Spectral equivalences through nonstandard samplings

The goal of this paper is to introduce a process that generates, given Hilbert space $H$ and symmetric operator $A$, an embedding of $H$ into an $L_2$-space through which $A$ is extended by a multiplication operator. This process will depend on two parameters, the nonstandard sampling and the standard-biased scale. We will use that process to prove diverse versions of the general spectral theorem, showing its appeal. Furthermore, through landmark examples, we will observe that by carefully tweaking the two parameters, we can make the resulting spaces, embeddings and operators quite explicit and natural.