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From Herglotz-Nevanlinna functions to completely monotonic functions

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Auteurs : Berg, Christian (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : A Herglotz-Nevanlinna function is a holomorphic function $f$, defined in the upper half-plane $\mathbb{H}:=\{z \in \mathbb{C} \mid \Im z>0\}$, such that $\Im f(z) \geq 0$ for all $z \in \mathbb{H}$, and they are the functions in focus at the present conference. These functions are also called Pick functions, and they are characterized as the functions of the form$$f(z)=\alpha z+\beta+\int_{-\infty}^{\infty} \frac{t z+1}{t-z} d \tau(t), \quad z \in \mathbb{H}$$where $\alpha \geq 0, \beta \in \mathbb{R}$ and $\tau$ is a positive finite measure on $\mathbb{R}$.
Since $\mathbb{H}$ is a simply connected domain, caracterization of this class of functions is the same as characterization of the set of non-negative harmonic functions in $\mathbb{H}$ and by conformal mapping this set is in one-to-one correspondence with the set of non-negative harmonic functions in the unit disc.
We shall discuss various subclasses of Pick functions and their relation to other important classes of functions such as the completely monotonic functions and the subclass of Stieltjes functions. We recall that these classes are the functions $f:] 0, \infty[\rightarrow \mathbb{R}$ of the form$$f(x)=\int_{0}^{\infty} e^{-s x} d \mu(s), \quad \text { resp. } f(x)=a+\int_{0}^{\infty} \frac{d \mu(s)}{x+s}$$where $a \geq 0$ and $\mu$ is a non-negative measure on $[0, \infty[$.At the 7 th OPSFA, Copenhagen 2003 , we posed the problem of determining the largest value $\alpha=\alpha^{*}>0$ for which $f_{\alpha}(x)=e^{\alpha}-(1+1 / x)^{\alpha x}, x>0$ is a completely monotonic function, and it was noticed that $1 \leq \alpha^{*}<3$ and that graphs suggested that $\alpha^{*}>2$. The value has now been calculated with 20 decimals starting with $\alpha^{*} \approx 2.29965$.This is based on a recent result obtained in collaboration with Massa and Peron from Brazil. We have found a family $\varphi_{\alpha}, \alpha>0$ of entire functions such that$$f_{\alpha}(x)=\int_{0}^{\infty} e^{-s x} \varphi_{\alpha}(s) d s, \quad x>0 .$$We showed that each function $\varphi_{\alpha}$ has an alternating power series expansion, whose coefficients are determined as an explicit sequence of polynomials in $\alpha$. It is therefore possible to calculate as accurately as desired for which values of $\alpha$ the function $\varphi_{\alpha}$ is non-negative on $\left[0, \infty\left[\right.\right.$. It turned out that the functions $\varphi_{\alpha}$ are 'close' to the well known Bessel function $J_{1}$ when $\alpha$ is large, and 'close' to the Lambert $W$ function, when $\alpha$ is small.

Keywords : Pick functions; completely monotonic functions

Codes MSC :
26A48 - Monotonic functions, generalizations
30E20 - Integration, integrals of Cauchy type, integral representations of analytic functions, See also {45Exx}
42A38 - Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

    Informations sur la Vidéo

    Réalisateur : Hennenfent, Guillaume
    Langue : Anglais
    Date de publication : 20/06/2022
    Date de captation : 24/05/2022
    Sous collection : Research talks
    arXiv category : Complex Variables
    Domaine : Analysis and its Applications
    Format : MP4 (.mp4) - HD
    Durée : 00:51:17
    Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2022-05-24_Berg.mp4

Informations sur la Rencontre

Nom de la rencontre : Herglotz-Nevanlinna Functions and their Applications to Dispersive Systems and Composite Materials / Fonctions de Herglotz-Nevanlinna et leurs applications aux systèmes dispersifs et aux matériaux composites
Organisateurs de la rencontre : Bonnet-Ben Dhia, Anne-Sophie ; Cassier, Maxence ; Gralak, Boris ; Luger, Annemarie ; Milton, Graeme
Dates : 23/05/2022 - 27/05/2022
Année de la rencontre : 2022
URL Congrès : https://conferences.cirm-math.fr/2225.html

Données de citation

DOI : 10.24350/CIRM.V.19919403
Citer cette vidéo: Berg, Christian (2022). From Herglotz-Nevanlinna functions to completely monotonic functions. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19919403
URI : http://dx.doi.org/10.24350/CIRM.V.19919403

Voir aussi

Bibliographie

  • BERG, Christian, MASSA, Eugenio, et PERON, Ana P. A Family of Entire Functions Connecting the Bessel Function $ J_1 $ and the Lambert W Function. Constructive Approximation, 2021, vol. 53, no 1, p. 121-154. - https://doi.org/10.1007/s00365-020-09499-x



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