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

• Emanuel Guariglia
• Sergei Silvestrov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 179)

## Abstract

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.

## Keywords

Wavelet basis Positive definite distribution Complex fractional derivative Gabor–Morlet wavelet

## Notes

### Acknowledgements

Emanuel Guariglia would like to thank the Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalens University for giving him the opportunity to work in an extremely favourable research environment.

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