Contact Structures and Distributions

The most important source of the theory of generalized functions has been problems that arise in the theory of hyperbolic differential equations (including analytic questions of field theory). In particular the generalized functions that we call Hadamard kernels (Chapt. 1) were introduced under the name “nonsingular integrals of a new type” to construct a fundamental solution of a hyperbolic equation (Hadamard 1932). The work of Sobolev (1936) is devoted to the solution of this same problem. In the paper of M. Riesz (1949) the important idea of analytic continuation of a family of generalized functions arose in the same context. A new class of singular functions arises in the analysis of a wave field in a neighborhood of a caustic. Among the pioneering works in this area is the paper of V.M. Babich (1961a), where new special functions are introduced to study a field near an elementary caustic.