Weighted Paley-Wiener spaces

Lyubarskii, Yurii; Seip, Kristian

doi:10.1090/S0894-0347-02-00397-1

Weighted Paley-Wiener spaces

Authors: Yurii I. Lyubarskii and Kristian Seip
Journal: J. Amer. Math. Soc. 15 (2002), 979-1006
MSC (2000): Primary 46E22; Secondary 30E05, 42A99
DOI: https://doi.org/10.1090/S0894-0347-02-00397-1
Published electronically: June 21, 2002
MathSciNet review: 1915824
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Abstract: We study problems of sampling and interpolation in a wide class of weighted spaces of entire functions. These weights are characterized by the property that their natural regularization as the envelop of the unit ball of the corresponding space is equivalent to the original weight. We give an independent description of such weights and also show that, in a sense, this is the widest class of weights and associated spaces for which results on sets of uniqueness, sampling, and interpolation related to the classical Paley-Wiener spaces can be extended in a direct and natural way, keeping the basic features of the theory intact. One of the basic tools for our study is the De Brange theory of spaces of entire functions.

1. N. Ahiezer, On weighted approximations of continuous functions by polynomials on the entire number axis. (Russian), Uspehi Mat. Nauk (N.S.) 11 (1956), 3-43. MR 18:802f
2. Arne Beurling, The collected works of Arne Beurling. Vol. 1, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1989. Complex analysis; Edited by L. Carleson, P. Malliavin, J. Neuberger and J. Wermer. MR 1057613
3. A. Beurling and P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291–309. MR 147848, https://doi.org/10.1007/BF02545792
4. Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
5. Kristin M. Flornes, Sampling and interpolation in the Paley-Wiener spaces 𝐿_{𝜋}^{𝑝},0<𝑝\le1, Publ. Mat. 42 (1998), no. 1, 103–118. MR 1628146, https://doi.org/10.5565/PUBLMAT_42198_04
6. F. Holland and R. Rochberg, Bergman Kernel Asymptotics for Generalized Fock Spaces, J. Anal. Math. 83 (2001), 207-242. CMP 2001:12
7. Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, https://doi.org/10.1090/S0002-9947-1973-0312139-8
8. S. V. Hruščëv, N. K. Nikol′skiĭ, and B. S. Pavlov, Unconditional bases of exponentials and of reproducing kernels, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 214–335. MR 643384
9. I. F. Krasichkov-Ternovskiĭ, Interpretation of the Beurling-Malliavin theorem on the radius of completeness, Mat. Sb. 180 (1989), no. 3, 397–423 (Russian); English transl., Math. USSR-Sb. 66 (1990), no. 2, 405–429. MR 993233, https://doi.org/10.1070/SM1990v066n02ABEH001178
10. B. Ya. Levin, Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko; Translated from the Russian manuscript by Tkachenko. MR 1400006
11. Peng Lin and Richard Rochberg, Trace ideal criteria for Toeplitz and Hankel operators on the weighted Bergman spaces with exponential type weights, Pacific J. Math. 173 (1996), no. 1, 127–146. MR 1387794
12. Yurii I. Lyubarskii and Kristian Seip, Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt’s (𝐴_{𝑝}) condition, Rev. Mat. Iberoamericana 13 (1997), no. 2, 361–376. MR 1617649, https://doi.org/10.4171/RMI/224
13. S. N. Mergelyan, Weighted approximations by polynomials, American Mathematical Society Translations, Ser. 2, Vol. 10, American Mathematical Society, Providence, R.I., 1958, pp. 59–106. MR 0094633
14. Joaquim Ortega-Cerdà and Kristian Seip, Multipliers for entire functions and an interpolation problem of Beurling, J. Funct. Anal. 162 (1999), no. 2, 400–415. MR 1682065, https://doi.org/10.1006/jfan.1998.3357
15. J. Ortega-Cerdà and K. Seip, On Fourier frames, To appear in Ann. Math., 2002.
16. Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142
17. A. L. Vol′berg, Thin and thick families of rational fractions, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 440–480. MR 643388