Harmonic analysis is the generalization of Fourier analysis, which studies representation of functions as superposition of basic waves. In classical Fourier analysis, we know that if f \in L^2(\mathbb{R}), then we can write f as

\begin{aligned} f(x) = \sum\limits_{n \in \mathbb{Z}} c_ne^{inx}\end{aligned},

where n runs over all integers. Moreover, we can find the coefficients c_n to be

\begin{aligned} c_n = \dfrac{1}{2\pi} \int\limits_0^{2\pi} f(x)e^{-inx} dx\end{aligned}.

Harmonic analysis is just Fourier analysis on a more general setting. Here, we consider our function to be defined in a topological group G. Topological group is just a group with a binary operation \cdot \colon G \times G \rightarrow G with extra condition that the binary operation and inverse function are continuous with respect to the topology.

I am currently working on harmonic analysis on Lie groups, mainly 3-sphere. Unlike 1-sphere (also known as circle) which is an abelian Lie group, 3-sphere is non-commutative, so the problem is not as easy as in 1-sphere which is basically just classical Fourier analysis. To get the idea of 3-sphere, my research will also include the study of 2-sphere and harmonic polynomials there.

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