A.Yu. Popov. On scientific contacts with Sergei Aleksandrovich Telyakovskii ... P. 164-176

The paper presents some memories of the joint research with the prominent specialist in the theory of functions of a real variable Sergei Aleksandrovich Telyakovskii.

Keywords: S.A. Telyakovskii

Received September 5, 2022

Revised October 18, 2022

Accepted October 24, 2022

Funding Agency: This work was carried out at Moscow State University and was supported by the Russian Science Foundation (project no. 22-11-00129).

Anton Yur’evich Popov, Dr. Phys.-Math. Sci., Lomonosov Moscow State University, Moscow Centre of Fundamental and Applied Mathematics, Moscow, 119991 Russia, station@list.ru


1.   Popov A.Yu., Telyakovskii S.A. On estimates for partial sums of Fourier series of functions of bounded variation. Russian Math. (Iz. VUZ), 2000, vol. 44, no. 1, pp. 50–54.

2.   Popov A.Yu., Telyakovskii S.A. Estimate for the integral of the absolute value of a sine series with monotone coefficients. Proc. Steklov Inst. Math., 2013, vol. 280, pp. 263–267. doi: 10.1134/S0081543813010197 

3.   Telyakovskii S.A. On partial sums of Fourier series of functions of bounded variation. Proc. Steklov Inst. Math., 1997, vol. 219, pp. 372–381.

4.   Belov A.S., Telyakovskii S.A. Refinement of the Dirichlet–Jordan and Young’s theorems on Fourier series of functions of bounded variation. Sb. Math., 2007, vol. 198, no. 6, pp. 777–791. doi: 10.1070/SM2007v198n06ABEH003860 

5.   Hardy G.H. Notes on some points in the integral calculus. LV: On the integration of Fourier series. Messenger Math., 1922, vol. 51, pp. 186–192.

6.   Young W.H. On the Fourier series of bounded functions. Proc. Lond. Math. Soc. Ser. 2, 1913, vol. 12, pp. 41–70. doi: 10.1112/plms/s2-12.1.41 

7.   Bari N.K. A Treatise on trigonometric series. Oxford; New York: Pergamon Press, 1964. Original Russian text published in Bari N.K. Trigonometricheskie ryady. Moscow: Fizmatgiz Publ., 1961, 936 p.

8.   Telyakovskij S.A. Certain properties of sine series with monotone coefficients. Anal. Math., 1992, vol. 18, no. 4, pp. 307–323. doi: 10.1007/BF02204778 (in Russian).

9.   Hartman P., Wintner A. On sine series with monotone coefficients. J. London Math. Soc., 1953, vol. 28, pp. 102–104. doi: 10.1112/JLMS/S1-28.1.102 

10.   Salem R. Détermination de l’ordre de grandeur à l’origine de certaines séries trigonométriques. C. R. Acad. Sci., Paris, 1928, vol. 186, pp. 1804–1806.

11.   Izumi S. Some trigonometrical series, xii. Proc. Japan Acad., 1955, vol. 31, no. 4, pp. 207–209. doi: 10.3792/pja/1195525743 

12.   Telyakovskii S.A. On the behavior of sine series near zero. Makedon. Akad. Nauk. Umet. Oddel. Mat.-Tehn. Nauk. Prilozi, 2000, vol. 21, no. 1-2, pp. 47–54 (2002).

13.   Aljančić S., Bojanić R., Tomić M.  Sur le comportement asymptotique au voisinage de zéro des séries trigonométriques de sinus à coefficients monotones. Publ. Inst. Math. (Beograd) (N.S.), 1956, vol. 10, pp. 101–120.

14.   Popov A.Yu. Estimates of the sums of sine series with monotone coefficients of certain classes. Math. Notes, 2003, vol. 74, no. 6, pp. 829–840. doi: 10.1023/B:MATN.0000009019.66625.fb 

15.   Solodov A.P. A sharp lower bound for the sum of a sine series with convex coefficients. Sb. Math., 2016, vol. 207, no. 12, pp. 1743–1777. doi: 10.1070/SM8633 

16.   Popov A.Yu., Solodov A.P. Exact lower estimate of the upper limit of the ratio of the sum of sine series with monotone coefficients to its majorant. Moscow Univ. Math. Bull., 2014, vol. 69, no. 4, pp. 169–173. doi: 10.3103/S0027132214040056 

17.   Popov A.Yu., Solodov A.P. Estimates with sharp constants of the sums of sine series with monotone coefficients of certain classes in terms of the Salem majorant. Math. Notes, 2018, vol. 104, no. 5, pp. 702–711. doi: 10.1134/S0001434618110111 

18.   Popov A.Yu. Estimates of the least positive root of the sum of a sine series with monotone coefficients. Math. Notes, 2014, vol. 96, no. 5, pp. 753–766. doi: 10.1134/S0001434614110145 

19.   Solodov A.P. Exact constants in Telyakovskii’s two-sided estimate of the sum of a sine series with convex sequence of coefficients. Math. Notes, 2020, vol. 107, no. 6, pp. 988–1001. doi: 10.1134/S0001434620050314 

20.   Solodov A.P. Sharp two-sided estimate for the sum of a sine series with convex slowly varying sequence of coefficients. Anal. Math., 2020, vol. 46, no. 3, pp. 579–603. doi: 10.1007/s10476-020-0047-5 

21.   Alferova E.D., Popov A.Yu. Two-sided estimates of the $L^{\infty}$-norm of the sum of a sine series with monotone coefficients ${b_k}$ via the $l^{\infty}$-norm of the sequence ${kb_k}$. Math. Notes, 2020, vol. 108, no. 4, pp. 471–476. doi: 10.1134/S0001434620090199 

22.   Alferova E.D., Popov A.Yu. On the positivity of average sums of sine series with monotone coefficients. Math. Notes, 2021, vol. 110, no. 3-4, pp. 623–627. doi: 10.1134/S0001434621090327 

23.   Popov A.Yu. Refinement of estimates of sums of sine series with monotone coefficients and cosine series with convex coefficients. Math. Notes, 2021, vol. 109, no. 5, pp. 808–818. doi: 10.1134/S0001434621050126 

24.   Popov A.Yu., Solodov A.P. The negative parts of the sums of sine series with quasimonotonic coefficients. Sb. Math., 2017, vol. 208, no. 6, pp. 878–901. doi: 10.1070/SM8764 

25.   Stechkin S.B. Trigonometric series with monotone type coefficients. Proc. Steklov Inst. Math. (Suppl.), 2001, vol. 7, suppl. 1, pp. S214–S224.

26.   Popov A.Yu., Solodov A.P. Optimal two-sided estimates on the interval [π∕2,π] of the sum of the sine series with convex coefficient sequence. Math. Notes, 2022, vol. 112, no. 2, pp. 328–331. doi: 10.1134/S0001434622070380 

27.   Jordan C. Sur la séries de Fourier. C.R.Acad.Sci., 1881, vol. 92, pp. 228–230.

28.   Stechkin S.B. The approximation of continuous functions by Fourier sums. Uspekhi Mat. Nauk, 1952, vol. 7, no. 4, pp. 139–141 (in Russian).

29.   Telyakovskii S.A. On the works of S.B. Stechkin on approximation of periodic functions by polynomials. Fundam. Prikl. Mat., 1997, vol. 3, no. 4, pp. 1059–1068 (in Russian).

30.   Zhuk V.V. Approksimatsiya periodicheskikh funktsii [Approximation of periodic functions]. Leningrad: Leningrad Univ. Publ., 1982, 368 p.

31.   Gavrilyuk V.T., Stechkin S.B. Approximation of continuous periodic functions by Fourier sums. Proc. Steklov Inst. Math., 1987, vol. 172, pp. 119–142.

32.   Shakirov I.A. About the optimal replacement of the Lebesque constant Fourier operator by a logarithmic function. Lobachevskii J. Math., 2018, vol. 39, no. 6, pp. 841–846. doi: 10.1134/S1995080218060185 

Cite this article as: A.Yu. Popov. On scientific contacts with Sergei Aleksandrovich Telyakovskii. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 4, pp. 164–176.