【摘要】
Reproducing kernel theory has developed over more than a century as one of the most fundamental structures in modern analysis. A reproducing kernel is not merely a technical device; it is a complete representation of a Hilbert space, encoding geometry, analytic structure, and operator behavior in a single function K(x,y).
This lecture presents a unified global perspective on reproducing kernels, beginning with classical origins (Bergman, Szegő, Schiffer) and extending to modern developments. We outline the essential aspects of reproducing kernels:
What is a reproducing kernel?
What is the theory of reproducing kernels?
Why are reproducing kernels fundamental?
General properties of reproducing kernels
Operator-theoretic interpretations
Bounded linear operator equations
Inverse problems
PDE solution constructions
Kernel geometry and domain function theory
Q. Guan’s identity and its applications
Discretization principles and infinite precision computation
The aim is to provide a panoramic view of the theory—accessible yet deep—showing how reproducing kernels unify classical analysis, modern operator theory, and current breakthroughs. This lecture forms the foundation for the second and third talks.
