Higher order Fourier analysis

Higher-Order Difference and Higher-Order Splitting Methods for 2D Parabolic Problems with Mixed Derivatives

In this article we discuss a combination between fourth-order finite difference methods and fourth-order splitting methods for 2D parabolic problems with mixed derivatives. The finite difference methods are based on higher-order spatial discretization methods, whereas the timediscretization methods are higher-order discretizations using CrankNicolson or BDF methods. The splitting methods are hi...

Higher-order Fourier Analysis and Applications

Image analysis by orthogonal Fourier-Mellin moments

In this research, an attempt to analyze images with the orthogonal Fourier-Mellin moments is conducted. It leads to the conclusions that the lower order of orthogonal Fourier-Mellin moments primarily contain the fundamental image information; the higher order moments preserve more detailed image information; and each finite set of the moments will contribute individually in the reconstruction p...

Formal Analysis of Continuous-time Systems using Fourier Transform

To study the dynamical behaviour of the engineering and physical systems, we often need to capture their continuous behaviour, which is modeled using differential equations, and perform the frequency-domain analysis of these systems. Traditionally, Fourier transform methods are used to perform this frequency domain analysis using paper-and-pencil based analytical techniques or computer simulati...

Limits of Boolean Functions over Finite Fields

In this thesis, we study sequences of functions of the form Fp → {0, 1} for varying n, and define a notion of convergence based on the induced distributions from restricting the functions to a random affine subspace. One of the key tools we use is the recently developed theory of ‘higher order Fourier analysis’, where the characters of standard Fourier analysis are replaced with exponentials of...