Weighted-𝐿² interpolation on non-uniformly separated sequences

Ostrovsky, Stanislav

doi:10.1090/S0002-9939-2010-10193-4

Weighted-$L^2$ interpolation on non-uniformly separated sequences
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by Stanislav Ostrovsky PDF

Proc. Amer. Math. Soc. 138 (2010), 4413-4422 Request permission

Abstract:

We define a weighted-$\ell ^2$-norm associated to a discrete sequence $\Gamma$ in $\mathbb {C}$ and a weight function $\varphi$. We then give a sufficient condition which ensures that we can always extend weighted-$\ell ^2$ data to global holomorphic functions which are also weighted-$L^2$. The condition is such that the so-called upper density of $\Gamma$ is strictly less than one.

References

Bo Berndtsson and Joaquim Ortega Cerdà, On interpolation and sampling in Hilbert spaces of analytic functions, J. Reine Angew. Math. 464 (1995), 109–128. MR 1340337
Peter Duren and Alexander Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. MR 2033762, DOI 10.1090/surv/100
Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
Niklas Lindholm, Sampling in weighted $L^p$ spaces of entire functions in ${\Bbb C}^n$ and estimates of the Bergman kernel, J. Funct. Anal. 182 (2001), no. 2, 390–426. MR 1828799, DOI 10.1006/jfan.2000.3733
Joaquim Ortega-Cerdà and Kristian Seip, Beurling-type density theorems for weighted $L^p$ spaces of entire functions, J. Anal. Math. 75 (1998), 247–266. MR 1655834, DOI 10.1007/BF02788702
Joaquim Ortega-Cerdà, Alexander Schuster, and Dror Varolin, Interpolation and sampling hypersurfaces for the Bargmann-Fock space in higher dimensions, Math. Ann. 335 (2006), no. 1, 79–107. MR 2217685, DOI 10.1007/s00208-005-0726-3
Kristian Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space. I, J. Reine Angew. Math. 429 (1992), 91–106. MR 1173117, DOI 10.1515/crll.1992.429.91
Kristian Seip, Beurling type density theorems in the unit disk, Invent. Math. 113 (1993), no. 1, 21–39. MR 1223222, DOI 10.1007/BF01244300
Kristian Seip and Robert Wallstén, Density theorems for sampling and interpolation in the Bargmann-Fock space. II, J. Reine Angew. Math. 429 (1992), 107–113. MR 1173118