Nearly shiftable scaling functions

The goal of the paper is to derive an approach for designing nearly shiftable scaling functions for multiresolution analyses (MRAs). Because this method does not increase the sampling density, the sparseness and efficiency of a dyadic grid is preserved. It contrasts with other attempts for the same problem which suffer either from oversampling or from being computationally expensive and data dependent. The algorithm reshapes a starting scaling function by modifying the Zak transform of its energy spectral density (ESD). The paper shows that although the modified signal does not strictly satisfy the 2-scale equation, the approximation error is sufficiently small. The result is a wavelet representation whose subband energy is "nearly" invariant to translations of its input. The paper illustrates this property with specific examples.