# Wavelet analog of: fourier transform of derivative of a function is multiplication with polynomial

so I have a question concerning the continuous wavelet transform (please forgive me if this is something very simple however i couldn't seem to find any answer so far):

We know that for the Fourier transform $$\mathcal{F}$$, under appropriate assumptions on a function $$f$$, it follows that derivation is equal to multiplication in the sense:

$$\mathcal{F}(\frac{d}{dx}f) \propto ix\mathcal{F}(f)$$ What I was asking myself now is if there exist wavelets for which something similar holds using the continuous wavelet transform.

Any help would be much appreciated.

• Most of the properties of the Fourier transform come from the fact that the $e^{i 2\pi f t}$ functions are the eigen functions of the delay operators. It must be the case for the derivative formula. Then, I have some doubt that you can get the same property exactly with another basis – Damien Jan 30 at 9:41
• thank you for your comment. Yes I also realised yesterday that it should only hold for the Fourier transform. However what I found is that when choosing Hermitian Wavelets it at least is again a transform with an hermitian wavelet - up to some constants. Maybe the same holds for other wavelet families ?? – RZA Chris Jan 31 at 11:32