Balayage and pseudo-differential operator frame inequalities

Using his formulation of the potential theoretic notion of balayage and his deep results about this idea, Beurling gave sufficient conditions for Fourier frames in terms of balayage. The analysis makes use of spectral synthesis, due to Wiener and Beurling, as well as properties of strict multiplicity, whose origins go back to Riemann. In this setting and with this technology, with the goal of formulating non-uniform sampling formulas, we show how to construct frames using pseudo-differential operators. This work fits within the context of the short time Fourier transform (STFT) and time-frequency analysis.