Continuous-order derivative of a function with a known Fourier transform

ABSTRACT We relate the definition of the continuous order derivative of a function, f(x) with Fourier transform F(v), with the continuous order derivative of monomials and polynomials, in x, as well as with that of functions expressible as Mclaunnseries,Keywords: Spatial filtering, Fourier optics, signal processing and mathematical operations. 1. INTRODUCTION Wave propagation and focus errors can suitably be analyzed using derivative operators', in a similar fashion to the timepropagation operator in quantum physics. Furthermore, for signal and image processing, it is useful to implement optically derivative operators, of integer and fractional order2.In a previous publication a continuous order derivative operation was defined in the realm of Fourier optics3. This typeof operation has been defined and used in other branches of mathematics4. Here we link the Fourier optics definition of acontinuous order derivative with the continuous order operator on monomials and functions expressible as Mclaurin series.