Wavelet transform have become a standard tool in the field of image processing and data compression. However, it is still not widely used in seismic processing. One reason is that researchers have just started starting to discover operators in the wavelet domain doing standard Fourier operations like shifting, and differentiating time series. Additionally, DWT fails to render seismic sections as sparse matrices, which frustrates data compression. Interpolation is a more successful application. As with Fourier methods, padding of zeroes in the wavelet domain results in denser sampling in the time domain. In contrast to extending the Fourier transform by padding, the wavelet base function is localized and energy in the extended spectrum part is small but nonzero.
I have shown how the choice of filter coefficients implements different Daubechies wavelets. In the future wavelets other than Daubechies' may prove more applicable to seismic data.
The next goal is to implement discrete Fourier-like operators for shifting, differentiating, and interpolating time series. Comparing these operators with the traditional Fourier operators will require establishing fundamental criteria such as the specification of what accurate differentiation of discrete seismic data means.