Radial Convergence along Decreasing Coordinate Radii in \texorpdfstring{$H^\infty(\T^\infty)$}{H-infinity(T-infinity)}

Abstract

Aleman, Olsen, and Saksman asked whether radial convergence may fail for bounded analytic functions on the infinite-dimensional polydisc when every radial point has non-increasing coordinates, and whether the approach may be chosen independently of the boundary point. We construct a point-dependent counterexample in which every fixed coordinate nevertheless increases to $1$. A finite discretization yields a boundary-point-independent coordinatewise decreasing approach for which convergence to the boundary function fails almost everywhere. In contrast, every boundary-point-independent approach whose fixed coordinates increase monotonically to $1$ satisfies an $L^p$ maximal inequality and the almost-everywhere Fatou theorem for $1<p<\infty$.