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We introduce a unified framework for the construction of convolutions and product formulas associated with a general class of regular and singular Sturm-Liouville boundary value problems. Our approach is based on the application of the Sturm-Liouville spectral theory to the study of the associated hyperbolic equation. As a by-product, an existence and uniqueness theorem for degenerate hyperbolic Cauchy problems with initial data at a parabolic line is established. The mapping properties of convolution operators generated by Sturm-Liouville operators are studied. Analogues of various notions and facts from probabilistic harmonic analysis are developed on the convolution measure algebra. Various examples are presented which show that many known convolution-type operators --- including those associated with the Hankel, Jacobi and index Whittaker integral transforms --- can be constructed using this general approach.

The Hyperbolic Maximum Principle Approach to the Construction of Generalized Convolutions