  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.
