Higher-order Fourier Analysis and Applications

Higher-order Fourier Analysis and Applications

Higher order Fourier analysis

Fourier series of higher-order Bernoulli functions and their applications

In this paper, we study the Fourier series related to higher-order Bernoulli functions and give new identities for higher-order Bernoulli functions which are derived from the Fourier series of them.

Using higher-order Fourier analysis over general fields

Higher-order Fourier analysis, developed over prime fields, has been recently used in different areas of computer science, including list decoding, algorithmic decomposition and testing. We extend the tools of higher-order Fourier analysis to analyze functions over general fields. Using these new tools, we revisit the results in the above areas. (i) For any fixed finite field K, we show that th...

On Higher-Order Fourier Analysis over Non-Prime Fields

The celebrated Weil bound for character sums says that for any low-degree polynomial P and any additive character χ, either χ(P ) is a constant function or it is distributed close to uniform. The goal of higher-order Fourier analysis is to understand the connection between the algebraic and analytic properties of polynomials (and functions, generally) at a more detailed level. For instance, wha...