The admittable Sobolev regularity is quantified for a function, w, which has a zero in the d-dimensional torus and whose reciprocal u " 1{w is a pp, qq-multiplier. Several aspects of this problem are addressed, including zero-sets of positive Hausdorff dimension, matrix valued Fourier multipliers, and non-symmetric versions of Sobolev regularity. Additionally, we make a connection between Fourier multipliers and approximation properties of Gabor systems and shift-invariant systems. We exploit this connection and the results on Fourier multipliers to refine and extend versions of the Balian-Low uncertainty principle in these settings.
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