Discontinuous information in the worst case and randomized settings

TL;DR

This paper proves that discontinuous linear information does not outperform continuous linear information in approximating operators, both in worst case and randomized settings, especially when function evaluations are only almost everywhere defined.

Contribution

It establishes the equivalence in power between discontinuous and continuous linear information for operator approximation in both worst case and randomized frameworks.

Findings

Discontinuous linear information is not more powerful than continuous in worst case.

In randomized settings, discontinuous information offers limited advantage over continuous.

Results apply even when function evaluations are only almost everywhere defined.

Abstract

We believe that discontinuous linear information is never more powerful than continuous linear information for approximating continuous operators. We prove such a result in the worst case setting. In the randomized setting we consider compact linear operators defined between Hilbert spaces. In this case, the use of discontinuous linear information in the randomized setting cannot be much more powerful than continuous linear information in the worst case setting. These results can be applied when function evaluations are used even if function values are defined only almost everywhere.