Extended Families of Cardinal Spline Wavelets

AbstractCardinal spline prewavelets ψm and ηm are well known. In this paper we present an extension of cardinal spline prewavelets and construct nonorthogonal cardinal spline wavelet systems. Both compactly supported spline wavelets ψm,l;c and globally supported spline wavelets ηm,l;c(x) = L(l)m+l;c(2x - 1) are given. When l = m and c = 0, we obtain cardinal spline prewavelets ψm and ηm. As l decreases, so does the support of the wavelet ψm,l;c. When l increases, the smoothness of the dual wavelets ψ̃m,l;c and η̃m,l;c improves. When c = 0 the wavelets ψm,l;0 = ψm,l and ηm,l;0 = ηm,l are symmetric or antisymmetric. The dual wavelets ψ̃m,l;0 = ψ̃m,l and η̃m,l;0 = η̃m,l are lth-order spline functions. We give an explicit analytic formula for the dual wavelet ψ̃m,l. The wavelets ψm,l and ηm,l have properties similar to spline prewavelets ψm and ηm except for the orthogonality among the wavelet spaces. Each wavelet is constructed by spline multiresolution analysis. The dual MRAs are given