Our goal is to construct smooth wavelet functions. In constructing such wavelet functions, we need a smooth scaling function to begin with. B-spline functions are suitable as our scaling function because they are piecewise polynomials with compact supports and are relatively smooth. B-wavelet functions are just dilations and translations of these B-spline functions. In addtion, we can find a recurrence relation of the B-spline functions with different order. Hence B-wavelets of any order can be constructed successively from the lower order ones.
Kumazawa '94, Rumi, "The Recurrence Relation of B-Wavelets" (1994). Honors Projects. Paper 11.