Wavelet Notes

Wavelets are a useful basis for constructing solutions of the integral and differential equations of scattering theory. Wavelet bases efficiently represent functions with smooth structures on different scales, and the matrix representation of operators in a wavelet basis are well-approximated by sparse matrices. The basis functions are related to solutions of a linear renormalization group equation, and the basis functions have structure on all scales. Numerical methods based on this renormalization group equation are discussed. These methods lead to accurate and efficient numerical approximations to the scattering equations. These notes provide a detailed introduction to the subject that focuses on numerical methods. We plan to provide periodic updates to these notes.