Weighted PaleyWiener spaces
Authors:
Yurii I. Lyubarskii and Kristian Seip
Journal:
J. Amer. Math. Soc. 15 (2002), 9791006
MSC (2000):
Primary 46E22; Secondary 30E05, 42A99
Published electronically:
June 21, 2002
MathSciNet review:
1915824
Fulltext PDF Free Access
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 PaleyWiener 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.
I. Ahiezer, On weighted approximations of continuous functions by
polynomials on the entire number axis, Uspehi Mat. Nauk (N.S.)
11 (1956), no. 4(70), 3–43 (Russian). MR 0084064
(18,802f)
 2.
Arne
Beurling, The collected works of Arne Beurling. Vol. 2,
Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA,
1989. Harmonic analysis; Edited by L. Carleson, P. Malliavin, J. Neuberger
and J. Wermer. MR 1057614
(92k:01046b)
 3.
A.
Beurling and P.
Malliavin, On Fourier transforms of measures with compact
support, Acta Math. 107 (1962), 291–309. MR 0147848
(26 #5361)
 4.
Louis
de Branges, Hilbert spaces of entire functions, PrenticeHall,
Inc., Englewood Cliffs, N.J., 1968. MR 0229011
(37 #4590)
 5.
Kristin
M. Flornes, Sampling and interpolation in the PaleyWiener spaces
𝐿_{𝜋}^{𝑝},\0<𝑝\le1, Publ. Mat.
42 (1998), no. 1, 103–118. MR 1628146
(2000e:42002), http://dx.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), 207242. 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 0312139
(47 #701), http://dx.doi.org/10.1090/S00029947197303121398
 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, BerlinNew York, 1981,
pp. 214–335. MR 643384
(84k:46019)
 9.
I.
F. KrasichkovTernovskiĭ, Interpretation of the
BeurlingMalliavin theorem on the radius of completeness, Mat. Sb.
180 (1989), no. 3, 397–423 (Russian); English
transl., Math. USSRSb. 66 (1990), no. 2,
405–429. MR
993233 (91h:42011)
 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
(97j:30001)
 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
(97d:47034)
 12.
Yurii
I. Lyubarskii and Kristian
Seip, Complete interpolating sequences for PaleyWiener spaces and
Muckenhoupt’s (𝐴_{𝑝}) condition, Rev. Mat.
Iberoamericana 13 (1997), no. 2, 361–376. MR 1617649
(99e:42004), http://dx.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
(20 #1146)
 14.
Joaquim
OrtegaCerdà and Kristian
Seip, Multipliers for entire functions and an interpolation problem
of Beurling, J. Funct. Anal. 162 (1999), no. 2,
400–415. MR 1682065
(2000c:30071), http://dx.doi.org/10.1006/jfan.1998.3357
 15.
J. OrtegaCerdà 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 (98a:01023)
 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, BerlinNew York, 1981,
pp. 440–480. MR 643388
(83j:30038)
 1.
 N. Ahiezer, On weighted approximations of continuous functions by polynomials on the entire number axis. (Russian), Uspehi Mat. Nauk (N.S.) 11 (1956), 343. MR 18:802f
 2.
 A. Beurling, The Collected Works of Arne Beurling, vol. 2, Ed. L. Carleson et al., Birkhäuser, Boston, 1989, pp. 341365. MR 92k:01046b
 3.
 A. Beurling and P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291309. MR 26:5361
 4.
 L. de Branges, Hilbert Spaces of Entire Functions, PrenticeHall, Englewood Cliffs, 1968. MR 37:4590
 5.
 K. M. Flornes, Sampling and interpolation in the PaleyWiener spaces , , Publ. Mat. 42 (1998), 103118. MR 2000e:42002
 6.
 F. Holland and R. Rochberg, Bergman Kernel Asymptotics for Generalized Fock Spaces, J. Anal. Math. 83 (2001), 207242. 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), 227251. 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, SpringerVerlag, Berlin/Heidelberg, 1981, pp. 214335. MR 84k:46019
 9.
 I. F. KrasichkovTernovskii, An interpretation of the BeurlingMalliavin theorem on the radius of convergence, Math. USSR Sbornik 66 (1990), 405429. 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), 127146. MR 97d:47034
 12.
 Yu. I. Lyubarskii and K. Seip, Complete interpolating sequences for PaleyWiener spaces and Muckenhoupt's condition, Rev. mat. iberoamericana 13 (1997), 361376. MR 99e:42004
 13.
 S. N. Mergelyan, Weighted approximations by polynomials, American Mathematical Society Translations, ser. 2, vol. 10, 1958, pp. 59106. MR 20:1146
 14.
 J. OrtegaCerdà and K. Seip, Multipliers for entire functions and an interpolation problem of Beurling, J. Funct. Anal. 162 (1999), 400415. MR 2000c:30071
 15.
 J. OrtegaCerdà 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, SpringerVerlag, Berlin/Heidelberg, 1981, pp. 440480. 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:
http://dx.doi.org/10.1090/S0894034702003971
PII:
S 08940347(02)003971
Keywords:
De Branges spaces,
PaleyWiener spaces,
interpolation,
sampling
Received by editor(s):
March 7, 2002
Published electronically:
June 21, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
