Let L2(G) denote the Hilbert
space of analytic functions f which are regular in a region G and have finite norms:
(∫∫G|f(z)|2dxdy)1∕2< ∞. It is well-known that the set {K(z,z1)|zI∈ G} of the
Bergman kernels for the class L2(G) is complete in L2(G). In this paper, for regular
regions Ġ in the plane, it is shown that the set {K(z,zI)|zI∈ G} is also complete in
the Hilbert space of analytic functions f which are regular in G and finite norms:
(∫∂G|f(z)|2ds)1∕2< ∞. The object of this paper is to discuss some problems of this
type.