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 , then we can write as
where runs over all integers. Moreover, we can find the coefficients to be
Harmonic analysis is just Fourier analysis on a more general setting. Here, we consider our function to be defined in a topological group . Topological group is just a group with a binary operation 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.