اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn certain non-unique solutions of the Stieltjes moment problem
105
0
0.0
(
0
)
تأليف
K. A. Penson
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ${rho}_{1}^{(r)}(n)=(2rn)!$ and ${rho}_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,...$, $n=0,1,2,...$, textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ satisfying $int_{0}^{infty}x^{n}W^{(r)}_{1,2}(x)dx = {rho}_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both ${rho}_{1,2}^{(r)}(n)$ give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing ${rho}_{1,2}^{(r)}(n)$, such as the product ${rho}_{1}^{(r)}(n)cdot{rho}_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,...$.
قيم البحث
اقرأ أيضاً
The Stieltjes moment problem is studied in the framework of general Gelfand-Shilov spaces defined via weight sequences. We characterize the injectivity and surjectivity of the Stieltjes moment mapping, sending a function to its sequence of moments, i
n terms of growth conditions for the defining weight sequence. Finally, a related moment problem at the origin is studied.
Let $ast_P$ be a product on $l_{rm{fin}}$ (a space of all finite sequences) associated with a fixed family $(P_n)_{n=0}^{infty}$ of real polynomials on $mathbb{R}$. In this article, using methods from the theory of generalized eigenvector expansion,
we investigate moment-type properties of $ast_P$-positive functionals on $l_{rm{fin}}$. If $(P_n)_{n=0}^{infty}$ is a family of the Newton polynomials $P_n(x)=prod_{i=0}^{n-1}(x-i)$ then the corresponding product $star=ast_P$ is an analog of the so-called Kondratiev--Kuna convolution on a Fock space. We get an explicit expression for the product $star$ and establish a connection between $star$-positive functionals on $l_{rm{fin}}$ and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).
These lecture notes are meant to accompany two lectures given at the CDM 2016 conference, about the Kadison-Singer Problem. They are meant to complement the survey by the same authors (along with Spielman) which appeared at the 2014 ICM. In the first
part of this survey we will introduce the Kadison-Singer problem from two perspectives ($C^*$ algebras and spectral graph theory) and present some examples showing where the difficulties in solving it lie. In the second part we will develop the framework of interlacing families of polynomials, and show how it is used to solve the problem. None of the results are new, but we have added annotations and examples which we hope are of pedagogical value.
Given a domain $Omega$ in $mathbb{C}^m$, and a finite set of points $z_1,ldots, z_nin Omega$ and $w_1,ldots, w_nin mathbb{D}$ (the open unit disc in the complex plane), the $Pick, interpolation, problem$ asks when there is a holomorphic function $f:O
mega rightarrow overline{mathbb{D}}$ such that $f(z_i)=w_i,1leq ileq n$. Pick gave a condition on the data ${z_i, w_i:1leq ileq n}$ for such an $interpolant$ to exist if $Omega=mathbb{D}$. Nevanlinna characterized all possible functions $f$ that $interpolate$ the data. We generalize Nevanlinnas result to an arbitrary set $Omega$. In this case, the function $f$ comes from the Schur-Agler class. The abstract result is then applied to three examples - the bidisc, the symmetrized bidisc and the annulus. In these examples, the Schur-Agler class is the same as the Schur class.
The theory of mean field games is a tool to understand noncooperative dynamic stochastic games with a large number of players. Much of the theory has evolved under conditions ensuring uniqueness of the mean field game Nash equilibrium. However, in so
me situations, typically involving symmetry breaking, non-uniqueness of solutions is an essential feature. To investigate the nature of non-unique solutions, this paper focuses on the technically simple setting where players have one of two states, with continuous time dynamics, and the game is symmetric in the players, and players are restricted to using Markov strategies. All the mean field game Nash equilibria are identified for a symmetric follow the crowd game. Such equilibria correspond to symmetric $epsilon$-Nash Markov equilibria for $N$ players with $epsilon$ converging to zero as $N$ goes to infinity. In contrast to the mean field game, there is a unique Nash equilibrium for finite $N.$ It is shown that fluid limits arising from the Nash equilibria for finite $N$ as $N$ goes to infinity are mean field game Nash equilibria, and evidence is given supporting the conjecture that such limits, among all mean field game Nash equilibria, are the ones that are stable fixed points of the mean field best response mapping.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
