Geodesic
Integrability of trigonometric series. The estimation of the integral modulus of continuity
Sbornik. Mathematics, Tome 20 (1973) no. 4, pp. 557-573 
S. A. Telyakovskii. Integrability of trigonometric series. The estimation of the integral modulus of continuity. Sbornik. Mathematics, Tome 20 (1973) no. 4, pp. 557-573. http://geodesic.mathdoc.fr/item/SM_1973_20_4_a4/
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     author = {S. A. Telyakovskii},
     title = {Integrability of trigonometric series. {The} estimation of the integral modulus of continuity},
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     volume = {20},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1973_20_4_a4/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $a_m$ tend to zero and let the quantities \begin{align*} B_n&=\sum_{m=1}^n\biggl(\frac mn\biggr)^k|\Delta a_m|+\sum_{m=n+1}^\infty|\Delta a_m|+ \\ &\qquad+\sum_{m=2}^n\biggl(\frac mn\biggr)^k\biggl|\sum_{i=1}^{[m/2]}\frac{\Delta a_{m-i}-\Delta a_{m+i}}i\biggr|+\sum_{m=n+1}^\infty\biggl|\sum_{i=1}^{[m/2]}\frac{\Delta a_{m-i}-\Delta a_{m+i}}i\biggr|. \end{align*} be finite. We put $f(x)=\frac{a_0}2+\sum_{m=1}^\infty a_m\cos mx$ and $g(x)=\sum_{m=1}^\infty a_m\sin mx$. It is shown that the integral modulus of continuity of $k$th order for the function $f$ satisfies the estimate $\omega_k\bigl(f,\frac1n\bigr)_L=O(B_n)$, and that if the series $\sum\frac{|a_m|}m$, converges then $$ \omega_k\biggl(g,\frac1n\biggr)_L=\frac{2^k}\pi\sum_{m=n}^\infty\frac{|a_m|}m+O(B_n). $$ Bibliography: 10 titles.

[1] S. Aljančić, M. Tomić, “Sur le module de continuité intégral des séries de Fourier à coefficients convexes”, C. r. Acad. Sci. Paris, 259:9 (1964), 1609–1611 | MR | Zbl

[2] S. Aljančić, M. Tomić, “Über den Stetigkeitsmodul von Fourier-Reihen mit monotonen Koeffizienten”, Math. Z., 88:3 (1965), 274–284 | DOI | MR | Zbl

[3] S. Aljančić, “Sur le module des séries de Fourier particuliéres et sur le module des séries de Fourier transformées par des miltiplicateurs de types divers”, Bull. Acad. Serbe Sc. Arts, XL:6 (1967), 13–38

[4] S. Aljančić, M. Tomić, “Sur la borne inférieure du module de continuité de la fonction exprimée par les coefficients de Fourier”, Bull. Acad. Serbe Sc. Arts, XL:6 (1967), 39–51

[5] C. S. Rees, “A bound for the integral modulus of continuity”, J. Math. Analysis and Appl., 19:3 (1967), 469–474 | DOI | MR | Zbl

[6] M. Izumi, S. Izumi, “Modulus of continuity of functions defined by trigonometric series”, J. Math. Analysis and Appl., 24:3 (1968), 564–581 | DOI | MR | Zbl

[7] S. A. Telyakovskii, “Otsenka integralnogo modulya nepreryvnosti funktsii s kvazivypuklymi koeffitsientami Fure”, Sib. matem. zh., 11:5 (1970), 1140–1145 | Zbl

[8] S. A. Telyakovskii, “Usloviya integriruemosti trigonometricheskikh ryadov i ikh prilozhenie k izucheniyu lineinykh metodov summirovaniya ryadov Fure”, Izv. AN SSSR, seriya matem., 28:6 (1964), 1209–1236 | Zbl

[9] S. A. Telyakovskii, “Asimptoticheskaya otsenka integrala ot modulya funktsii, zadannoi ryadom iz sinusov”, Sib. matem. zh., 8:6 (1967), 1416–1422

[10] S. A. Telyakovskii, “Otsenka normy funktsii cherez ee koeffitsienty Fure, udobnaya v zadachakh teorii approksimatsii”, Trudy Matem. in-ta im. V. A. Steklova, CIX (1971), 65–97