Interpolation in a classical Hilbert space of entire functions

Author:
Robert M. Young

Journal:
Trans. Amer. Math. Soc. **192** (1974), 97-114

MSC:
Primary 30A98; Secondary 30A80, 46E20

DOI:
https://doi.org/10.1090/S0002-9947-1974-0357823-6

MathSciNet review:
0357823

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let *H* denote the Paley-Wiener 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****[2]**R. P. Boas Jr.,*Expansions of analytic functions*, Trans. Amer. Math. Soc.**48**(1940), 467–487. MR**2594**, https://doi.org/10.1090/S0002-9947-1940-0002594-3**[3]**R. P. Boas Jr.,*A general moment problem*, Amer. J. Math.**63**(1941), 361–370. MR**3848**, https://doi.org/10.2307/2371530**[4]**Ralph Philip Boas Jr.,*Entire functions*, Academic Press Inc., New York, 1954. MR**0068627****[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**7173**, https://doi.org/10.1090/S0002-9904-1942-07797-4**[6]**R. J. Duffin and A. C. Schaeffer,*A class of nonharmonic Fourier series*, Trans. Amer. Math. Soc.**72**(1952), 341–366. MR**47179**, https://doi.org/10.1090/S0002-9947-1952-0047179-6**[7]**A. E. Ingham,*Some trigonometrical inequalities with applications to the theory of series*, Math. Z.**41**(1936), no. 1, 367–379. MR**1545625**, https://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****[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**133446**, https://doi.org/10.2307/2372892**[11]**E. C. Titchmarsh,*The zeros of certain integral functions*, Proc. London Math. Soc. (2)**25**(1926), 283-302.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30A98,
30A80,
46E20

Retrieve articles in all journals with MSC: 30A98, 30A80, 46E20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0357823-6

Keywords:
Paley-Wiener space,
interpolating sequence,
complete interpolating sequence,
nonharmonic Fourier series expansion,
entire functions of exponential type

Article copyright:
© Copyright 1974
American Mathematical Society