The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines

E.V. Asadova (Oles Honchar Dnipro National University)
V.A. Kofanov (Oles Honchar Dnipro National University),


For given $$$n, r \in \mathbb{N}$$$; $$$p, A > 0$$$ and any fixed interval $$$[a,b] \subset \mathbb{R}$$$ we solve the extremal problem $$$\int\limits_a^b |x(t)|^q dt \rightarrow \sup$$$, $$$q \geqslant p$$$, over sets of trigonometric polynomials $$$T$$$ of order $$$\leqslant n$$$ and $$$2\pi$$$-periodic splines $$$s$$$ of order $$$r$$$ and minimal defect with knots at the points $$$k\pi / n$$$, $$$k \in \mathbb{Z}$$$, such that $$$\| T \| _{p, \delta} \leqslant A \| \sin n (\cdot) \|_{p, \delta} \leqslant A \| \varphi_{n,r} \|_{p, \delta}$$$, $$$\delta \in (0, \pi / n]$$$, where $$$\| x \|_{p, \delta} := \sup \{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a < \delta\}$$$ and $$$\varphi_{n, r}$$$ is the $$$(2\pi / n)$$$-periodic spline of Euler of order $$$r$$$. In particular, we solve the same problem for the intermediate derivatives $$$x^{(k)}$$$, $$$k = 1, ..., r-1$$$, with $$$q \geqslant 1$$$.


Bojanov-Naidenov problem; polynomial; spline; rearrangement; comparison theorem

Full Text:



Korneichuk N.P., Babenko V.F., Kofanov V.A., Pichugov S.A. Inequalities for derivatives and their applications, Nauk. dumka, Kyiv, 2003; 590 p. (in Russian)

Babenko V.F. "Researches of Dnipropetrovsk mathematicians on inequalities for derivatives of periodic functions and their applications", Ukrainian Math. J., 2000, 52(1): pp. 5-29. (in Russian)

Kwong M.K., Zettl A. "Norm inequalities for derivatives and differences", Lecture notes in mathematics, Berlin: Springer-Verlag, 1992; 1536: 150 p.

Bojanov B., Naidenov N. "An extension of the Landau-Kolmogorov inequality. Solution of a problem of Erdös", J. d'Analyse Mathematique, 1999; 78: pp. 263-280.

Erdös P. "Open problems", Open Problems in Approximation Theory (B. Bojanov, ed.), SCT Publishing, Singapur, 1994; pp. 238-242.

Pinkus A., Shisha O. "Variations on the Chebyshev and $$$L^q$$$-Theories of Best Approximation", Journal of Approx. Theory, 1982; 35(2): pp. 148-168.

Kofanov V.A. "Some extremum problems in different metrics for differentiable functions on the real domain", Ukrainian Math. J., 2009; 61(6): pp. 765-776. (in Russian)

Kofanov V.A. "Some extremal problems in various metrics and sharp inequalities of Nagy-Kolmogorov type", East. J. Approx., 2010; 16(4): p. 313-334.

Kofanov V.A. "Exact upper bounds of norms of functions and their derivatives on the classes of functions with given comparison function", Ukrainian Math. J., 2011; 63(7): pp. 969-984. (in Russian)

Kofanov V.A. "Bojanov-Naidenov problem for differentiable functions on the axis and inequalities of various metrics", Ukrainian Math. J., 2019; 71(6): pp. 786-800.

Kofanov V.A. "Inequalities of various metrics for differentiable periodic functions", Ukrainian Math. J., 2015; 67(2): pp. 202-212. (in Russian)

Kolmogorov A.N. "On inequalities between upper bounds of consecutive derivatives of the function on infinite interval", Izbr. tr. Matematika, mekhanika, Nauka, Moscow, 1985; pp. 252-263. (in Russian)

Korneichuk N.P., Babenko V.F., Ligun A.A. Extremum properties of polynomials and splines, Naukova dumka, Kyiv, 1992. (in Russian)




  • There are currently no refbacks.

Copyright (c) 2019 E.V. Asadova, V.A. Kofanov

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Registered in

ISSN (Online): 2664-5009
ISSN (Print): 2664-4991