Theory of reproducing kernels

TLDR

The paper surveys the general theory of reproducing kernels, covering their construction, properties, and applications to various domains such as Bergman, harmonic, and Szegö kernels, and discusses limits and sums of kernels. Reproducing kernels are constructed via the projection formula presented in §12, I. The manuscript contains sections on differences, products, limits, and construction of reproducing kernels, with detailed page references and a concluding remarks section.

Abstract

May 7. Difference of reproducing kernels.354 8. Product of reproducing kernels.357 9. Limits of reproducing kernels.362 10.Construction of a r.k. by resolution of identity.368 11.Operators in spaces with reproducing kernels.371 12.The reproducing kernel of a sum of two closed subspaces.375 13.Final remarks in the general theory.380 Part II.Examples.384 1.I ntroductory remarks.384 (1) Bergman's kernels.384 (2) Harmonic kernels.386 2. Comparison domains.387 3. The difference of kernels.388 4. The square of a kernel introduced by Szegö.391 5.The kernel H{z, zi) for an ellipse.393 6. Construction of H(z, z¡) for a strip.394 7. Limits of increasing sequences of kernels.396 8. Construction of reproducing kernels by the projection-formula of §12, I.