Nonuniform Sampling Expansions of Two-Dimensional Bandlimited Signals

Abstract

Let L²(ℝ) (L²(ℝ²), respectively) be the space of all square integrable functions on ℝ (ℝ)², i.e., $$\rm \|f\|^2_{L^2(R)}:= {1\over\sqrt{2\pi}} {\int\limits^\infty_{-\infty}}\mid f(x)\mid^2dx \infty (\|f\|^2_{L^2(R^2)}:= {1\over{2\pi}} {\int\limits_{R^2}} \mid f(x,y)\mid^2d(x,y) \infty).$$ The L² (ℝ) (L² (ℝ²)) Fourier transform is defined by $$f\hat (u)\ :=\ \ {\mathop {\rm l.i.m.}\limits_{\rho \rightarrow\infty}}\ {1\over\sqrt{2\pi}}\ {\int\limits^\rho_{-\rho}}\ {\rm f(x)e^{-iux}dx (f\hat (u,v):= {\mathop {\rm l.i.m.}\limits_{\rho \rightarrow\infty}}\ {1\over{2\pi}}\ {\int\limits^\rho_{-\rho}} {\int\limits^\rho_{-\rho}}\ f(x,y)e^{-i(ux+vy)}dxdy)}.$$