New compactly supported scaling and wavelet functions derived from Gegenbauer polynomials

A new family of scaling and wavelet functions is introduced; it is derived from Gegenbauer polynomials. The link of ordinary 2nd order differential equations to multiresolution filters is employed to construct these new functions. These functions, termed as ultraspherical harmonic or Gegenbauer scaling and wavelet functions, possess compact support and generalized linear phase. This is an interesting property since, from the computational point of view, only half the number of filter coefficients is required to be computed. By using an alpha factor that is within the orthogonality range of such polynomials, scaling and wavelet functions are generated by frequency selective FIR filters. Potential applications of such wavelets include fault defection in transmission lines of power systems.