Integral Transforms
This set of notes discusses a variety of details regarding the theory and numerics of integral transforms. We start with the basics: how should one evaluate an integral on the computer? This is followed by discussion regarding four types of integral transforms which have been used by the group in previous work. This is capped by the general theory of integral equations and transform/inverse pairs.
Topics are generally arranged from most elementary to most technical, although each section does devolve into the weeds eventually. Some of these details are little more than useful facts to keep in mind (most of the final section) or only tangentially related to the topic (quantum Fourier transforms), but it’s material that someone in the group has considered in the past, and seems relevant enough to document.
Lastly, much of the numerical discussion features code. In an attempt to remain language neutral, most of it is pseudocode, with a few exceptions describing specific approaches in a particular language (like Mathematica). Implementations of some of the more complicated algorithms or procedures can be found through the group Teams and will be identified in-text.