Spectral analysis and integral representations of the tempered fractional Riesz derivative

Fethi Bouzeffour; Fethi Bouzeffour

doi:10.3934/math.2025918

AIMS Mathematics

2025, Volume 10, Issue 9: 20571-20585. doi: 10.3934/math.2025918

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Spectral analysis and integral representations of the tempered fractional Riesz derivative

Fethi Bouzeffour ^,

Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 26 May 2025 Revised: 22 August 2025 Accepted: 28 August 2025 Published: 08 September 2025
MSC : 26A33, 42B10

In this work, we develop the harmonic analysis associated with the second-order differential operator $ \mathscr{L}_\gamma = -\frac{d^2}{dx^2} -2\gamma\, \frac{d}{dx} - \gamma^2 $. Fractional powers of $ \mathscr{L}_\gamma $ are defined via spectral representation, and a singular integral representation is provided. Furthermore, we establish the equivalence between the fractional powers of $ \mathscr{L}_\gamma $ and a tempered Riesz derivative.
- tempered fractional derivative,
- Fourier transform
Citation: Fethi Bouzeffour. Spectral analysis and integral representations of the tempered fractional Riesz derivative[J]. AIMS Mathematics, 2025, 10(9): 20571-20585. doi: 10.3934/math.2025918

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Abstract

In this work, we develop the harmonic analysis associated with the second-order differential operator $ \mathscr{L}_\gamma = -\frac{d^2}{dx^2} -2\gamma\, \frac{d}{dx} - \gamma^2 $. Fractional powers of $ \mathscr{L}_\gamma $ are defined via spectral representation, and a singular integral representation is provided. Furthermore, we establish the equivalence between the fractional powers of $ \mathscr{L}_\gamma $ and a tempered Riesz derivative.

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