ON EXISTENCE THEOREMS OF POTENTIAL THEORY AND

A further point which we gain is that the kernel functions associated with the various norms of function theory and elliptic partial differential equations are well adapted to numerical computation, in view of their relation to orthogonal functions, so that we obtain at one and the same time an existence proof and a computational algorithm. The most serious drawback in our method is, perhaps, that we must make assumptions upon the smoothness of the boundary of the domains we consider, so that the general case is reached only after a topological approximation argument is given. From the broader point of view, our treatment is important in that it yields a unified attack upon various existence problems. An extensive class of existence theorems of function theory of one or more variables and of partial differential equations of elliptic type can be developed using one basic procedure. This procedure can be outlined as follows. We set up the reproducing kernel function in a given function space by solving a suitable extremal problem, or alternatively by means of orthogonal functions, and we investigate the scalar product of the kernel function with what may be described as the fundamental singularity associated with this space. A local argument is used to study the behavior of this scalar product as the infinity of the fundamental singularity crosses the boundary of the domain of definition of the functions in our class, and the desired existence theorems follow from the geometric properties thus obtained. The local investigations depend in a general way upon a knowledge of our problem in simple domains, such as, for example, the unit circle.