A class of fractional refinable functions

Starting from a family of centrally symmetric, totally positive and compactly supported (GP) refinable functions and introducing a fractional exponent in the discrete Fourier transform, new functions, that are proved to be still refinable, are generated. Even if, for non integer, they are not compactly supported anymore they exhibit a decay that allows them to belong to L2(R); moreover, for certain values of their parameters they reduce to the fractional B-splines, while, for integer, they interpolate the GP refinable functions. Also, these refinable functions can be characterized by a convolution relation between suitable minimally supported GP refinable functions and suitable fractional B-splines.