On Density Conditions for Interpolation in the Ball

Marco, Nicolas; Massaneda, Xavier

doi:10.4153/cmb-2003-053-x

Cited by 3 publications

(2 citation statements)

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“…The techniques used in the present paper seem not to be able to yield strict versions of Theorem 1.3. Lastly, it is expected that complementing sufficient conditions to Theorem 1.3 do not hold for domains ⊂ C d in dimension d > 1; see [13,24] for some possible generalizations of the results [40] to the unit ball. For the failure of sufficiency of the lattice density conditions [4,38] underlying Theorem 1.1 for coherent states in the Bargmann-Fock spaces on C d with d > 1, see, e.g., [25,31].…”

Section: Frames and Riesz Sequencesmentioning

confidence: 99%

Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domains

Caspers

Velthoven

2022

Math. Z.

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Let $$\pi _{\alpha }$$ π α be a holomorphic discrete series representation of a connected semi-simple Lie group G with finite center, acting on a weighted Bergman space $$A^2_{\alpha } (\Omega )$$ A α 2 ( Ω ) on a bounded symmetric domain $$\Omega $$ Ω , of formal dimension $$d_{\pi _{\alpha }} > 0$$ d π α > 0 . It is shown that if the Bergman kernel $$k^{(\alpha )}_z$$ k z ( α ) is a cyclic vector for the restriction $$\pi _{\alpha } |_{\Gamma }$$ π α | Γ to a lattice $$\Gamma \le G$$ Γ ≤ G (resp. $$(\pi _{\alpha } (\gamma ) k^{(\alpha )}_z)_{\gamma \in \Gamma }$$ ( π α ( γ ) k z ( α ) ) γ ∈ Γ is a frame for $$A^2_{\alpha }(\Omega )$$ A α 2 ( Ω ) ), then $${{\,\mathrm{vol}\,}}(G/\Gamma ) d_{\pi _{\alpha }} \le |\Gamma _z|^{-1}$$ vol ( G / Γ ) d π α ≤ | Γ z | - 1 . The estimate $${{\,\mathrm{vol}\,}}(G/\Gamma ) d_{\pi _{\alpha }} \ge |\Gamma _z|^{-1}$$ vol ( G / Γ ) d π α ≥ | Γ z | - 1 holds for $$k^{(\alpha )}_z$$ k z ( α ) being a $$p_z$$ p z -separating vector (resp. $$(\pi _{\alpha } (\gamma ) k^{(\alpha )}_z)_{\gamma \in \Gamma / \Gamma _z}$$ ( π α ( γ ) k z ( α ) ) γ ∈ Γ / Γ z being a Riesz sequence in $$A^2_{\alpha } (\Omega )$$ A α 2 ( Ω ) ). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for $$G ={\mathrm {PSU}}(1, 1)$$ G = PSU ( 1 , 1 ) .

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Section: Frames and Riesz Sequencesmentioning

confidence: 99%

Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domains

Caspers

Velthoven

2022

Math. Z.

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“…Seip [13] gave a characterization of interpolating sequences of Bergman spaces on the unit disk. For the case of several variables, Marco and Massaneda [6] showed that Seip's condition for a sequence in the unit disk is sufficient to be of interpolating, however a complete generalization of Seip's characterization is not known yet. As for harmonic case, Choe and Yi [3] studied interpolations of harmonic Bergman spaces on the upper half-space of R n+1 and obtained a result similar to Amar.…”

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confidence: 99%