On a sampling problem for a Bargmann-Fock space

Year 2024, Volume: 53 Issue: 5, 1305 - 1311, 15.10.2024

Mohammed El Aıdı

https://doi.org/10.15672/hujms.1339373

Abstract

The purpose of the present article is to provide geometric sufficient conditions for discrete points to be a sampling sequence for a generalized Hilbert Bargmann-Fock space in several complex variables.

Keywords

Beurling density , generalized Bargmann-Fock spaces , plurisubharmonic functions , relatively separated sequence , sampling sequences

References

• [1] B. Berndtsson and J. Ortega-Cerdà, On interpolating and sampling in Hilbert spaces of analytic functions, J. reine angew Math. 464, 109-128, 1995.
• [2] B. Berndtsson and M. Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier. 32 (3), 91-110, 1982.
• [3] K. Fritzsche and H. Grauert, From Holomorphic Functions to Complex Manifolds, Springer New York, NY, 2002.
• [4] H. Führ, K. Gröchenig, A. Haimi, A. Klotz and J.L. Romero, Density of sampling and interpolation in reproducing kernel Hilbert spaces, J. Lond. Math. Soc. 96 (3), 663-686, 2017.
• [5] J. Garnett,Bounded analytic functions, Springer-Verlag New York, 2007.
• [6] K. Gröchenig, A. Haimi, J. Ortega-Cerdà and J.L. Romero, Strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions, J. Funct. Anal. 277 (12), 34 pp, 2019.
• [7] W. K. Hayman, P. B. Kennedy, Subharmonic Functions, Academic Press, London 1976.
• [8] L.L. Helms, Potential Theory, Springer Dordrecht Heidelberg London New York, 2009.
• [9] C.O. Kiselman, Plurisubharmonic functions and potential theory in several complex variables, Development of mathematics 1950-2000, 655-714, Birkhäuser, Basel, 2000.
• [10] M. Klimek, Pluripotential theory, London Mathematical Society Monographs, Clarendon Press, 266 p, 1991.
• [11] N. Lindholm, Sampling in weighted $L^p$ spaces of entire functions in $\mathbb C^n$ and estimates of the Bergman kernel, J. Funct. Anal. 182 (2), 390-426, 2001.
• [12] Yu. Lyubarskii and K. Seip, Sampling and interpolation of entire functions and exponential systems in convex domains, Ark. Mat. 32 (1), 157-193, 1994.
• [13] J. Ortega-Cerdà and K. Seip, Beurling-type density theorems for weighted $L^p$ spaces of entire functions, J. Anal. Math. 75 (1), 247-266, 1998.
• [14] K. Seip, Interpolation and sampling in spaces of analytic functions, 33, University Lecture Series. American Mathematical Society, Providence, RI, 2004.
• [15] K. Seip. Density theorem for sampling and interpolating in the Bargmann-Fock spaces III, Math. Scand. 73, 112-126, 1993.
• [16] K. Seip and R. Wallstén, Density theorems for sampling and interpolation in the Bargmann-Fock space II, J. reine angew. Math. 429, 107-113, 1992.
• [17] K. Seip, Density theorem for sampling and interpolating in the Bargmann-Fock spaces I, J. reine angrew Math. 429, 91-106, 1992.
• [18] K. Seip, Reproducing formulas and double orthogonality in Bargmann and Bergman spaces, SIAM J. Math. Anal. 22 (3), 856-876, 1991.