Wavelets and differential-dilation equations

It is shown how differential-dilation equations can be constructed using iterations, similar to the iterations with which wavelets and dilation equations are constructed. A continuous-time wavelet is constructed starting from a differential-dilation equation. It has compact support and excellent time domain and frequency domain localization properties. The wavelet is infinitely differentiable and therefore cannot be obtained using digital filter banks. In addition, the wavelet has excellent approximation properties. New sampling and differentiation techniques are also introduced. Results on image interpolation using the solution of the differential-dilation equation are presented. Examples are given, demonstrating the suitability of the new wavelet function for signal analysis.