A Selection of Distributions and Their Fourier Transforms with Applications in Magnetic Resonance Imaging

TL;DR

This paper provides a rigorous mathematical overview of distributions and their Fourier transforms relevant to MRI, emphasizing the underlying topology and explicit definitions, with key results like the Poisson summation formula and Gaussian Fourier transforms.

Contribution

It offers a detailed, self-contained mathematical treatment of distributions and Fourier transforms specifically tailored for applications in MRI, clarifying foundational concepts.

Findings

Explicit definitions of distributions and Fourier transforms in MRI context

Derivation of the Poisson summation formula within this framework

Fourier transform of Gaussian functions via ODE approach

Abstract

This note presents a rigorous introduction to a selection of distributions along with their Fourier transforms, which are commonly encountered in signal processing and, in particular, magnetic resonance imaging (MRI). In contrast to many textbooks on the principles of MRI, which place more emphasis on the signal processing aspect, this note will take a more mathematical approach. In particular, we will make explicit the underlying topological space of interest and clarify the exact sense in which these distributions and their Fourier transforms are defined. Key results presented in this note involve the Poisson summation formula and the Fourier transform of a Gaussian function via an ordinary differential equation (ODE) argument, etc. Although the readers are expected to have prior exposure to functional analysis and distribution theory, this note is intended to be self-contained.