Generalized Functions, Volume 3

This volume explores the application of generalized functions to the Cauchy problem for partial differential equations, examining solution uniqueness and well-posedness, along with spectral decomposition related to generalized eigenfunctions.

Generalized Functions, Volume 3 is part of a comprehensive six-volume series authored by I. M. Gelfand and his collaborators, focusing on the theory and applications of generalized functions, also known as distributions. This volume specifically delves into the applications of these functions in relation to the Cauchy problem for systems of partial differential equations (PDEs) with constant coefficients. The groundwork for the theory of generalized functions was laid in the early 1950s, with earlier contributions from mathematicians and physicists, notably in the context of Green's functions and Paul Dirac's work in quantum electrodynamics.

In this volume, Gelfand explores the uniqueness classes of solutions to the Cauchy problem, providing insights into the conditions under which solutions exist and are well-defined. The book emphasizes the importance of understanding the characteristics of functions that ensure the well-posedness of the Cauchy problem, which is crucial for both theoretical and practical applications in mathematics and physics.

The final chapter of the volume presents significant findings related to the spectral decomposition of differential operators associated with generalized eigenfunctions. This aspect of the study not only enhances the theoretical framework of generalized functions but also opens avenues for further research in various fields, including analysis, stochastic processes, and representation theory. Overall, Generalized Functions, Volume 3 serves as an essential resource for those interested in advanced mathematical concepts and their applications.