Non-local fractional derivatives. Discrete and continuous

Abadías, Luciano; León-Contreras, Marta De; Torrea, José L.

doi:10.1016/j.jmaa.2016.12.006

Cited by 26 publications

(46 citation statements)

References 18 publications

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“…To compute (−∆) s u(x) for each point x ∈ R n one could try to take the inverse Fourier transform in (1). In fact, since |ξ| 2s u(ξ) ∈ L 1 (R n ), one can make sense to (−∆) s u(x) = F −1 (|ξ| 2s u(ξ))(x).…”

Section: Fractional Laplacian: Semigroups Pointwise Formulas and Limitsmentioning

confidence: 99%

User’s guide to the fractional Laplacian and the method of semigroups

Stinga¹

2019

Fractional Differential Equations

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The method of semigroups is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators L and their regularity properties in related functional spaces. The approach was introduced by the author and José L. Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian L s = (−∆) s , 0 < s < 1. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointwise formula for (−∆) s u(x). One of the key advantages is that our technique relies on the use of heat kernels, which allows for applications in settings where the Fourier transform is not the most suitable tool. In addition, it provides explicit constants that are key to prove, under minimal conditions on u, the validity of the pointwise limitsThe formula for the solution to the Poisson problem (−∆) s u = f is found through the semigroup approach as the inverse of the fractional Laplacian u(x) = (−∆) −s f (x) (fundamental solution). We then present the Caffarelli-Silvestre extension problem, whose explicit solution is given by the semigroup formulas that were first discovered by the author and Torrea. With the extension technique, an interior Harnack inequality and derivative estimates for fractional harmonic functions can be obtained. The classical Hölder and Schauder estimates (−∆) ±s : C α → C α∓2s are proved with the method of semigroups in a rather quick, elegant way. The crucial point for this will be the characterization of Hölder and Zygmund spaces with heat semigroups.

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Section: Fractional Laplacian: Semigroups Pointwise Formulas and Limitsmentioning

confidence: 99%

User’s guide to the fractional Laplacian and the method of semigroups

Stinga¹

2019

Fractional Differential Equations

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“…The proof is quite long and can be find in [47], Theorem 20.4. Moreover, about this topic we recall the very recent contribution [1]. By the way, this last paper can be considered also as further signal of the renascent interest for Marchaud derivative.…”

Section: Grünwald-letnikov Derivativementioning

confidence: 88%

“…In fact for giving a different perspective of the Marchaud derivative we have to introduce the Grünwald-Letnikov derivative. 1 To do this we need some new notation.…”

Section: Grünwald-letnikov Derivativementioning

confidence: 99%

Weyl and Marchaud Derivatives: A Forgotten History

Ferrari

2018

Mathematics

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“…Fractional calculus has recently been applied to the theory of meromorphic functions (see, e.g., [8][9][10][11]). In particular, the α-order fractional derivative of the Riemann ζ function given by ζ (α)…”

Section: Introductionmentioning

confidence: 99%

Riemann zeta fractional derivative—functional equation and link with primes

Guariglia

2019

Adv Differ Equ

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This paper outlines further properties concerning the fractional derivative of the Riemann ζ function. The functional equation, computed by the introduction of the Grünwald-Letnikov fractional derivative, is rewritten in a simplified form that reduces the computational cost. Additionally, a quasisymmetric form of the aforementioned functional equation is derived (symmetric up to one complex multiplicative constant). The second part of the paper examines the link with the distribution of prime numbers. The Dirichlet η function suggests the introduction of a complex strip as a fractional counterpart of the critical strip. Analytic properties are shown, particularly that a Dirichlet series can be linked with this strip and expressed as a sum of the fractional derivatives of ζ. Finally, Theorem 4.3 links the fractional derivative of ζ with the distribution of prime numbers in the left half-plane.

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Non-local fractional derivatives. Discrete and continuous

Cited by 26 publications

References 18 publications

User’s guide to the fractional Laplacian and the method of semigroups

User’s guide to the fractional Laplacian and the method of semigroups

Weyl and Marchaud Derivatives: A Forgotten History

Riemann zeta fractional derivative—functional equation and link with primes

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