A MERCERIAN THEOREM FOR SLOWLY VARYING SEQUENCES

N. Tanovi\'c-Miller

Abstract: The purpose of this note is to investigate a Mercerian problem for triangular matrix transformations of slowly varying sequences. A statement of this type for the nonnegative arithmetical means $M_p$, was recently proved by S. Aljanci\'c [1], using the evaluation of the inverse of the associated Mercerian transformation. In this note a corresponding result is proved for nonnegative triangular matrix transformations satisfying a certain condition, which can be applied to the arithmetical means $M_p$ $p_n \geq 0$, the Cesàro transformation $C_\alpha$ of order $\alpha$, $0<\alpha\leq1$, other Nörlund transformations $N_p$, $p_n>0$ and $(p_{n+1}/p_n)$ nondecreasing, as well as to some other standard methods. The proof is based on the properties rather than on the evaluation, of the inverse of the associated Mercerian transformation.

Full text of the article:

• Compressed PostScript file (46 kilobytes)
• PDF file (121 kilobytes)