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
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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 C Dα [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
1 Introduction In recent years, wavelet analysis and fractional calculus have shown to be a powerful tool in several areas of mathematics. Indeed, the time-frequency localization property provided by the wavelet approach gives the possibility to use a wavelet basis as a mathematical microscope in order to better investigate the behavior of a function by the well-known Heisenberg box [18] located in the time-frequency plane. Wavelet expansions are used to characterize different function spaces, such as L p spaces, Sobolev spaces, Morrey–Campanato spaces, etc. [19]. In particular, several key concepts of wavelet analysis, such as the wavelet transform, can be extended to the space of tempered distributions S (R). E. Guariglia (B) Department of Physics “E. R. Caianiello”, University of Salerno, Via Giovanni Paolo II, 84084 Fisciano, Italy e-mail: [email protected] E. Guariglia · S. Silvestrov Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Box 883, 721 23 Västerås, Sweden e-mail: [email protected] © Springer International Publishing Switzerland 2016 S. Silvestrov and M. Ranˇci´c (eds.), Engineering Mathematics II, Springer Proceedings in Mathematics & Statistics 179, DOI 10.1007/978-3-319-42105-6_16
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In this chapter, a wavelet expansion for the family of positive definite distributions is presented. It has many applications in different areas of both pure and applied mathematics (Lie groups, maximum entropy methods, etc.) [7, 14] and can be generalized for the class of tempered distributions [16]. Furthermore, an open problem about the reconstruction formula for Shannon wavelets in the distribution sense is proposed. In [3, 4] a generalization to complex functions of the distribution theory is presented. In particular, these two papers provide a generalization of the classical Dirac delta in the complex plane which gives the possibility to rewrite the complex fractional derivative in the distribution sense. Indeed, the fractional derivative of complex functions is provided by the Ortigueira-Caputo fractional derivative C Dα [13], which can be written as a convolution of the given complex function with a suitable function that defines a regular distribution on C (see (30) and (31)). The authors have co
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