[19] The kernel function of an orthonormal system

Abstract

This is the first in a series of well-known papers (the remainder written with Stefan Bergman) dealing with the reproducing kernel in a Hilbert space of solutions of an elliptic partial differential equation. Here the elliptic operator is simply $\frac{\partial } {\partial \overline{z}},$ and so the kernel function $K(z,\overline{\zeta })$ is the classical Bergman kernel. Schiffer offers a new proof of the relation

$$\displaystyle{K(z,\overline{\zeta }) = -\frac{2} {\pi } \,\frac{{\partial }^{2}g(z,\zeta )} {\partial z\partial \overline{\zeta }} }$$

between $K(z,\overline{\zeta })$ and Green’s function g(z, ζ) of the domain and then applies it to study the variation of the kernel function with respect to that of the domain. A similar investigation is also carried out for the reproducing kernel in the space of analytic functions that possess single-valued antiderivatives. “Green’s functions” associated with the latter kernel were later used by V. A. Zmorovich (Dopovidi Akad. Nauk Ukraïn. RSR, 1958, 489–492, 1958, Ukrainian) and P. M. Tamrazov (Dopovidi Akad. Nauk Ukraïn. RSR, 1962, 853–856 1962, Ukrainian) in a geometric construction of analogues of Blaschke products in finitely connected domains. (For a more streamlined exposition of the results in Zmorovič and Tamrazov (Dopovidi Akad. Nauk Ukraïn. RSR, 1958, 489–492, 1958, Ukrainian; Dopovidi Akad. Nauk Ukraïn. RSR, 1962, 853–856, 1962, Ukrainian), see Coifman and Guido Weiss, Studia Math., 28, 31–68, 1966/1967 and Khavinson, Pacific J. Math., 108 295–318, 1983).