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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?

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They can be used for time-varying filtering of non-stationary signals, under names like Gabor multipliers, STFT filters for instance. You can find a review reference in the chapter Linear Time-Frequency Filters, 11.1.3 Implicit design and generalization in Spreading function representation of operators and Gabor multiplier approximation.

You also have the 2004 paper The convolution theorem for the continuous wavelet transform. My experience in this domain is limited to localized matched filtering, by multiplying coefficients by scalar complex numbers to locally modify amplitude and phase (with complex continuous wavelet frames), see Adaptive multiple subtraction with wavelet-based complex unary Wiener filters.

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.

Finally, you can find across-scale multiplication for ROI (region-of interest) coding with JPEG 2000.

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  • $\begingroup$ Hi, thanks! I ask because I want to model 3D data with a covariance that's diagonal in wavelet space. I'm struggling to understand how multiplying wavelet basis functions affects the structure in real space. $\endgroup$ – cgreen Jun 17 '16 at 23:56
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    $\begingroup$ @cgreen aside from the experience that Laurent brought in (really awesome!) a good trick is to really write down what you're doing with respect to formalized methods – in this case, linear algebra, representing the DWT as linear operators, ie. matrix multiplications. That way, point-wise multiplication can often also be understood as diagonal matrix – and, depending on the structure of the wavelet you're actually looking at, this might be sufficient or not to make the statements you need. $\endgroup$ – Marcus Müller Jul 17 '16 at 22:25

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