Geodesic
Density of the sums of shifts of a single function in the $L_2^0$ space on a compact Abelian group
Sbornik. Mathematics, Tome 215 (2024) no. 6, pp. 743-754 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $G$ be a nontrivial compact Abelian group. The following result is proved: a real-valued function on $G$ such that the sums of shifts of it are dense in the $L_{2}$-norm in the corresponding real space of mean zero functions exists if and only if the group $G$ is connected and has an infinite countable character group. Bibliography: 13 titles.
Keywords: density, sums of shifts, compact groups, space $L_{2}$. 
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N. A. Dyuzhina. Density of the sums of shifts of a single function in the $L_2^0$ space on a compact Abelian group. Sbornik. Mathematics, Tome 215 (2024) no. 6, pp. 743-754. http://geodesic.mathdoc.fr/item/SM_2024_215_6_a1/

[1] P. A. Borodin, “Approximation by sums of shifts of a single function on the circle”, Izv. Math., 81:6 (2017), 1080–1094 | DOI | MR | Zbl

[2] P. A. Borodin and S. V. Konyagin, “Convergence to zero of exponential sums with positive integer coefficients and approximation by sums of shifts of a single function on the line”, Anal. Math., 44:2 (2018), 163–183 | DOI | MR | Zbl

[3] P. A. Borodin, “Density of sums of shifts of a single vector in sequence spaces”, Proc. Steklov Inst. Math., 303 (2018), 31–35 | DOI | MR | Zbl

[4] N. A. Dyuzhina, “Multidimensional analogs of theorems about the density of sums of shifts of a single function”, Math. Notes, 113:5 (2023), 731–735 | DOI | MR | Zbl

[5] K. Shklyaev, “Approximation by sums of shifts and dilations of a single function and neural networks”, J. Approx. Theory, 291 (2023), 105915, 17 pp. | DOI | MR | Zbl

[6] P. A. Borodin and K. S. Shklyaev, “Density of quantized approximations”, Russian Math. Surveys, 78:5 (2023), 797–851 | DOI | MR

[7] E. Hewitt and K. A. Ross, Abstract harmonic analysis, v. 1, Grundlehren Math. Wiss., 115, Springer-Verlag; Academic Press, Inc., Publishers, Berlin–Göttingen–Heidelberg, 1963, viii+519 pp. | DOI | MR | Zbl

[8] W. Rudin, Fourier analysis on groups, Intersci. Tracts Pure Appl. Math., 12, Interscience Publishers (a division of John Wiley Sons), New York–London, 1962, ix+285 pp. | DOI | MR | Zbl

[9] P. R. Halmos, Measure theory, D. Van Nostrand Co., Inc., New York, 1950, xi+304 pp. | DOI | MR | Zbl

[10] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli and A. I. Rubinshtein, Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, Èlm, Baku, 1981, 180 pp. (Russian) | MR | Zbl

[11] J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Math. and Math. Phys., 45, Cambridge Univ. Press, New York, 1957, x+166 pp. | MR | Zbl

[12] P. A. Borodin, “Density of a semigroup in a Banach space”, Izv. Math., 78:6 (2014), 1079–1104 | DOI | MR | Zbl

[13] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, v. II, Ergeb. Math. Grenzgeb., 97, Springer-Verlag, Berlin–New York, 1979, x+243 pp. | MR | Zbl