L2 (R) Solutions of Dilation Equations and

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L2(R) Solutions of Dilation Equations and Fourier-like Transforms David Malone∗ December 6, 2000 Abstract We state a novel construction of the Fourier transform on L2(R) based on translation and dilation properties which makes use of the multiresolu- tion analysis structure commonly used in the design of wavelets. We ex- amine the conditions imposed by variants of these translation and dilation properties. This allows other characterisations of the Fourier transform to be given, and operators which have similar properties are classified.

• sets of the mul- tiresolution analysis
• dilation equations
• dilation property
• spirit of multiresolution analysis
• dilation property for scales
• various cases
• fourier transform
• linear
• structure