The regularity of wavelet transform with the high order vanishing moments

The regularity of wavelet transform has been widely applied in signal analysis, image processing, seismic exploration, network anomaly analysis and other fields. For the high order vanishing moment of rapid attenuation wavelet function do the continuous wavelet transform, building a relationship in the Hölder continuity of higher-order differentiable function and the attenuation of absolute value of wavelet transform. Namely under the condition of rapid attenuation wavelet function with high order vanishing moments, given the high order differentiable function with Hölder continuity is a sufficient condition of the absolute value of the continuous wavelet transform with rapid attenuation. Under the condition of given compactly supported wavelet function with high order vanishing moments, this paper gives that the high order differentiable function with Hölder continuity is the necessary condition of the absolute value of the continuous wavelet transform with rapid attenuation. By means of wavelet transform attenuation of coefficient depict the global regularity of function.