Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 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. A. Beurling, The Collected Works of Arne Beurling, vol. 2, Ed. L. Carleson et al., Birkhäuser, Boston, 1989, pp. 341-365. MR 92k:01046b
  • 3. A. Beurling and P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291-309. MR 26:5361
  • 4. L. de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood Cliffs, 1968. MR 37:4590
  • 5. K. M. Flornes, Sampling and interpolation in the Paley-Wiener spaces $L^{p}_{\pi }$, $0<p\leq 1$, Publ. Mat. 42 (1998), 103-118. MR 2000e:42002
  • 6. F. Holland and R. Rochberg, Bergman Kernel Asymptotics for Generalized Fock Spaces, J. Anal. Math. 83 (2001), 207-242. CMP 2001:12
  • 7. R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251. MR 47:701
  • 8. S. V. Khrushchev, N. K. Nikol'skii, and B. S. Pavlov, Unconditional bases of exponentials and reproducing kernels, in ``Complex Analysis and Spectral Theory'', Lecture Notes in Math. Vol. 864, Springer-Verlag, Berlin/Heidelberg, 1981, pp. 214-335. MR 84k:46019
  • 9. I. F. Krasichkov-Ternovskii, An interpretation of the Beurling-Malliavin theorem on the radius of convergence, Math. USSR Sbornik 66 (1990), 405-429. MR 91h:42011
  • 10. B. Ya. Levin, Lectures on Entire Functions, American Mathematical Society, Providence, R.I., 1996. MR 97j:30001
  • 11. P. Lin and R. Rochberg, Trace ideal criteria for Toeplitz and Hankel operators on the weighted Bergman spaces with exponential type weights, Pacific J. Math. 173 (1996), 127-146. MR 97d:47034
  • 12. Yu. I. Lyubarskii and K. Seip, Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's $(A_{p})$ condition, Rev. mat. iberoamericana 13 (1997), 361-376. MR 99e:42004
  • 13. S. N. Mergelyan, Weighted approximations by polynomials, American Mathematical Society Translations, ser. 2, vol. 10, 1958, pp. 59-106. MR 20:1146
  • 14. J. Ortega-Cerdà and K. Seip, Multipliers for entire functions and an interpolation problem of Beurling, J. Funct. Anal. 162 (1999), 400-415. MR 2000c:30071
  • 15. J. Ortega-Cerdà and K. Seip, On Fourier frames, To appear in Ann. Math., 2002.
  • 16. R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, American Mathematical Society, New York, 1934. (Reprint MR 98a:01023)
  • 17. A. Volberg, Thin and thick families of rational functions, in ``Complex Analysis and Spectral Theory'', Lecture Notes in Math. Vol. 864, Springer-Verlag, Berlin/Heidelberg, 1981, pp. 440-480. MR 83j:30038

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 46E22, 30E05, 42A99

Retrieve articles in all journals with MSC (2000): 46E22, 30E05, 42A99


Additional Information

Yurii I. Lyubarskii
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, N–7491 Trondheim, Norway
Email: yura@math.ntnu.no

Kristian Seip
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, N–7491 Trondheim, Norway
Email: seip@math.ntnu.no

DOI: https://doi.org/10.1090/S0894-0347-02-00397-1
Keywords: De Branges spaces, Paley-Wiener spaces, interpolation, sampling
Received by editor(s): March 7, 2002
Published electronically: June 21, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society