Lagrange M-th band filters and the construction of smooth M-band wavelets

This paper develops a new explicit formula for the construction of M-th band maximally flat lowpass filters based on Lagrange interpolation. Both the minimum length and arbitrary length solutions are obtained. This leads to the construction of M-band wavelet filters with N vanishing wavelet moments as spectral factors of the new M-th band filters. One can then vary the free parameters of the general (non-minimal length) M-th band filter to obtain filters with better stopband attenuation than the minimal length filters, and wavelet bases with greater smoothness than the corresponding minimal length solutions. Remarkably, in the M-band case (M>2), one can simultaneously improve the stopband attenuation of the lowpass filters and obtain smoother wavelet bases