Letter to the Editor: Linear Independence of Time-Frequency Shifts Up To Extreme Dilations
Tóm tắt
Given $$f \in C_0({{\,\mathrm{\mathbb {R}}\,}}^n)$$ and a finite set $$\Lambda \subset {{\,\mathrm{\mathbb {R}}\,}}^{2n}$$ we demonstrate the linear independence of the set of time-frequency translates $$\mathcal {G}(f, \Lambda ) = \{\pi (\lambda )f\}_{\lambda \in \Lambda }$$ when the time coordinates of points in $$\Lambda $$ are far apart relative to the decay of f. As a corollary, we prove that for any $$f \in C_0({{\,\mathrm{\mathbb {R}}\,}}^n)$$ and finite $$\Lambda \subset {{\,\mathrm{\mathbb {R}}\,}}^{2n}$$ there exist infinitely many dilations $$D_r$$ such that $$\mathcal {G}(D_rf, \Lambda )$$ is linearly independent. Furthermore, we prove that $$\mathcal {G}(f, \Lambda )$$ is linearly independent for functions like $$f(t) = \frac{cos(t)}{|t|}$$ which have a singularity and are bounded away from any neighborhood of the singularity.
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