A note on a trigonometric moment problem

Author:
Robert M. Young

Journal:
Proc. Amer. Math. Soc. **49** (1975), 411-415

MSC:
Primary 42A80

MathSciNet review:
0367548

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Abstract: A sequence is said to be an *interpolating sequence* for if the system of equations

*complete interpolating sequence*. It is shown that if the imaginary part of is uniformly bounded and if , then is a complete interpolating sequence and is a Schauder basis for . It is also shown that this result is sharp in the sense that the condition is not sufficient to guarantee that is an interpolating sequence.

**[1]**Ralph Philip Boas Jr.,*Entire functions*, Academic Press Inc., New York, 1954. MR**0068627****[2]**R. J. Duffin and A. C. Schaeffer,*A class of nonharmonic Fourier series*, Trans. Amer. Math. Soc.**72**(1952), 341–366. MR**0047179**, 10.1090/S0002-9947-1952-0047179-6**[3]**A. E. Ingham,*Some trigonometrical inequalities with applications to the theory of series*, Math. Z.**41**(1936), no. 1, 367–379. MR**1545625**, 10.1007/BF01180426**[4]**M. Ĭ. Kadec′,*The exact value of the Paley-Wiener constant*, Dokl. Akad. Nauk SSSR**155**(1964), 1253–1254 (Russian). MR**0162088****[5]**Norman Levinson,*Gap and Density Theorems*, American Mathematical Society Colloquium Publications, v. 26, American Mathematical Society, New York, 1940. MR**0003208****[6]**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****[7]**Robert M. Young,*Interpolation in a classical Hilbert space of entire functions*, Trans. Amer. Math. Soc.**192**(1974), 97–114. MR**0357823**, 10.1090/S0002-9947-1974-0357823-6**[8]**Robert M. Young,*Inequalities for a perturbation theorem of Paley and Wiener*, Proc. Amer. Math. Soc.**43**(1974), 320–322. MR**0340948**, 10.1090/S0002-9939-1974-0340948-4

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0367548-5

Keywords:
Interpolating sequence,
frames,
Paley-Wiener space

Article copyright:
© Copyright 1975
American Mathematical Society