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 (p, q)-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.

可以为函数w量化可允许的Sobolev正则性w，该函数在d维环面中为零，并且倒数\（u = 1 / w \）为（p，  q）乘数。解决了此问题的几个方面，包括正Hausdorff维数的零集，矩阵值傅立叶乘数和Sobolev正则性的非对称形式。此外，我们在傅立叶乘数与Gabor系统和平移不变系统的逼近性质之间建立了联系。我们利用这种联系以及在傅立叶乘法器上的结果在这些情况下完善和扩展了Balian–Low不确定性原理的版本。