Nonlinear Sampling and Lebesgue's Integral Sums

TL;DR

This paper explores nonlinear, event-dependent sampling methods for constructing Lebesgue integral sums, highlighting their physical measurement implications and potential for clockless frequency detection.

Contribution

It introduces a novel perspective on nonlinear sampling in Lebesgue integration and discusses its practical applications and theoretical implications.

Findings

Nonlinear sampling can be used to construct Lebesgue integral sums.

Such sampling methods are relevant for physical measurements and systems without clocks.

A method for frequency detection using nonlinear sampling is proposed.

Abstract

We consider nonlinear, or "event-dependent", sampling, i.e. such that the sampling instances {tk} depend on the function being sampled. The use of such sampling in the construction of Lebesgue's integral sums is noted and discussed as regards physical measurement and also possible nonlinearity of singular systems. Though the limit of the sums, i.e. Lebesgue's integral, is linear with regard to the function being integrated, these sums are nonlinear in the sense of the sampling. A relevant method of frequency detection not using any clock, and using the nonlinear sampling, is considered. The mathematics and the realization arguments essentially complete each other.