Nonlinear Sampling and Lebesgues Integral Sums

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.

💡 Research Summary

The paper introduces the concept of “non‑linear (event‑dependent) sampling,” in which the instants at which a signal is sampled are determined by the signal itself rather than by an external clock. This idea is then employed to construct Lebesgue integral sums, a formulation that partitions the domain according to the range of the function instead of the time axis. The authors begin by contrasting the classical Riemann approach—where a fixed or adaptively chosen time grid is imposed and the function is evaluated at those points—with the Lebesgue approach, where one first selects a set of value intervals (