Euler spaces of analytic functions

Abstract:

A formula due to Euler and Legendre is used to construct finite-difference counterparts to the Dirichlet space. The spaces have integral representations and characterizations in terms of area integrals. Their reproducing kernels are logarithms of the reproducing kernels of the Newton spaces, which are counterparts to the Hardy class. A Hilbert space with reproducing kernel \[ \log [(1/\overline w z) \log \;1/(1 - \overline w z)]\] is also shown to exist and to be related to Bernoulli numbers and combinatorial theory.