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Density theorems for sampling and interpolation in the Bargmann-Fock space
Author:
Kristian Seip
Journal:
Bull. Amer. Math. Soc. 26 (1992), 322-328
MSC (2000):
Primary 30D15; Secondary 46E22
MathSciNet review:
1136138
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Abstract: We give a complete description of sampling and interpolation in the Bargmann-Fock space, based on a density concept of Beurling. Roughly speaking, a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von Neumann lattice, and similarly, a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von Neumann lattice.
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Bergman spaces, SIAM J. Math. Anal. 22 (1991),
no. 3, 856–876. MR 1091688
(92e:44006), http://dx.doi.org/10.1137/0522054
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Kristian
Seip, Regular sets of sampling and
interpolation for weighted Bergman spaces, Proc. Amer. Math. Soc. 117 (1993), no. 1, 213–220. MR 1111222
(93c:30051), http://dx.doi.org/10.1090/S0002-9939-1993-1111222-5
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Seip, Density theorems for sampling and
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Robert
M. Young, An introduction to nonharmonic Fourier series, Pure
and Applied Mathematics, vol. 93, Academic Press Inc. [Harcourt Brace
Jovanovich Publishers], New York, 1980. MR 591684
(81m:42027)
- [1]
- H. Bacry, A. Grossmann, and J. Zak, Proof of the completeness of lattice states in the kq-representation, Phys. Rev. B 12 (1975), 1118-1120.
- [2]
- V. Bargmann, P. Butero, L. Girardello, and J. R. Klauder, On the completeness of coherent states, Rep. Mod. Phys. 2 (1971), 221-228. MR 0290680 (44:7860)
- [3]
- A. Beurling, The collected works of Arne Beurling, vol. 2 Harmonic analysis (L. Carleson, P. Malliavin, J. Neuberger, and J. Wermer, eds.), Birkhäuser, Boston, MA, 1989, pp. 341-365. MR 1057614 (92k:01046b)
- [4]
- I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), 961-1005. MR 1066587 (91e:42038)
- [5]
- I. Daubechies and A. Grossmann, Frames in the Bargmann space of entire functions, Comm. Pure Appl. Math., 41 (1988), 151-164. MR 924682 (89e:46028)
- [6]
- I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271-1283. MR 836025 (87e:81089)
- [7]
- R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366. MR 0047179 (13:839a)
- [8]
- G. B. Folland, Harmonic analysis in phase space, Princeton Univ. Press, Princeton, NJ, 1989. MR 983366 (92k:22017)
- [9]
- K. Gröchenig and D. Walnut, A Riesz basis for the Bargmann-Fock space related to sampling and interpolation, Ark. Mat. (to appear). MR 1289756 (95h:46038)
- [10]
- S. Janson, J. Peetre, and R. Rochberg, Hankel forms and the Fock space, Revista Mat. Iberoamer. 3 (1987), 61-138. MR 1008445 (91a:47029)
- [11]
- H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967), 37-52. MR 0222554 (36:5604)
- [12]
- Y. Lyubarskii, Frames in the Bargmann space of entire functions, manuscript, 1990.
- [13]
- J. von Neumann, Foundations of quantum mechanics, Princeton Univ. Press, Princeton, NJ, 1955. MR 0066944 (16:654a)
- [14]
- A. M. Perelomov, On the completeness of a system of coherent states, Theor. Math. Phys. 6 (1971), 156-164. MR 0475444 (57:15050)
- [15]
- K. Seip, Reproducing formulas and double orthogonality in Bargmann and Bergman spaces, SIAM J. Math. Anal. 22 (1991), 856-876. MR 1091688 (92e:44006)
- [16]
- -Regular sets of sampling and interpolation for weighted Bergman spaces, Proc. Amer. Math. Soc. (to appear). MR 1111222 (93c:30051)
- [17]
- -Density theorems for sampling and interpolation in the Bargmann-Fock space I, J. Reine Angew. Math, (to appear). MR 1173117 (93g:46026a)
- [18]
- K. Seip, Beurling type density theorems in the unit disk, manuscript, 1992. MR 1223222 (94g:30033)
- [19]
- K. Seip and R. Wallstén, Density theorems for sampling and interpolation in the Bargmann-Fock space II, J. Reine Angew. Math. (to appear). MR 1136138 (93d:30036)
- [20]
- R. M. Young, An introduction to nonharmonic Fourier series, Academic Press, New York, 1980. MR 591684 (81m:42027)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0273-0979-1992-00290-2
PII:
S 0273-0979(1992)00290-2
Article copyright:
© Copyright 1992 American Mathematical Society
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