2019
Том 71
№ 8

Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means

Usenko E. G.

Abstract

We indicate criteria for the coincidence of the Knopp kernels K(f) K(A f), and K (R f) of bounded functions f(t); here, $$R_f \left( t \right) = \frac{1}{{P\left( x \right)}}\int\limits_{\left[ {0;\left. t \right)} \right.} {f\left( x \right)dP and A_f \left( t \right)} = \frac{1}{{\int_0^\infty {e^{{{ - x} \mathord{\left/ {\vphantom {{ - x} t}} \right. \kern-\nulldelimiterspace} t}} dP} }}\int\limits_0^\infty {f\left( x \right)} e^{{{ - x} \mathord{\left/ {\vphantom {{ - x} t}} \right. \kern-\nulldelimiterspace} t}} dP$$ . In Particular, we prove that K(f) = K(A f) ⇔ K(f) = K(R f).

English version (Springer): Ukrainian Mathematical Journal 50 (1998), no. 12, pp 1952-1955.

Citation Example: Usenko E. G. Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means // Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1712–1714.

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