A unified construction of product formulas and convolutions for Sturm-Liouville operators
This is a voluminous article on unification of product formulas and convolutions for Sturm-Liouville operators. The abtsract of the article reflects the content in the best way. ``We establish a positive product formula for the solutions of the Sturm-Liouville equation \(l(u) = \lambda u\), where \(l\) belongs to a general class which includes singular and degenerate Sturm-Liouville operators. Our technique relies on a positivity theorem for possibly degenerate hyperbolic Cauchy problems and on a regularization method which makes use of the properties of the diffusion semigroup generated by the Sturm-Liouville operator. We show that the product formula gives rise to a probability preserving convolution algebra structure on the space of finite measures which satisfies the basic axioms for developing harmonic analysis on the convolution algebra. Unlike previous works, our framework includes a subfamily of Sturm-Liouville operators for which the support of the convolution of Dirac measures is noncompact. The connection with hypergroup theory is discussed. Convolution-type integral equations on weighted Lebesgue spaces are also discussed, and a solvability condition is established.''
scientific article; zbMATH DE number 7355863
scientific article