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

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).