Geodesic
Quasiconvex uniform-convergence factors for Fourier series of functions with a~given modulus of continuity
Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 619-623.

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It is proved that a quasiconvex sequencelambda $\{\lambda_\nu\}$ of convergence factors transforms Fourier series of functions whose moduli of continuity do not exceed a given modulus of continuity $\omega(\delta)$ into uniformly convergent series if and only iflambda $\lambda_n\omega(1/n)\log n\to0$. The sufficiency of this condition is already known. 
@article{MZM_1970_8_5_a8,
     author = {S. A. Telyakovskii},
     title = {Quasiconvex uniform-convergence factors for {Fourier} series of functions with a~given modulus of continuity},
     journal = {Matemati\v{c}eskie zametki},
     pages = {619--623},
     publisher = {mathdoc},
     volume = {8},
     number = {5},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a8/}
}
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S. A. Telyakovskii. Quasiconvex uniform-convergence factors for Fourier series of functions with a~given modulus of continuity. Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 619-623. http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a8/