Geodesic
Approximation by sums of shifts of a~single function on the circle
Izvestiya. Mathematics , Tome 81 (2017) no. 6, pp. 1080-1094

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We study approximation properties of the sums $\sum_{k=1}^nf(t-a_k)$ of shifts of a single function $f$ in real spaces $L_p(\mathbb{T})$ and $C(\mathbb{T})$ on the circle $\mathbb{T}=[0,2\pi)$, and also in complex spaces of functions analytic in the unit disc. We obtain sufficient conditions in terms of the trigonometric Fourier coefficients of $f$ for these sums to be dense in the corresponding subspaces of functions with zero mean. We investigate the accuracy of these conditions. We also suggest a simple algorithm for the approximation by sums of plus or minus shifts of one particular function in $L_2(\mathbb{T})$ and obtain bounds for the rate of approximation.
Keywords: approximation, sums of shifts, semigroup.
Mots-clés : Fourier coefficients 
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     author = {P. A. Borodin},
     title = {Approximation by sums of shifts of a~single function on the circle},
     journal = {Izvestiya. Mathematics },
     pages = {1080--1094},
     publisher = {mathdoc},
     volume = {81},
     number = {6},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2017_81_6_a2/}
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P. A. Borodin. Approximation by sums of shifts of a~single function on the circle. Izvestiya. Mathematics , Tome 81 (2017) no. 6, pp. 1080-1094. http://geodesic.mathdoc.fr/item/IM2_2017_81_6_a2/