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Poisson type generators for \(L^{1}(\mathbb R)\). (English) Zbl 1185.42035

The authors consider sequences of the form \( \{\phi(t-\lambda), \lambda \in \Lambda\}, \) where \(\Lambda \in \mathbb{R}\) is a discrete set and \(\widehat{\phi}(\xi)\) behaves like \(\exp(-2 \pi |\xi|)\), and in their Theorem 1.2 find a necessary and sufficient condition on the set \(\Lambda\) for such a sequence to be dense in \(L(\mathbb{R})\). This result is a particular case of Theorem 2 of the reviewer’s article [J. Math. Anal. Appl. 82, 361–369 (1981; Zbl 0499.30009)]. However, a generalization of their result (Theorem 3.1), stated with only a sketch of proof, is not included in the reviewer’s theorem.
For other results on the closure of translates see the references cited in the reviewer’s article.

MSC:

42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
41A30 Approximation by other special function classes
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
30B60 Completeness problems, closure of a system of functions of one complex variable

Citations:

Zbl 0499.30009
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References:

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[3] Bruna, J., Olevskii, A., Ulanovskii, A.: Completeness in L 1(\(\mathbb{R}\)) of discrete translates. Rev. Mat. Iberoam. 22(1), 1–16 (2006) · Zbl 1104.42019
[4] Korenblum, B.: An extension of the Nevanlinna theory. Acta Math. 135(3–4), 187–219 (1975) · Zbl 0323.30030 · doi:10.1007/BF02392019
[5] Olevskii, A.: Completeness in L 2(\(\mathbb{R}\)) of almost integer translates. C. R. Acad. Sci. Paris Sér. I Math. 324(9), 987–991 (1997) · Zbl 0897.46016
[6] Olevvskii, A., Ulanovskii, A.: Almost integer translates. Do nice generators exist?. J. Fourier Anal. Appl. 10(1), 93–104 (2004) · Zbl 1071.42020 · doi:10.1007/s00041-004-8006-2
[7] Zalik, R.A.: On approximation by shifts and a theorem of Wiener. Trans. Am. Math. Soc. 243, 299–308 (1978) · Zbl 0403.41008 · doi:10.1090/S0002-9947-1978-0493077-1
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