Geodesic
Riemann's Localization Theorem. An Estimate for the Rate of Convergence
Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 178-181.

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An estimate for the Riemann localization principle for trigonometric series is established. At the same time, there is an error in treating this problem in the paper [1] being now corrected. 
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S. A. Telyakovskii. Riemann's Localization Theorem. An Estimate for the Rate of Convergence. Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 178-181. http://geodesic.mathdoc.fr/item/CMFD_2007_25_a13/

[1] Bari N. K., Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR

[2] Zigmund A., Trigonometricheskie ryady, T. 2, Mir, M., 1965 | MR

[3] Timan A. F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960

[4] Freud G., Zentr.-Bl. MATH., 57 (1956), 53

[5] Hille E., Klein G., “Riemann's localization theorem for Fourier series”, Duke Math. J., 21 (1954), 587–591 | DOI | MR | Zbl

[6] Izumi Shin-ichi, “Some trigonometrical series, XIV”, Proc. Japan Acad., 31 (1955), 324–326 | DOI | MR | Zbl