2013/03/09 by Antti Haimi, Haakan Hedenmalm, Håakan Hedenmalm · 23 citations
Mathematics · #Analytic function #Asymptotic expansion #Bergman kernel #Bergman space #Bounded function #Function (biology) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Hilbert space #Holomorphic and Operator Theory #Kernel (algebra) #Mathematical analysis #Mathematics #Metric (unit) #Pure mathematics #Reproducing kernel Hilbert space #Space (punctuation) #Square-integrable function #math.AP #math.CV #msc:30A94 #msc:30G30 #msc:32A36 #msc:46E22 #msc:58J37
paper · pdf · doi:10.1016/j.jfa.2014.09.002
published in Journal of Functional Analysis 267(12), 4667-4731 (Elsevier BV) · 40 pages
arxiv created 2013/03/09 · openalex publication_date 2014/09/18 · arxiv updated 2015/09/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider mainly the Hilbert space of bianalytic functions on a given domain in the plane, square integrable with respect to a weight. We show how to obtain the asymptotic expansion of the corresponding bianalytic Bergman kernel for power weights, under the standard condition on those weights. This is known only in the analytic setting, from the work of e.g. Tian, Yau, Zelditch, Catlin, et al. We remark that a bianalytic function may be identified with a vector-valued analytic function, supplied with a locally singular metric on the vectors. We also apply our findings to two bianalytic Bergman metrics introduced here.