. Using this approach, we derive new pointwise error bounds and unimprovability results. For s < η -1 in (1), the basis functions are sufficiently fast decaying such that the following Peano representation of the error functional can be stated for the standard class of functions (1) ke ζ ¥>/b,fc(t) = <Pk(h~lt),IwtWI < C{ 1 + |t|)