Phase unwinding, or invariant subspace decompositions of Hardy spaces

TL;DR

This paper explores invariant subspace decompositions of Hardy spaces, establishing convergence in Lp, and constructs explicit multiscale wavelet bases and decompositions for specific inner functions.

Contribution

It introduces new invariant subspace decompositions related to phase unwinding and provides explicit constructions for wavelet bases and inner functions.

Findings

Proves convergence of decompositions in Lp spaces

Constructs explicit multiscale wavelet bases

Provides explicit decompositions for specific inner functions

Abstract

We consider orthogonal decompositions of invariant subspaces of Hardy spaces, these relate to the Blaschke based phase unwinding decompositions. We prove convergence in Lp. In particular we build an explicit multiscale wavelet basis. We also obtain an explicit unwindinig decomposition for the singular inner function, exp 2i\pi/x.