Sensing continuous Functions

When we examine the physical environment around us, we may want to record our observations. Recording everything we observe is rarely possible however and therefore we often try to single out one elementary aspect and focus on this aspect alone. The most basic type of such a recording is that of a single quantity that we express as a real number \(r \in {\mathbb{R}}\). To obtain such a value \(r\), we often use some sensor that we place somewhere in our physical environment and then we read the sensor’s value. In most situations the sensor will show a different value, when we place it somewhere else and even when we repeat the process and read the value from the sensor in the same location another time, the value may be different. In the latter situation the value is likely to be similar however and in the former situation the value is likely to be similar, if the new location is very close to the previous location. We assume that there is a certain subspace \(X\) of the environment around us within our reach and interest, where we can place the sensor. As a subspace of our environment, \(X\) inherits a topology and the above observations lead to the intuition, that there is a proper continuous function \(f \colon X \rightarrow {\mathbb{R}}\) whose values we are reading of the sensor in proximity. This is also called a scalar field. Now we may never learn this function \(f\), but we can interpolate between our samples from a sufficiently dense set of measuring locations, to define another continuous function \(g \colon X \rightarrow {\mathbb{R}}\) that is close to \(f\) in the sense that \(-\varepsilon \leq f(p) - g(p) \leq \varepsilon\) for all \(p \in X\) and a real number \(\varepsilon > 0\) of moderate size. More specifically we can make \(\varepsilon\) as small as we like by increasing both, the density of our samples and the accuracy of our sensor. We concede that the problem of approximating \(f\) is non-trivial and that interpolation is a vast field outside the scope of this document.