Synthesis of Complete Orthonormal Fractional Basis Functions With Prescribed Poles

In this paper, fractional orthonormal basis functions that generalize the well-known fixed pole rational basis functions are synthesized. For a range of noninteger differentiation orders and under mild restrictions on the choice of the basis poles, the synthesized basis functions are shown to be complete in the space of functions which are analytic on the open right-half plane and square-integrable on the imaginary axis. This presents an extension of the completeness results for the fractional Laguerre and Kautz bases to fractional orthonormal bases with prescribed pole locations.