略论丁夏畦,丁毅近著《Hermite展开与广义函数》

Remark on Xiaqi Ding and Yi Ding's book Hermite Expansions and Generalized Functions

对丁夏畦、丁毅著《Hermite展开与广义函数》一书进行简单的评述.该书是Schwartz广义函数理论的最新研究发展.其理论的独创之处是运用Hermite函数展开理论研究Schwartz速降函数空间及其上的Schwartz广义函数.

In the present note, we shortly comment a recent book Hermite Expansions and Generalized Functions written by Xiaqi Ding and Yi Ding. This book describes the latest research progress on the Schwartz＇s distribution theory. The originality in this book is to apply Hermite function expansion theory to investigate Schwartz quickly decreasing function space and Schwartz distributions. As pointed out by the authors, this idea is a continuation and creativity of Prof. Luogen Hua＇s related research works. Using the expansion theory of Hermite functions, the authors prove that any formal series （denoted by weak function by the authors） is a classical Schwartz distribution if and only if its n-th expansion coefficient increases not faster than a polynomial increase of n. Obviously, this is an extremely important and deep work in which the Schwartz distributions are clearly and neatly described. Weak functions also include an important subclass, i.e. parabolic Hua＇s class H. In fact, this is a new type of hyperfunctions. Another advantage of the idea by the expansions with Hermite functions is that we can introduce new generalized number and generalized weak function in the most general sense as an extension of Schwartz＇s classical theory. With the help of the extension theory, the product of any two generalized functions can be well defined. This new theory can be applied to the classical analysis such as Riemann Zeta function theory. More importantly, it can be used to investigate some nonlinear partial differential equations with singular solutions. For instance, the authors use the new theory to construct global weak solutions for some nonlinear hyperbolic systems of conservation laws, while these problems can not be solved in the classical framework.