Hilbert space models and Blaschke frames
Connor Evans
TL;DR
The paper develops Hilbert-space model methods to resolve the redundancy problem for Blaschke frames, sequences of the form $b_m = B(V^*)e_m$ with a finite Blaschke product $B$ and an isometry $V$. By constructing a model via the Riesz-Dunford calculus and Malmquist–Takenaka functions, it proves Blaschke frames are Parseval frames with frame operator identity, and provides an explicit norm-based criterion for redundancy. The redundancy analysis, leveraging the Wold decomposition, yields a complete classification: unitary $V$ gives a Riesz basis with no redundancy, a pure shift case $B(z)=z^d$ yields partial redundancy, and the presence of a nonzero Blaschke root forces full redundancy (fully insured frames). These results underscore the power of Hilbert-space models in frame theory and provide explicit tools for redundancy analysis in Blaschke frames.
Abstract
For a finite Blaschke product $B$ and for an isometry $V$ on an infinite-dimensional separable complex Hilbert space $\mathcal{H}$ we study a sequence $(b_m)_{m=1}^\infty$ of vectors in $\mathcal{H}$, defined by $b_m = B(V^*)e_m$, where $(e_m)_{m=1}^\infty$ is an orthonormal basis in $\mathcal{H}$. We call $(b_m)_{m=1}^\infty$ a Blaschke frame for $B$ with isometry $V$ on $\mathcal{H}$. We show how instrumental the use of Hilbert space models are in frame theory by completely solving the question of redundancy for a Blaschke frame, that is, what vectors can be removed from the frame $(b_{m})_{m=1}^{\infty}$ such that $(b_{m})_{m\neq k}$ is still a frame? Using the Wold decomposition, we prove that a Blaschke frame can have no redundant vectors (a Riesz basis), have some redundant vectors, or every vector is redundant (a fully insured frame). These unique cases depend on the choice of finite Blaschke product and which isometry one chooses in the construction of a Blaschke frame.