Is there an equivalent of Plancherel's theorem with wavelets transform?

Question

For the Fourier transform, we know that:

$$||f||_2=c||\hat{f}||_2$$ where $c$ depends on the normalization. Is there an equivalent with wavelet transform?

Thanks.

Answer 1

There is a multidimensional generalization:

$$ ||f||^2_{L^2(R^n)}=||g||^{-2}||\hat{f}||^2_{L^2(R_n)} $$

where $R^n$ is the space-time domain and $R_n$ is the wavenumber-frequency domain. We normalize g by $||g||=1$.

Answer 2

Yes, there is. It is usually called orthogonality relations. The best reference I know at the moment would be this, page 4 on top. This claims even more than the norm relation you cite. Indeed that can be extended to inner products to find: $$ \langle f, g \rangle = c_h \langle W_h f, W_h g \rangle, $$ where $W_h$ denotes the wavelet transform with respect to $h$ as mother wavelet. Note that the inner products above are in two different spaces.

Please also note that this relation holds not only for wavelets but rather for all transforms that have an underlying group structure (e.g. short time Fourier transform). If you have a look at the reference I gave you will see that in this paper the topic is treated in great generality as they use group representations. If that is not interesting for you you can just take away that the answer to your question is yes. If I find the time I'll try to find a more specific wavelet reference.