Mittag-Leffler analysis I: Construction and characterization

Grothaus, Martin; Jahnert, Florian; Riemann, Felix; Silva, José Luís da

doi:10.1016/j.jfa.2014.12.007

Cited by 34 publications

(51 citation statements)

References 40 publications

(52 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 2.9. It is shown in [GJRdS15] that f ∈ L 2 (µ β ) does not admit a chaos expansion if β = 1. This means that there is no system of polynomials (I n (ξ)) n∈N , ξ ∈ N , on N ′ fulfilling the following properties simultaneously:…”

Section: Prerequisitesmentioning

confidence: 99%

“…Instead of using a system of orthogonal polynomials, it was proposed in [GJRdS15] to use Appell systems, compare [KSWY98]. These are biorthogonal systems allowing to construct a test function and a distribution space.…”

Section: Distributions and Donsker's Deltamentioning

confidence: 99%

“…In Mittag-Leffler analysis, the measures µ β , 0 < β < 1, are defined via their characteristic function given by Mittag-Leffler functions. It is shown in [GJRdS15] that the Mittag-Leffler measures belong to the class of measures for which Appell systems exist. Furthermore the (weak) integrable functions with values in the distribution space (N ) −1 µ β are characterized and a distribution is constructed which is a generalization of Donsker's delta in Gaussian analysis.…”

Section: Introductionmentioning

confidence: 99%

“…We further develop the theory of Mittag-Leffler analysis in this work by characterizing the convergent sequences in the distribution space (N ) −1 µ β . This is done in Section 2, where we first repeat the main steps for the construction of a Mittag-Leffler analysis from [GJRdS15] and we introduce the test function space (N ) 1 and the distribution space (N ) −1 µ β . In this way we give an approximation of Donsker's delta by a sequence of Bochner integrals in the Hilbert space L 2 (µ β ).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Mittag-Leffler analysis II: Application to the fractional heat equation

Grothaus

Jahnert

2016

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

Mittag-Leffler analysis is an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which generalizes the powerful theory of Gaussian analysis and in particular white noise analysis. In this paper we further develop the Mittag-Leffler analysis by characterizing the convergent sequences in the distribution space. Moreover we provide an approximation of Donsker's delta by square integrable functions. Then we apply the structures and techniques from Mittag-Leffler analysis in order to show that a Green's function to the time-fractional heat equation can be constructed using generalized grey Brownian motion (ggBm) by extending the fractional Feynman-Kac formula [Sch92]. Moreover we analyse ggBm, show its differentiability in a distributional sense and the existence of corresponding local times. (2010): Primary: 46F25, 60G22. Secondary: 26A33, 33E12. Mathematics Subject ClassificationRemark 2.1. N is a perfect space, i.e. every bounded and closed set in N is compact. As a consequence strong and weak convergence coincide in both N and N ′ , see page 73 in [GV64] and Section I.6.3 and I.6.4 in [GS68].Example 2.2. Consider the white noise setting, where N = S(R) is the space of Schwartz test functions, H = L 2 (R, dx) and N ′ = S ′ (R) are the tempered distributions. S(R) is dense in L 2 (R, dx) and can be represented as the projective limit of certain Hilbert spaces H p , p > 0, with norms denoted by |·| p , see e.g. [Kuo96]. Thus the white noise setting is an example for the nuclear triple described above. The Mittag-Leffler measureAs Mittag-Leffler measures µ β , 0 < β < 1, we denote the probability measures on N ′ whose characteristic functions are given via Mittag-Leffler functions. The Mittag-Leffler function was introduced by Gösta Mittag-Leffler in [ML05] and we also consider a generalization first appeared in [Wim05]. Definition 2.3. For 0 < β < ∞ the Mittag-Leffler function is an entire function defined by its power series E β (z) := ∞ n=0 z n Γ(βn + 1), z ∈ C.

show abstract

Section: Prerequisitesmentioning

confidence: 99%

Section: Distributions and Donsker's Deltamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Mittag-Leffler analysis II: Application to the fractional heat equation

Grothaus

Jahnert

2016

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…Actually we could start by giving the characteristic functional as in (6) and show the conditions of Minlos-Sazonov's theorem. This approach leads to the Mittag-Leffler analysis (see [7]) where the law of the process is a probability measure on a space of generalized functions.…”

Section: Periodic Grey Brownian Motionmentioning

confidence: 99%