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Register a new userInjectivity and surjectivity of the Stieltjes moment mapping in Gelfand-Shilov spaces
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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, in terms of growth conditions for the defining weight sequence. Finally, a related moment problem at the origin is studied.
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We consider r-ramification ultradifferentiable classes, introduced by J. Schmets and M. Valdivia in order to study the surjectivity of the Borel map, and later on also exploited by the authors in the ultraholomorphic context. We characterize quasianalyticity in such classes, extend the results of Schmets and Valdivia about the image of the Borel map in a mixed ultradifferentiable setting, and obtain a version of the Whitney extension theorem in this framework.
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 $[(rn)!]^{p}$, $p=3,4,...$.
We are concerned with surjectivity of perturbations of maximal monotone operators in non-reflexive Banach spaces. While in a reflexive setting, a classical surjectivity result due to Rockafellar gives a necessary and sufficient condition to maximal monotonicity, in a non-reflexive space we characterize maximality using a ``enlarged version of the duality mapping, introduced previously by Gossez.
We study the injectivity and surjectivity of the Borel map in three instances: in Roumieu-Carleman ultraholomorphic classes in unbounded sectors of the Riemann surface of the logarithm, and in classes of functions admitting, uniform or nonuniform, asymptotic expansion at the corresponding vertex. These classes are defined in terms of a log-convex sequence $mathbb{M}$ of positive real numbers. Injectivity had been solved in two of these cases by S. Mandelbrojt and B. Rodriguez-Salinas, respectively, and we completely solve the third one by means of the theory of proximate orders. A growth index $omega(mathbb{M})$ turns out to put apart the values of the opening of the sector for which injectivity holds or not. In the case of surjectivity, only some partial results were available by J. Schmets and M. Valdivia and by V. Thilliez, and this last author introduced an index $gamma(mathbb{M})$ (generally different from $omega(mathbb{M})$) for this problem, whose optimality was not established except for the Gevrey case. We considerably extend here their results, proving that $gamma(mathbb{M})$ is indeed optimal in some standard situations (for example, as far as $mathbb{M}$ is strongly regular) and puts apart the values of the opening of the sector for which surjectivity holds or not.
We prove the exponential law $mathcal A(E times F, G) cong mathcal A(E,mathcal A(F,G))$ (bornological isomorphism) for the following classes $mathcal A$ of test functions: $mathcal B$ (globally bounded derivatives), $W^{infty,p}$ (globally $p$-integrable derivatives), $mathcal S$ (Schwartz space), $mathcal D$ (compact sport, $mathcal B^{[M]}$ (globally Denjoy_Carleman), $W^{[M],p}$ (Sobolev_Denjoy_Carleman), $mathcal S_{[L]}^{[M]}$ (Gelfand_Shilov), and $mathcal D^{[M]}$. Here $E, F, G$ are convenient vector spaces (finite dimensional in the cases of $W^{infty,p}$, $mathcal D$, $W^{[M],p}$, and $mathcal D^{[M]})$, and $M=(M_k)$ is a weakly log-convex weight sequence of moderate growth. As application we give a new simple proof of the fact that the groups of diffeomorphisms $operatorname{Diff} mathcal B$, $operatorname{Diff} W^{infty,p}$, $operatorname{Diff} mathcal S$, and $operatorname{Diff}mathcal D$ are $C^infty$ Lie groups, and $operatorname{Diff} mathcal B^{{M}}$, $operatorname{Diff}W^{{M},p}$, $operatorname{Diff} mathcal S_{{L}}^{{M}}$, and $operatorname{Diff}mathcal D^{[M]}$, for non-quasianalytic $M$, are $C^{{M}}$ Lie groups, where $operatorname{Diff}mathcal A = {operatorname{Id} +f : f in mathcal A(mathbb R^n,mathbb R^n), inf_{x in mathbb R^n} det(mathbb I_n+ df(x))>0}$. We also discuss stability under composition.
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