My ultimate goal is to see how the point of infinity and an arbitrary transform in Riemann sphere can lead to what consequences in dynamical systems, and it seems that harmonic analysis plays a crucial role in between since it connects Fourier transform and spherical harmonics, Hilbert space and functional analysis, topology, group, graph and representation theory in one place. In the term of complex systems theory, this could be the super high-degree node.
What road map should I take? The suggested related topics are (1) Fourier transform of distributions, (2) spectral theorem (spectrum of the Laplacian), (3) harmonic analysis and (4) representations of locally compact groups. I have finished complex analysis (Needham), dynamical systems theory (Strogatz), and the first 6 chapters of Kreyszig's on functional analysis. I have some choices starting from here:
Continue reading functional analysis:
- Keep reading Kreyszig. The rest of the book dedicates to spectral theory and I'm happy to keep reading, but since there are many other things to learn is this the optimized path?
- Switch to Rudin's book. It dedicates two parts specifically for (1) and (2), but it seems to mostly use Banach spaces?
- Switch to harmonic analysis:
- Dym & McKean, Fourier Series and Integrals. But it's too focused on Fourier analysis? It's also pretty old (about 50 years).
- Folland, A Course in Abstract Harmonic Analysis. Seem ideal, but is it too soon to read it now? And it doesn't talk about distributions.
Related questions on harmonic analysis: Undergrad level: Math SE; grad level: Math Overflow, Reddit. Also What's a good primer from linear algebra to spherical harmonics?
My question on Physics: How would behaviors of Riemann spheres represent characteristics of physical dynamical systems?