Frames for compactly supported functions with irrational density

Key Points

• To establish sufficient conditions under which a Gabor system forms a frame for compactly supported functions with irrational densities.
• Identified conditions on function $g$ such that the Gabor system forms a frame for all $eta m$ and $eta n$, with specific inequalities on $eta$ and $eta$ being irrational.
• Provided examples of functions satisfying these conditions, such as $g(x) = ext{exp}(rac{1}{x^{4}-1}) ext{chi}_{(-1,1)}(x)$.
• Examined implications for the frame properties of function $g$.
• The Gabor system forms a frame for almost all compactly supported functions meeting the specified conditions.
• Functions including $g(x) = ext{exp}(rac{1}{x^{4}-1}) ext{chi}_{(-1,1)}(x)$ serve as concrete examples of frames.
• Explicit conditions enable identification of various functions that can generate a Gabor frame.