An Anisotropic Balian-Low Phenomenon: Geometric Obstructions to Wavelet Frames

Abstract

We investigate the analytic stability of wavelet frames in anisotropic Hardy spaces associated with expansive dilation matrices. The main result establishes a deterministic operator-norm lower bound on the reconstruction error of the mixed frame operator, uniformly across the Hardy range, whenever a real eigenvalue of the adjoint dilation exceeds a band-limited threshold imposed on a radial generator. The obstruction is identified as an anisotropic Balian--Low phenomenon and rests on a geometric incompatibility index measuring the deviation of the Calderón sum from the admissibility identity. The bound carries no dependence on the conditioning of the dilation.