2019
Том 71
№ 8

# Usenko E. G.

Articles: 2
Article (Ukrainian)

### Coincidence criteria for the kernel of a function and the kernel of its integral almost positive means

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1267–1275

We establish necessary and sufficient conditions for a point A of the Knopp kernelK(f) of a functionf to belong to the kernelK(M) of a functionM(t):=∫ S fdμ t , where the so-called almost positive measures μ t determine a regular method of summation. In particular, this gives coincidence criteria for the kernelsK(f) andK(M).

Brief Communications (Ukrainian)

### 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

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).