Wavelet transform on the torus: a group theoretical approach. (English) Zbl 1309.42046
A continuous wavelet transform (CWT) on the torus is constructed. A CWT on the sphere is considered before. A CWT on the torus is based on the theory of coherent states of quantum physics (formulated in terms of group representation theory). The Euclidean limit reproduces wavelets on the plane with two dilations, which can be defined through the natural tensor product representation of usual wavelets. Restricting ourselves to a single dilation imposes severe conditions on the mother wavelet that can be overcome by adding extra modular group \(\mathrm{SL}(2,\mathbb Z)\) transformations, thus leading to the concept of modular wavelets.
Reviewer: V. Leontiev (Ul’yanovsk)
MSC:
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 44A20 | Integral transforms of special functions |
Keywords:
continuous wavelet transform on the sphere; continuous wavelet transform on the torus; tensor representation of wavelets; harmonic analysis on groups; modular transformations on the torusReferences:
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