Geodesic
Uniformly convergent Fourier series and multiplication of functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 186-192

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Let $U(\mathbb T)$ be the space of all continuous functions on the circle $\mathbb T$ whose Fourier series converges uniformly. Salem's well-known example shows that a product of two functions in $U(\mathbb T)$ does not always belong to $U(\mathbb T)$ even if one of the factors belongs to the Wiener algebra $A(\mathbb T)$. In this paper we consider pointwise multipliers of the space $U(\mathbb T)$, i.e., the functions $m$ such that $mf\in U(\mathbb T)$ whenever $f\in U(\mathbb T)$. We present certain sufficient conditions for a function to be a multiplier and also obtain some Salem-type results.
Keywords: uniformly convergent Fourier series, function spaces, multiplication operators. 
@article{TM_2018_303_a13,
     author = {V. V. Lebedev},
     title = {Uniformly convergent {Fourier} series and multiplication of functions},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {186--192},
     publisher = {mathdoc},
     volume = {303},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_303_a13/}
}
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%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
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V. V. Lebedev. Uniformly convergent Fourier series and multiplication of functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 186-192. http://geodesic.mathdoc.fr/item/TM_2018_303_a13/