# Exact Constants in Telyakovskii’s Two-Sided Estimate of the Sum of a Sine Series with Convex Sequence of Coefficients

## Abstract

It is known that the sum of the sine series $$g\left( {{\rm{b,}}\,x} \right) = \sum\nolimits_{k = 1}^\infty {{b_k}}$$bk sin kx whose coefficients constitute a convex sequence b is positive on the interval (0, π). To estimate its values in a neighborhood of zero, Telyakovskii used the piecewise continuous function

$$\sigma \left( {{\mathop{\rm b}\nolimits} ,\,x} \right) = {1 \over {m\left( x \right)}}\sum\limits_{k = 1}^{m\left( x \right) - 1} {{k^2}\left( {{b_k} - {b_{k + 1}}} \right),\,\,\,\,\,\,\,\,\,\,\,m\left( x \right) = \left[ {{{\rm{\pi }} \over x}} \right].}$$

He showed that the difference g(b, x) − (bm(x)/2)cot(x/2) in a neighborhood of zero admits a two-sided estimate in terms of the function a(b,x) with absolute constants. The exact values of these constants for the class of convex sequences b are obtained in this paper.

This is a preview of subscription content, log in to check access.

## References

1. 1.

S. A. Telyakovskiĭ, “On the behavior near the origin of the sine series with convex coefficients,” Publ. Inst. Math. (Beograd) (N. S.) 58(72), 43–50 (1995).

2. 2.

S. A. Telyakovskii, “On the behavior of the sine series near zero,” Makedon. Akad. Nauk. Umet. Oddel. Mat.-Tehn. Nauk. Prilozi 21, 47–54 (2002).

3. 3.

S. Aljančić, R. Bojanić, and M. Tomić, “Sur le comportement asymptotique au voisinage de zéro des séries trigonométriques de sinus à coefficients monotones,” Acad. Serbe Sci. Publ. Inst. Math. 10, 101–120 (1956).

4. 4.

A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1959, 1960; Mir, Moscow, 1965), Vols. 1, 2.

5. 5.

E. Seneta, Regularly Varying Functions (Springer-Verlag, Berlin-Heidelberg-New York, 1976; Nauka, Moscow, 1985).

6. 6.

S. A. Telyakovskii, “On the behavior of sine series with convex coefficients near the origin,” Dokl. Dokl. Akad. Nauk SSSR 56(3), 913–914 (1997) [Soviet Math. Dokl. 357 (4), 462–463 (1997)].

7. 7.

A. Yu. Popov, “Estimates of the sums of sine series with monotone coefficients of certain classes,” Mat. Zametki 74(6), 877–888 (2003) [Math. Notes 74 (6), 829–840 (2003)].

8. 8.

A. P. Solodov, “A sharp lower bound for the sum of a sine series with convex coefficients,” Mat. Sb. 207(12), 124–158 (2016) [Sb. Math. 207 (12), 1743–1777 (2016)].

## Funding

This work was supported by the Russian Foundation for Basic Research under grant 20-01-00584.

## Author information

Authors

### Corresponding author

Correspondence to A. P. Solodov.