Extension of Lipschitz functions

Extension of P-adic Definable Lipschitz Functions

Write OK for the valuation ting, MK for the maximal ideal of K and kK for the residue field. Let us fix $ some uniformizer of K. We denote by acm : K → OK/(MK) the map sending some nonzero x ∈ K to x$−ord(x) mod MK , and sending zero to zero. This is a definable map. We denote by RV the union of K×/(1 +MK) and {0} and by rv : K → RV the quotient map. More generally, if m ∈ N∗, we set RVm = K×/(...

On the Extension of Lipschitz, Lipschitz- Hölder Continuous, and Monotone Functions

Interpolation of Lipschitz functions

This paper describes a new computational approach to multivariate scattered data interpolation. It is assumed that the data is generated by a Lipschitz continuous function f. The proposed approach uses the central interpolation scheme, which produces an optimal interpolant in the worst case scenario. It provides best uniform error bounds on f, and thus translates into reliable learning of f. Th...

Controlling Lipschitz functions

Given any positive integers m and d, we say the a sequence of points (xi)i∈I in Rm is Lipschitz-d-controlling if one can select suitable values yi (i ∈ I) such that for every Lipschitz function f : Rm → Rd there exists i with |f(xi)−yi| < 1. We conjecture that for every m ≤ d, a sequence (xi)i∈I ⊂ Rm is d-controlling if and only if sup n∈N |{i ∈ I : |xi| ≤ n}| nd =∞. We prove that this conditio...

Bandlimited Lipschitz Functions

We study the space of bandlimited Lipschitz functions in one variable. In particular we provide a geometrical description of interpolating and sampling sequences for this space. We also give a description of the trace of such functions to sequences of critical density in terms of a cancellation condition.