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Kolmogorov-type inequalities and the best formulas for numerical differentiation

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Abstract

For a certain class of complex-valued functionsf(x), −∞ <x<∞, is found the best approximation

$$u_N = \mathop {\inf }\limits_{\parallel A\parallel \leqslant N_\parallel f^{(n)} \parallel _{L_2 \leqslant } 1} \parallel f^{(k)} - A(f)\parallel C$$

of a differential operator by linear operators A with the norm ∥A∥ CL2 ≤N,N,>0. Using the value uN, the smallest constant Q in the inequality

$$\parallel f^{(k)} \parallel _Q \leqslant Q\parallel f\parallel _{L_2 }^\alpha \parallel f^{(n)} \parallel _{L_2 }^\beta $$

is found.

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Literature cited

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Authors and Affiliations

  1. Sverdlovsk Branch of the V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR, USSR

    L. V. Taikov

Authors
  1. L. V. Taikov
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Additional information

Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 233–238, August, 1968.

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Taikov, L.V. Kolmogorov-type inequalities and the best formulas for numerical differentiation. Mathematical Notes of the Academy of Sciences of the USSR 4, 631–634 (1968). https://doi.org/10.1007/BF01094964

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  • DOI: https://doi.org/10.1007/BF01094964

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