Generalized weighted Bergman–Dirichlet and Bargmann–Dirichlet spaces: explicit formulae for reproducing kernels and asymptotics

Abstract

We introduce new functional spaces that generalize the weighted Bergman and Dirichlet spaces on the disk D(0, R) in the complex plane and the Bargmann-Fock spaces on the whole complex plane. We give a complete description of the considered spaces. Mainly, we are interested in giving explicit formulas for their reproducing kernel functions and their asymptotic behavior as R goes to infinity.

Weighted Bergman–Dirichlet and Bargmann–Dirichlet Spaces in High Dimension

“…Our purpose in the present paper is to introduce the spaces A 2,α m (B n ), the analogue of the considered Bergman-Dirichlet spaces in high dimension. What we do in the construction of A 2,α m (B n ) works mutatis mutandis to introduce their counterparts on the whole n-complex space C n , the Bargmann-Dirichlet spaces F 2,ν m (C n ) of order m. We investigate their spectral properties and generalize the results obtained in [4] to high dimensions n ≥ 1. A part of some special techniques introduced in the calculation, the approach used here to prove our main results is quite similar to the one-dimensional setting.…”

“…Recently, two new classes of analytic function spaces of Sobolev type, labeled by a nonnegative integer m, have been introduced and studied in [4]. The first one is the Bergman-Dirichlet space generalizing the weighted Bergman and Dirichlet spaces on the disk D(0, R) in the complex plane C. The second is the Bargmann-Dirichlet space generalizing the Segal-Bargmann space on the complex plane C = D(0, +∞).…”