How to Refine Polynomial Functions

Refinable functions are functions that are in a sense self-similar: If you add shrunken translates of a refinable function in a weighted way, then you obtain that refinable function again. For instance, see Figure 1 for how a quadratic B-spline can be decomposed into four small B-splines and how the so called Daubechies-2 generator function is decomposed into four small variants of itself. All B-splines with successive integral nodes are refinable, but there are many more refinable functions that did not have names before the rise of the theory of refinable functions. In fact we can derive a refinable function from the weights of the linear combination in the refinement under some conditions. Refinable functions were introduced in order to develop a theory of real wavelet functions that complements the discrete sub-band coding theory. Following the requirements of wavelet applications, existing literature on wavelets focuses on refinable functions that are L2-integrable and thus have a well-defined Fourier transform, are localized (finite variance) or even better of compact support. It is already known, that polynomial functions are refinable as well. In this paper we want to explore in detail the connection between polynomials and the respective weights for refinement. Our results can be summarized as follows: