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Approximation of differentiable functions by functions of large smoothness

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Abstract

The order of the quantity\(\delta (L) = \mathop {\sup }\limits_{\chi _{_{_{_1 } } } } \mathop {\inf }\limits_{\chi _2 } \parallel \chi _1 - \chi _2 \parallel L_S [0,2 = ]\) as L → ∞ is studied for the classes of periodic functionsx 1εW n p (I), andx 2 εW m q (L). Necessary and sufficient conditions under which the inequality

$$\parallel x^{(n)} \parallel _{L_p } \leqslant C\parallel x\parallel _{L_q }^x \parallel x^{(m)} \parallel _{L_s }^\beta $$

with the constant independent of x holds for all periodic functions x(t) with\(\int_0^{2\pi } \chi (l)dl = 0\) andx (m) (t εL s [0, 2π] are found.

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Authors and Affiliations

  1. Moscow Energy Institute, USSR

    B. E. Klots

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  1. B. E. Klots
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Additional information

Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 21–32, January, 1977.

In conclusion the author thanks V. M. Tikhomirov for advice and guidance in the preparation of this article.

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Klots, B.E. Approximation of differentiable functions by functions of large smoothness. Mathematical Notes of the Academy of Sciences of the USSR 21, 12–19 (1977). https://doi.org/10.1007/BF02317028

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  • DOI: https://doi.org/10.1007/BF02317028

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