Harmonic analysis

Definition

Let $X$ be a space with a group of symmetries, such as the circle $\mathbb{T} = \R / 2\pi\Z$ or the space $\R^n$ with translations. Harmonic analysis is the branch of analysis that studies functions on such $X$ through the following questions:

1. how a function decomposes into basic oscillations, such as the exponentials $e^{inx}$ on $\mathbb{T}$ or $e^{2\pi i x \cdot \xi}$ on $\R^n$;
2. in which sense, and in which function spaces, the function is recovered from this decomposition;
3. how operators that commute with translations, such as convolutions and differential operators with constant coefficients, act on the decomposition, and which norm estimates they satisfy.