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arXiv · 2610.29198

On the existence of independent inner functions

Xiangdi Fu·Kunyu Guo·Dilong Li·Yanqi Qiu
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Abstract

The existence of independent inner functions is established: there exist inner functions $φ$ and $ψ$, both vanishing at the origin, such that $$ \frac{1}{2π}\int_0^{2π} φ^m(e^{iθ}) \overline{ψ^n(e^{iθ})}\,dθ= 0, \qquad m,n \geq 1. $$ In fact, both $φ$ and $ψ$ can be chosen to be Blaschke products. The proof proceeds by an iterative construction, which suggests a general approach to produce inner functions with a desired joint distribution.

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Xiangdi Fu, Kunyu Guo, Dilong Li, Yanqi Qiu. 2026-10-21. On the existence of independent inner functions. https://arxiv.org/abs/2610.29198

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