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Uniform regularization of the problem of calculating the values of an operator

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Abstract

Let X and Y be linear normed spaces, W a set in X, A an operator from W into Y, and\(\mathfrak{W}\) the set

of all operators or the set ℒ of linear operators from X into Y. With δ>0 we put

$$v\left( {\delta ,\mathfrak{M}} \right) = \mathop {\inf }\limits_{T \in \mathfrak{M}} \mathop {\sup }\limits_{x \in W} \mathop {\sup }\limits_{\left\| {\eta - x} \right\|_X \leqslant \delta } \left\| {Ax - T\eta } \right\|_Y $$

.

We discuss the connection of\(v\left( {\delta , \mathfrak{M}} \right)\) with the Stechkin problem on best approximation of the operator A in W by linear bounded operators. Estimates are obtained for\(v\left( {\delta , \mathfrak{M}} \right)\) e.g., we write the inequality

, where H(Y) is Jung's constant of the space Y, and Ω(t) is the modulus of continuity of A in W.

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, UNTs, Academy of Sciences, USSR

    V. V. Arestov

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Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 231–244, August, 1977.

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Arestov, V.V. Uniform regularization of the problem of calculating the values of an operator. Mathematical Notes of the Academy of Sciences of the USSR 22, 618–626 (1977). https://doi.org/10.1007/BF01780971

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