Discrete spectra of convolutions of compactly supported functions on SE(2) using Sturm–Liouville theory

This paper introduces a systematic study for analytic aspects of discrete spectra methods for functions supported on some compact domains of SE(2), according to Sturm–Liouville theory. We then apply these discrete spectra methods to approximate convolution of functions supported on these compact domains. We shall also investigate different aspects of the presented theory in the cases of zero-value boundary condition and derivative boundary condition. The paper is concluded by some Plancherel formulas associated to Sturm–Liouville theory and special cases of boundary conditions.