Asymptotics of reproducing kernels on a plane domain

Engliš, Miroslav

doi:10.1090/s0002-9939-1995-1277107-0

Cited by 5 publications

(1 citation statement)

References 4 publications

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“…5. Ω a domain in C of hyperbolic type, G = 1, F (φ(z)) = (1 − |z| 2 ) · |φ (z)| where φ : D → Ω is any uniformization map (that is, ds 2 = F (z) |dz| 2 is the Poincaré metric on Ω) ( [19], [9]). 6.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Asymptotic behaviour of reproducing kernels of weighted Bergman spaces

Engliš

1997

Trans. Amer. Math. Soc.

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Abstract. Let Ω be a domain in C n , F a nonnegative and G a positive function on Ω such that 1/G is locally bounded, A 2 α the space of all holomorphic functions on Ω square-integrable with respect to the measure F α G dλ, where dλ is the 2n-dimensional Lebesgue measure, and Kα(x, y) the reproducing kernel for A 2 α . It has been known for a long time that in some special situations (such as on bounded symmetric domains Ω with G = 1 and F = the Bergman kernel function) the formula lim α→+∞

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Section: Introduction and Resultsmentioning

confidence: 99%

Asymptotic behaviour of reproducing kernels of weighted Bergman spaces

Engliš

1997

Trans. Amer. Math. Soc.

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On holomorphick-differentials on open Riemann surfaces

Askaripour

Foth

2012

Complex Variables and Elliptic Equations

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Berezin Quantization and Reproducing Kernels on Complex Domains

Engliš¹

1996

Trans. Amer. Math. Soc.

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Abstract. Let Ω be a non-compact complex manifold of dimension n, ω = ∂∂Ψ a Kähler form on Ω, and Kα(x, y) the reproducing kernel for the Bergman space A 2 α of all analytic functions on Ω square-integrable against the measure e −αΨ |ω n |. Under the conditionF. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109-1163] was able to establish a quantization procedure on (Ω, ω) which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just Ω = C n and Ω a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as α → +∞. This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in C n . Along the way, we also fix two gaps in Berezin's original paper, and discuss, for Ω a domain in C n , a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure |ω n |.

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Asymptotics of reproducing kernels on a plane domain

Cited by 5 publications

References 4 publications

Asymptotic behaviour of reproducing kernels of weighted Bergman spaces

Asymptotic behaviour of reproducing kernels of weighted Bergman spaces

On holomorphick-differentials on open Riemann surfaces

Berezin Quantization and Reproducing Kernels on Complex Domains

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