Vitalii V. Arestov     (Ural Federal University, Ekaterinburg, Russian Federation)


We give a characterization of elements of a subspace of a complex Banach space with the property that the norm of a bounded linear functional on the subspace is attained at those elements. In particular, we discuss properties of polynomials that are extremal in sharp pointwise Nikol'skii inequalities for algebraic polynomials in a weighted \(L_q\)-space on a finite or infinite interval.


Complex Banach space, Bounded linear functional on a subspace, Algebraic polynomial, Pointwise Nikol'skii inequality

Full Text:



Arestov V.V., Deikalova M.V. Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere // Proc. Steklov Inst. Math., 2014. Vol. 284. Suppl. 1. P. S9–S23. DOI: 10.1134/S0081543814020023

Arestov V., Deikalova M. Nikol'skii inequality between the uniform norm and \(L_q\)-norm with ultraspherical weight of algebraic polynomials on an interval // Comput. Methods Funct. Theory, 2015. Vol. 15, no. 4. P. 689–708. DOI: 10.1007/s40315-015-0134-y

Arestov V., Deikalova M. Nikol'skii inequality between the uniform norm and \(L_q\)-norm with Jacobi weight of algebraic polynomials on an interval // Analysis Math., 2016. Vol. 42, no. 2. P. 91–120. DOI: 10.1007/s10476-016-0201-2

Arestov V., Deikalova M., Horváth Á. On Nikol'skii type inequality between the uniform norm and the integral \(q\)-norm with Laguerre weight of algebraic polynomials on the half-line // J. Approx. Theory, 2017. Vol. 222. P. 40–54. DOI: 10.1016/j.jat.2017.05.005

Babenko V.F., Korneichuk N.P., Ligun A.A. Extremal properties of polynomials and splines. New York: Nova Science, 1996.

Day M.M. Normed linear space. Berlin; Göttingen; Heidelberg: Springer, 1958.

Diestel J. Geometry of Banach spaces: selected topics. Berlin: Springer, 1975.

Dunford N., Schwartz J. Linear operators: general theory. New York: Interscience, 1958.

Gol'shtein E.G. Duality theory in mathematical programming and its applications. Moscow: Nauka, 1971. 351 p. [in Russian].

Handbook of the Geometry of Banach Spaces. Ed. W.B. Johnson and J. Lindenstrauss. Elsevier, 2001. Vol. 1.

James R. Characterizations of reexivity // Studia Mathematica, 1964. Vol. 23, iss. 3. P. 205–216.

Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. Moscow: Fizmatlit, 2004.

Korneichuk N.P. Extremal problems of approximation theory. Moscow: Nauka, 1976. [in Russian].

Milovanović G.V., Mitrinović D.S., Rassias Th.M. Topics in polynomials: extremal problems, inequalities, zeros. Singapore: World Scientific, 1994. 821 p.

Nikol'skii S.M. Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables // Trudy Mat. Inst. Steklova, 1951. Vol. 38. P. 244–278 [in Russian].

Simonov I.E., Glazyrina P.Yu. Sharp Markov–Nikol'skii inequality with respect to the uniform norm and the integral norm with Chebyshev weight // J. Approx. Theory, 2015. Vol. 192. P. 69–81. DOI: 10.1016/j.jat.2014.10.009

Singer I. Best approximation in normed linear spaces by elements of linear subspaces. Berlin: Springer, 1970.

Szegő G., Zygmund A. On certain mean values of polynomials // J. Anal. Math., 1953. Vol. 3, no. 1. P. 225–244.

DOI: http://dx.doi.org/10.15826/umj.2017.2.004

Article Metrics

Metrics Loading ...


  • There are currently no refbacks.