# Best Approximation of a Differentiation Operator on the Set of Smooth Functions with Exactly or Approximately Given Fourier Transform

• Vitalii V. Arestov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)

## Abstract

Let $$Y^n,$$ , be the set of continuous bounded functions on the numerical axis with the following two properties: (1) the Fourier transform of a function is a function of bounded variation on the axis (in particular, a summable function); (2) a function is $$n-1$$ times continuously differentiable, its derivative of order $$n-1$$ is locally absolutely continuous, and the nth order derivative is bounded, more exactly, belongs to the space $$L_\infty$$. In the space $$Y^n,$$ consider the class $$\mathcal {Q}^n$$ of functions, for which the $$L_\infty$$-norm of the nth order derivative is bounded by a constant, for example, by 1. The following two approximation problems are discussed: the best approximation of the differentiation operator $$D^k$$ of order k, , by bounded operators on the class $$\mathcal {Q}^n$$ and the optimal calculation of the differentiation operator $$D^k$$ on functions from the class $$\mathcal {Q}^n$$ under the assumption that their Fourier transform is given with a known error in the space of functions of bounded variation, in particular, in the space L of functions summable on the axis. In interrelation with these two problems, we discuss the exact Kolmogorov type inequality in the space $$Y^n$$ between the uniform norm of the kth order derivative of a function, the variation of the Fourier transform of the function, and the $$L_\infty$$-norm of its derivative of order n.

## Keywords

Functions with exactly or approximately given Fourier transform Kolmogorov inequality Optimal differentiation method

## References

1. 1.
Arestov, V.V.: On the best approximation of differentiation operators. Math. Notes 1(2), 100–103 (1967).
2. 2.
Arestov, V.V.: On the best approximation of differentiation operators in the uniform norm. Cand. Diss., Sverdlovsk (1969). (in Russian)Google Scholar
3. 3.
Arestov, V.V.: Approximation of unbounded operators by bounded operators and related extremal problems. Russ. Math. Surv. 51(6), 1093–1126 (1996).
4. 4.
Arestov, V.V.: Best uniform approximation of the differentiation operator by operators bounded in the space $$L_2$$. Trudy Inst. Mat. Mekhaniki URO RAN 24(4), 34–56 (2018). . (in Russian)
5. 5.
Babenko, V.F., Korneichuk, N.P., Kofanov, V.A., Pichugov, S.A.: Neravenstva dlya proizvodnykh i ikh prilozheniya [Inequalities for Derivatives and Their Applications]. Naukova Dumka Publisher, Kiev (2003). (in Russian)Google Scholar
6. 6.
Bosse, Yu.G. (Shilov, G.E.): On inequalities between derivatives. In: Collection of Works of Student Scientific Societies of Moscow State University, vol. 1, pp. 68–72 (1937). (in Russian)Google Scholar
7. 7.
Buslaev, A.P.: Approximation of a differentiation operator. Math. Notes 29(5), 372–378 (1981).
8. 8.
Domar, Y.: An extremal problem related to Kolmogoroff’s inequality for bounded functions. Arkiv för Mat. 7(5), 433–441 (1968).
9. 9.
Gabushin, V.N.: Best approximations of functionals on certain sets. Math. Notes 8(5), 780–785 (1970).
10. 10.
Hadamard, J.: Sur le module maximum d’une fonction et de ses dérivées. Soc. Math. France, Comptes rendus des Séances 41, 68–72 (1914)Google Scholar
11. 11.
Hardy, G.H., Littlewood, J.E.: Contribution to the arithmetic theory of series. Proc. Lond. Math. Soc. 11(2), 411–478 (1912)
12. 12.
Ivanov, V.K., Vasin, V.V., Tanana, V.P.: Theory of Linear Ill-Posed Problems and Its Applications. VSP, Utrecht (2002)
13. 13.
Kolmogorov, A.N.: On inequalities between upper bounds of consecutive derivatives of an arbitrary function defined on an infinite interval. In: Selected works. Mathematics and Mechanics, pp. 252–263. Nauka Publisher, Moscow (1985). (Moskov. Gos. Univ., Uchenye Zap. (Mat. 3) 30, 3–16 (1939)). (in Russian)Google Scholar
14. 14.
Korneichuk, N.P.: Extremal Problems of Approximation Theory. Nauka, Moscow (1976). (in Russian)Google Scholar
15. 15.
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Elsevier/Academic Press, London, Oxford (2007)
16. 16.
Stechkin, S.B.: Inequalities between norms of derivatives of arbitrary functions. Acta Sci. Math. 26(3–4), 225–230 (1965). (in Russian)
17. 17.
Stechkin, S.B.: Best approximation of linear operators. Math. Notes 1(2), 91–99 (1967).
18. 18.
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)

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## Authors and Affiliations

• Vitalii V. Arestov
• 1
• 2
1. 1.Ural Federal UniversityEkaterinburgRussia
2. 2.Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of SciencesEkaterinburgRussia