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On a Tauberian theorem of Keldysh. (Russian, English) Zbl 1080.41011

Zap. Nauchn. Semin. POMI 315, 63-89 (2004); translation in J. Math. Sci., New York 134, No. 4, 2272-2287 (2006).
M. V. Keldysh has introduced the new class of Tauberian theorems for operators \[ \Phi(x) = \int^\infty_0 K(x,t)d\varphi(t) \quad\text{ and }\quad \Psi(x) = \int^\infty_0 K(x,t)d\psi(t). \] He has found conditions when from \(\lim_{x \to \infty} \frac{\Phi(x)}{\Psi(x)}=1\) follows \(\lim_{x \to \infty} \frac{\varphi(x)}{\psi(x)}=1.\)
In the paper for the operators \[ \Phi(x) = \frac{1}{\pi}\int^\infty_0 \frac{\varphi(t)dt}{(1+\frac{t}{x})^{m+1}},\quad \Psi(x) = \frac{1}{\pi}\int^\infty_0 \frac{\psi(t)dt}{(1+\frac{t}{x})^{m+1}}, \] \(m>-1,\) are proved Tauberian theorems of Keldysh type with more common conditions.

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
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