Vitalii V. Arestov     (Ural Federal University, Institute of Mathematics and Computer Science, Deparment of Mathematical Analysis and Function Theory, Ekaterinburg, Russian Federation)


In this paper we give a solution of the problem of the best approximation in the uniform norm of the differentiation operator of order k by bounded linear operators in the class of functions with the property that the Fourier transforms of their derivatives of order n (0 < k <n) are finite measures. We also determine the exact value of the best constant in the corresponding inequality for derivatives.


Differentiation operator; Stechkin's problem; Kolmogorov inequality

Full Text:



Arestov V.V. On sharp inequalities between the norms of functions and their derivatives // Acta Sci. Math. 1972. Vol. 33. P. 243–267 [in Russian].

Arestov V.V. Approximation of operators that are invariant under a shift // Proc. Steklov Inst. Math. 1977. Vol. 138. P. 45–74.

Arestov V.V. Approximation of operators of convolution type by linear bounded operators // Proc. Steklov Inst. Math. 1981. Vol. 145. P. 1–118.

Buslaev A.P. Approximation of a differentiation operator // Math. Notes. 1981. Vol. 29. P. 372–378.

Gabushin V.N. On the best approximation of the differentiation operator on the half-line // Math. Notes. 1969. Vol. 6. P. 804–810.

Gabushin V.N. Best approximation of functionals on certain sets // Math. Motes. 1970. Vol. 8. P. 780–785.

Kolmogorov A.N. On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval // Transl. Amer. Math. Soc. 1962. Vol. 2, no. 1. P. 233–243; translated from Uchen. Zap. Moskov. Univ. Mat. 1939. Vol. 30. P. 3–16.

Stechkin S.B. Best approximation of linear operators // Math. Notes. 1967. Vol. 1. P. 91–99.

Subbotin Yu.N., Taikov L.V. Best approximation of a differentiation operator in L_2-space // Math. Notes. 1968. Vol. 3. P. 100–105.

Taikov L.V. Kolmogorov-type inequalities and best formulas for numerical differentiation // Math. Notes. 1968. Vol. 4. P. 631–634.

Tikhomirov V.M. Some problems in approximation theory. Moscow: Izd. Mosk. Univ., 1976. 304 p. [in Russian].

DOI: http://dx.doi.org/10.15826/umj.2015.1.002

Article Metrics

Metrics Loading ...


  • There are currently no refbacks.