Interpolation in a classical Hilbert space of entire functions
Author:
Robert M. Young
Journal:
Trans. Amer. Math. Soc. 192 (1974), 97114
MSC:
Primary 30A98; Secondary 30A80, 46E20
MathSciNet review:
0357823
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Abstract: Let H denote the PaleyWiener space of entire functions of exponential type which belong to on the real axis. A sequence of distinct complex numbers will be called an interpolating sequence for H if , where T is the mapping defined by . If in addition is a set of uniqueness for H, then is called a complete interpolating sequence. The following results are established. If and if the imaginary part of is sufficiently small, then is an interpolating sequence. If and if the imaginary part of is uniformly bounded, then is a complete interpolating sequence and is a basis for . These results are used to investigate interpolating sequences in several related spaces of entire functions of exponential type.
 [1]
N.
K. Bari, Biorthogonal systems and bases in Hilbert space,
Moskov. Gos. Univ. Učenye Zapiski Matematika 148(4)
(1951), 69–107 (Russian). MR 0050171
(14,289b)
 [2]
R.
P. Boas Jr., Expansions of analytic
functions, Trans. Amer. Math. Soc. 48 (1940), 467–487. MR 0002594
(2,80e), http://dx.doi.org/10.1090/S00029947194000025943
 [3]
R.
P. Boas Jr., A general moment problem, Amer. J. Math.
63 (1941), 361–370. MR 0003848
(2,281d)
 [4]
Ralph
Philip Boas Jr., Entire functions, Academic Press Inc., New
York, 1954. MR
0068627 (16,914f)
 [5]
R.
J. Duffin and J.
J. Eachus, Some notes on an expansion theorem of
Paley and Wiener, Bull. Amer. Math. Soc. 48 (1942), 850–855.
MR
0007173 (4,97e), http://dx.doi.org/10.1090/S000299041942077974
 [6]
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), http://dx.doi.org/10.1090/S00029947195200471796
 [7]
A.
E. Ingham, Some trigonometrical inequalities with applications to
the theory of series, Math. Z. 41 (1936), no. 1,
367–379. MR
1545625, http://dx.doi.org/10.1007/BF01180426
 [8]
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)
 [9]
J. Rosenbaum, Interpolation in Hilbert spaces of analytic functions, Thesis, University of Michigan, Ann Arbor, Mich., 1965.
 [10]
H.
S. Shapiro and A.
L. Shields, On some interpolation problems for analytic
functions, Amer. J. Math. 83 (1961), 513–532.
MR
0133446 (24 #A3280)
 [11]
E. C. Titchmarsh, The zeros of certain integral functions, Proc. London Math. Soc. (2) 25 (1926), 283302.
 [1]
 N. K. Bari, Biorthogonal systems and bases in Hilbert space, Moskov. Gos. Univ. Učen. Zap. 148 Mat. 4 (1951), 69107. (Russian) MR 14, 289. MR 0050171 (14:289b)
 [2]
 R. P. Boas, Jr., Expansions of analytic functions, Trans. Amer. Math. Soc. 48 (1940), 467487. MR 2, 80. MR 0002594 (2:80e)
 [3]
 , A general moment problem, Amer. J. Math. 63 (1941), 361370. MR 2, 281. MR 0003848 (2:281d)
 [4]
 , Entire functions. Academic Press, New York, 1954. MR 16, 914. MR 0068627 (16:914f)
 [5]
 R. J. Duffin and J. J. Eachus, Some notes on an expansion theorem of Paley and Wiener, Bull. Amer. Math. Soc. 48 (1942), 850855. MR 4, 97. MR 0007173 (4:97e)
 [6]
 R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341366. MR 13, 839. MR 0047179 (13:839a)
 [7]
 A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z. 41 (1936), 367379. MR 1545625
 [8]
 R. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., vol. 9, Amer. Math. Soc., Providence, R.I., 1934. MR 1451142 (98a:01023)
 [9]
 J. Rosenbaum, Interpolation in Hilbert spaces of analytic functions, Thesis, University of Michigan, Ann Arbor, Mich., 1965.
 [10]
 H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513532. MR 24 #A3280. MR 0133446 (24:A3280)
 [11]
 E. C. Titchmarsh, The zeros of certain integral functions, Proc. London Math. Soc. (2) 25 (1926), 283302.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403578236
PII:
S 00029947(1974)03578236
Keywords:
PaleyWiener space,
interpolating sequence,
complete interpolating sequence,
nonharmonic Fourier series expansion,
entire functions of exponential type
Article copyright:
© Copyright 1974
American Mathematical Society
