From quasiperiodic tilings with τ-inflation to τ-wavelets

We construct wavelets on quasiperiodic tilings T with stone-inflation symmetry. All tiles of T when scaled by the factor τ can be packed face-to-face from the original ones. For the planar Tübingen triangle and the Penrose–Robinson tiling, we give the geometric generators of the deflation as affine Euclidean pairs (t,g) of (translations, reflection/rotations) with t from the fivefold module and g from the Coxeter group I2(5). Deflation generates levels of subdivisions of the tilings. On these subdivisions, we give the Haar wavelet construction. On each level of deflation, we transform from the set of characteristic functions to explicit orthogonal wavelet bases.