Higher order Lipschitz Sandwich theorems

Abstract We investigate the consequence of two functions, in the sense of Stein, being close throughout a subset of their domain. A particular consequence of our results is the following. Given and , there is a constant for which the following is true. Let be closed and be functions whose norms are both bounded above by . Suppose is closed and that and coincide throughout . Then, over the set of points in whose distance to is at most , we have that the norm of the difference is bounded above by . More generally, we establish that this phenomenon remains valid in a less restrictive Banach space setting under the weaker hypothesis that the two functions and are only close in a pointwise sense throughout the closed subset . We require only that the subset be closed; in particular, the case that is finite is covered by our results. The restriction that is sharp in the sense that our result is false for .