Let M ∈ N0, K ∈ N , and u−M ≤··· ≤ u0 ∈ R, set                                     h(x) = max j:uj≤x Let p = q + r for some q ∈ {0,...,M}, r ∈ (0, 1]. Let Q be a bounded quasi interpolant of order q....

Let M ∈ N0, K ∈ N , and u−M ≤··· ≤ u0 <><>∈ R, set

                                    h(x) = max j:uj≤x

Let p = q + r for some q ∈ {0,...,M}, r ∈ (0, 1]. Let Q be a bounded quasi interpolant of order q. Show that for every (p, C)-smooth function f : R→R,

                                |(Qf)(x) − f(x)|≤Q · C q! · h(x) p (x ∈ [u0, uK)).

Hint: Proceed as in the proof of Theorem 14.3, but use (11.7) instead of (14.44).