What does it mean for a Wavelet transform to commute with translations?

It states "A wavelet transform commutes with translations, and is therefore not translation invariant". Now I understand why it is a problem that the result is not translation invariant, however, I'm confused as to why it is.

What does it mean for a transform to commute with translation and why does the Wavelet transform commute with translations (i.e. why is the Wavelet transformation shift invariant)?.

• Doesn't shift = translation? – Cherny Apr 15 at 10:52
• @Cherny Correct, one way I could pose the question is "What does it mean for the wavelet transform to commute with shifts?" – Izzo Apr 15 at 13:42

Transform G is shift invariant if - $$\forall x:\sigma^nG(x) = G(x)$$ $$\sigma^n$$ being shift by n. Examples for transforms that are invarient to shifts are histogram and the amplitude of Fourier transform.
Commuting with shift is - $$\forall x:\sigma^nG(x) = G(\sigma^nx)$$ So it can't be shift invariant (unless G(x) is constant).