Fractional GP refinable functions

A new class of refinable functions extending the GP class introduced in [12] is presented. It is characterized by a symbol with fractional exponent that gives rise to non-compactly supported refinable functions. Nevertheless, the decay and stability properties of these refinable functions allow them to generate a multiresolution analysis (MRA) of L2(R). For suitable values of their parameters these refinable functions reduce to the fractional B-splines introduced in [16], while, for integer α, they interpolate the GP refinable functions. Furthermore, this class of refinable functions is proved to be closed with respect to convolution and fractional differentiation, allowing for its convenient the applicability to Sobolev spaces. The fractional refinable functions introduced here show an useful order of polynomial exactness.