2015
DOI: 10.1007/s10455-015-9480-2
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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.

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Cited by 1 publication
(2 citation statements)
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“…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.…”
Section: Introduction and Statement Of Main Resultsmentioning
confidence: 93%
See 1 more Smart Citation
“…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.…”
Section: Introduction and Statement Of Main Resultsmentioning
confidence: 93%
“…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, +∞).…”
Section: Introduction and Statement Of Main Resultsmentioning
confidence: 99%