A New Density Hierarchy Based on Riesz and Hölder Summability

Let f(n) be an Erdős–Ulam function satisfying f(n+1)/f(n)→1 as n→∞, and let Af denote the associated Riesz matrix. Furthermore, let Af* be the Riesz matrix generated by the weights f(n)/f(1)+f(2)+⋯+f(n), n≥1. The main result establishes that the matrix density induced by an arbitrary power of Af—or, equivalently, by a generalized Hölder mean of arbitrary order—is stronger than the matrix density induced by Af*.