Completeness of discrete translates in the Hardy space 𝐻¹(ℝ)

Dharra, Bhawna; Sivananthan, S.

doi:10.1090/proc/16070

Completeness of discrete translates in the Hardy space $H^1(\mathbb {R})$
HTML articles powered by AMS MathViewer

by Bhawna Dharra and S. Sivananthan PDF

Proc. Amer. Math. Soc. 150 (2022), 5281-5291 Request permission

Abstract:

We provide a characterization of discrete sets $\Lambda \subset \mathbb {R}$ that admit a function whose $\Lambda$-translates are complete in the Hardy space $H^1(\mathbb {R})$. In particular, we show that such a set cannot be uniformly discrete. We then give a uniformly discrete $\Lambda \subset \mathbb {R}$ which admits a pair of functions such that their $\Lambda$-translates are complete in $H^1(\mathbb {R})$.

References

A. Atzmon and A. Olevskiĭ, Completeness of integer translates in function spaces on $\textbf {R}$, J. Approx. Theory 87 (1996), no. 3, 291–327. MR 1420335, DOI 10.1006/jath.1996.0106
Arne Beurling, On a closure problem, Ark. Mat. 1 (1951), 301–303. MR 43248, DOI 10.1007/BF02591366
A. Beurling and P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291–309. MR 147848, DOI 10.1007/BF02545792
Arne Beurling and Paul Malliavin, On the closure of characters and the zeros of entire functions, Acta Math. 118 (1967), 79–93. MR 209758, DOI 10.1007/BF02392477
N. Blank, Generating sets for Beurling algebras, J. Approx. Theory 140 (2006), no. 1, 61–70. MR 2226677, DOI 10.1016/j.jat.2005.12.001
Joaquim Bruna, Alexander Olevskii, and Alexander Ulanovskii, Completeness in $L^1(\Bbb R)$ of discrete translates, Rev. Mat. Iberoam. 22 (2006), no. 1, 1–16. MR 2267311, DOI 10.4171/RMI/447
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
H. G. Feichtinger and A. Gumber, Completeness of sets of shifts in Invariant Banach Spaces of Tempered Distributions via Tauberian Theorems, arXiv:2012.11127, 2020.
R. L. Johnson and C. R. Warner, The convolution algebra $H^1(R)$, J. Funct. Spaces Appl. 8 (2010), no. 2, 167–179. MR 2667874, DOI 10.1155/2010/524036
Nir Lev and Alexander Olevskii, Wiener’s ‘closure of translates’ problem and Piatetski-Shapiro’s uniqueness phenomenon, Ann. of Math. (2) 174 (2011), no. 1, 519–541. MR 2811607, DOI 10.4007/annals.2011.174.1.15
Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
Nikolai Nikolski, Remarks concerning completeness of translates in function spaces, J. Approx. Theory 98 (1999), no. 2, 303–315. MR 1692243, DOI 10.1006/jath.1999.3365
Alexander M. Olevskii and Alexander Ulanovskii, Functions with disconnected spectrum, University Lecture Series, vol. 65, American Mathematical Society, Providence, RI, 2016. Sampling, interpolation, translates. MR 3468930, DOI 10.1090/ulect/065
Alexander Olevskii and Alexander Ulanovskii, Discrete translates in $L^p(\Bbb R)$, Bull. Lond. Math. Soc. 50 (2018), no. 4, 561–568. MR 3870942, DOI 10.1112/blms.12159
A. Olevskii and A. Ulanovskii, Discrete translates in function spaces, Anal. Math. 44 (2018), no. 2, 251–261. MR 3814066, DOI 10.1007/s10476-018-0209-x
Shan Zhen Lu, Four lectures on real $H^p$ spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1342077, DOI 10.1142/9789812831194
Elias M. Stein and Guido Weiss, On the theory of harmonic functions of several variables. I. The theory of $H^{p}$-spaces, Acta Math. 103 (1960), 25–62. MR 121579, DOI 10.1007/BF02546524
Akihito Uchiyama, Hardy spaces on the Euclidean space, Springer Monographs in Mathematics, Springer-Verlag, Tokyo, 2001. With a foreword by Nobuhiko Fujii, Akihiko Miyachi and Kôzô Yabuta and a personal recollection of Uchiyama by Peter W. Jones. MR 1845883, DOI 10.1007/978-4-431-67905-9
Norbert Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1–100. MR 1503035, DOI 10.2307/1968102
A. Zygmund, On the boundary values of functions of several complex variables. I, Fund. Math. 36 (1949), 207–235. MR 35832, DOI 10.4064/fm-36-1-207-235
A. Zygmund, Trigonometric Series, Cambridge University Press, 1959.