# Multiplication in the wavelet domain, what does it look like in real space?

Say you wavelet transform a signal, multiply by some values, and untransform. Is there anything we can say about the effect in real space? I know that it's not exactly a convolution, because the convolution theorem doesn't really exist for the wavelet transform. So if it's not a convolution, what is it?

Also, is the effect different if you multiply in an orthogonal wavelet basis (at, say, dyadic frequencies) versus a continuous transform at all frequencies?

The redundancy of the continuous transform certainly helps a lot. With DWT (a variant of them) I have also used masks to remove localized high amplitude disturbances in some discrete wavelet subbands. The masks were initially binary ($0$ on the places to remove, $1$ outside), but they were tapered to have a smooth profile between $0$ and $1$. Not very theoretical, but efficient to separate directional data.