Size and shape of frequency domain of frame scaling functions

Due to resilience to background noise, stability of sparse reconstruction, and ability to capture local time-frequency information, various frame scaling functions and derived framelets are becoming widely used tools in analysis and approximation of signals from stochastic nonlinear systems, where their frequency domain plays a key role in deleting narrow-band noise effects. In this study, we focus on the characterization of size and shape of frequency domain of frame scaling functions. We give the non-improvable bounds for the size of frequency domain of frame scaling functions, present the sufficient and necessary conditions for bandlimited frame scaling functions with cube-shaped frequency domains and discuss frequency domains of derived framelets. These results can help to design suitable frame scaling functions and framelets to analyze the behavior of nonlinear systems and stochastic systems under noise interference scenario.