Spectral equivalences through nonstandard samplings

TL;DR

This paper introduces a novel process for embedding Hilbert spaces into L2-spaces using nonstandard sampling, enabling new spectral theorem variants and explicit operator representations.

Contribution

It presents a new embedding method dependent on two parameters, providing diverse spectral theorem versions and explicit, natural operator models.

Findings

New embedding process for Hilbert spaces into L2 using nonstandard sampling.

Derivation of multiple spectral theorem variants with this process.

Explicit operator models achieved by parameter tuning.

Abstract

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.