Fractional-Wavelet Analysis of Positive definite Distributions and Wavelets on $$\varvec{\mathscr {D'}}(\mathbb {C})$$

In this chapter we describe a wavelet expansion theory for positive definite distributions over the real line and define a fractional derivative operator for complex functions in the distribution sense. In order to obtain a characterization of the complex fractional derivative through the distribution theory, the Ortigueira-Caputo fractional derivative operator \(_{\text {C}}\text {D}^{\alpha }\) [13] is rewritten as a convolution product according to the fractional calculus of real distributions [8]. In particular, the fractional derivative of the Gabor–Morlet wavelet is computed together with its plots and main properties.