The Anisotropic Balian-Low Phenomenon and the Variational Construction of Wavelet Frames

TL;DR

This paper explores the stability and construction of wavelet frames in anisotropic spaces, revealing geometric obstructions to tight frames and providing variational methods to optimize dual wavelets, with applications to Sobolev embeddings.

Contribution

It introduces a matrix algebra approach for wavelet frame analysis, identifies an anisotropic Balian-Low phenomenon, and derives sharp constants for Sobolev embeddings based on dilation geometry.

Findings

Boundedness of frame operators established

Existence of fundamental geometric obstructions identified

Sharp constants for Sobolev embeddings derived

Abstract

This paper investigates the stability of wavelet frames within anisotropic function spaces. By replacing classical integral estimates with a matrix algebra approach, we establish the boundedness of frame operators and derive optimal dual wavelets via variational principles. Our analysis reveals fundamental geometric obstructions, identified here as an anisotropic Balian-Low phenomenon, which preclude the existence of tight frames for isotropic generators in high-shear regimes. Furthermore, we apply these results to determine sharp constants for Sobolev embeddings, explicitly quantifying the impact of dilation geometry on analytic stability.