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Scale-limitedness in wavelet transforms. (English) Zbl 1018.42024

Summary: The notion of scale-limitedness is introduced with reference to wavelet transforms. In the case of wavelet transforms, the variable corresponding to the frequency is the inverse of ‘scale’. Therefore, scale-limitedness is accordingly defined as the scale value below which the wavelet transform is zero. Scale-limitedness is similar to band-limitedness, but has some additional properties which makes it very useful. As an extension ‘practical scale-limitedness’ and ‘practical scale-limitedness on a partition’ have also been defined. The connection between scale-limitedness and the higher-order vanishing moment property of the wavelet has also been briefly discussed; and it is shown that different scale-limits for different regions has sense due to this property of the mother wavelet. Finally, it is shown that all band-limited signals are scale-limited with respect to a class of wavelet families. Further, the existence of time-limited signals which are also scale-limited is established. Moreover, it is proved that, the larger the value of the scale-limit, the smoother is the signal.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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