Speaker: Nelson Jose Rodrigues Faustino, Center for Research and Development in Mathematics and Applications, University of Aveiro

Time : 15:00 - 16:00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom : A Zoom link will appear here, an hour before the talk

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: Space-fractional Dirac equations, Hardy spaces, Paley–Wiener theorems

Abstract:Space--fractional models driven by stable Levy processes often lack an associated function theory that plays the role of classical complex analysis. In this talk, we introduce a fractional analogue of the Cauchy--Riemann operator ∂x₀ + D, in hypercomplex variables, built from a Riesz--Feller type fractional extension of the Dirac operator D, inspired by the approach of Mainardi--Luchko--Pagnini [6]. The resulting Dirac--like operator D^α_θ , of order α and skewness θ, generates a semigroup in the sense of Feller [3] that seamlessly extends nonsymmetric stable Levy processes and naturally leads to the Hardy space decomposition of L^p encoding the boundary values of solutions to space-fractional Dirac equations on the upper and lower half--spaces R_±ⁿ⁺¹ (cf. [1, 2]). Within this setting, we establish Paley--Wiener type theorems that characterize bandlimited Clifford--valued functions with compact Fourier support using exponential growth conditions, thus providing a fractional analogue of Bernstein spaces. A key new result is a closed--form formula for the spectral bandwidth, obtained from the sequence ((D^α_θ )^k f_± )_k∈N0 , where f_± are the boundary values of an L^p -function f_± in R_±ⁿ⁺¹, extending the earlier approaches by Kou--Qian [5], Pesenson [7] and Franklin--Hogan--Larkin [4]. These results open the door to sampling and interpolation schemes based on analytic-type solutions generated by fractional PDEs.

Biography: Nelson Jose Rodrigues Faustino is an Assistant Researcher at the University of Aveiro and currently holds the CIDMA Chair in Hypercomplex Analysis. After completing his PhD at the University of Aveiro, Portugal, in 2009, he was a postdoctoral researcher at the Centre for Mathematics, University of Coimbra (CMUC) from 2010 to 2013 and subsequently at the Institute of Mathematics, Statistics, and Scientific Computing (IMECC) at the State University of Campinas, Brazil, from 2013 to 2016, visiting professor at the Federal University of ABC (UFABC), from 2016 to 2020, and integrated member of Center for Research and Development in Mathematics and Applications (CIDMA), since 2020. He is a lifetime member of the International Society for Analysis, its Applications and Computation (ISAAC) and, in 2017, was awarded an Honorary Fellowship by the European Society of Computational Methods in Sciences and Engineering (ESCMSE) in recognition of his contributions to applied and computational mathematics.

Bibliography

[1]  Swanhild Bernstein and Nelson Faustino. ”Paley-Wiener Type Theorems Associated to Dirac Operators of Riesz-Feller type”. In: Journal of Fourier Analysis and Applications 31.52, (2025).
[2] Nelson Faustino. ”On fundamental solutions of higher-order space-fractional Dirac equations”. In: Mathematical Methods in the Applied Sciences 47.10 (2024), pp. 7988–8001.
[3] William Feller. ”On a Generalization of Marcel Riesz' Potentials and the Semi-Groups Generated by Them”. In: Comm. S´em. Math´em. Universit´e de Lund, Tome Suppl.d´edi´e a M. Riesz, Lund, Gleerup, (1952), pp. 73–81.
[4] David J. Franklin, Jeffrey A. Hogan, Kieran G. Larkin. ”Hardy, Paley-Wiener and Bernstein spaces in Clifford analysis”. In: Complex Variables and Elliptic Equations 62.9 (2017), pp. 1314–1328
[5] Kit-Ian Kon, Tao Qian. ”The Paley-Wiener theorem in Rn with the Clifford analysis setting”. In: Journal of Functional Analysis, 189.1 (2002), pp. 227–241.
[6] Francesco Mainardi, Yuri Luchko, Gianni Pagnini. ”The fundamental solution of space-time fractional diffusion equation”. In: Fractional Calculus and Applied Analysis 4.2 (2001), pp. 153–192.
[7] Isaac Z. Pesenson. ”Sampling formulas for groups of operators in Banach spaces”. In: Sampling Theory in Signal and Image Processing 14.1 (2015), pp. 1–16.