Weighted Fourier–Laplace transforms in reproducing kernel Hilbert spaces on the sphere

We study the action of a weighted Fourier–Laplace transform on the functions in the reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier–Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier–Laplace transforms: fractional derivatives and Laplace–Beltrami derivatives.

Pseudodifferential operators on Hilbert space riggings with associated psi *-algebras and generalized Hörmander classes

The present thesis is concerned with certain aspects of differential and pseudodifferential operators on infinite dimensional spaces. We aim to generalize classical operator theoretical concepts of pseudodifferential operators on finite dimensional spaces to the infinite dimensional case. At first we summarize some facts about the canonical Gaussian measures on infinite dimensional Hilbert space riggings. Considering the naturally unitary group actions in $L^2(H_-,gamma)$ given by weighted shifts and multiplication with $e^{iSkp{t}{cdot}_0}$ we obtain an unitary equivalence $F$ between them. In this sense $F$ can be considered as an abstract Fourier transform. We show that $F$ coincides with the Fourier-Wiener transform. Using the Fourier-Wiener transform we define pseudodifferential operators in Weyl- and Kohn-Nirenberg form on our Hilbert space rigging. In the case of this Gaussian measure $gamma$ we discuss several possible Laplacians, at first the Ornstein-Uhlenbeck operator and then pseudo-differential operators with negative definite symbol. In the second case, these operators are generators of $L^2_gamma$-sub-Markovian semi-groups and $L^2_gamma$-Dirichlet-forms. In 1992 Gramsch, Ueberberg and Wagner described a construction of generalized Hörmander classes by commutator methods. Following this concept and the classical finite dimensional description of $Psi_{ro,delta}^0$ ($0leqdeltaleqroleq 1$, $delta< 1$) in the $C^*$-algebra $L(L^2)$ by Beals and Cordes we construct in both cases generalized Hörmander classes, which are $Psi^*$-algebras. These classes act on a scale of Sobolev spaces, generated by our Laplacian. In the case of the Ornstein-Uhlenbeck operator, we prove that a large class of continuous pseudodifferential operators considered by Albeverio and Dalecky in 1998 is contained in our generalized Hörmander class. Furthermore, in the case of a Laplacian with negative definite symbol, we develop a symbolic calculus for our operators. We show some Fredholm-criteria for them and prove that these Fredholm-operators are hypoelliptic. Moreover, in the finite dimensional case, using the Gaussian-measure instead of the Lebesgue-measure the index of these Fredholm operators is still given by Fedosov's formula. Considering an infinite dimensional Heisenberg group rigging we discuss the connection of some representations of the Heisenberg group to pseudo-differential operators on infinite dimensional spaces. We use this connections to calculate the spectrum of pseudodifferential operators and to construct generalized Hörmander classes given by smooth elements which are spectrally invariant in $L^2(H_-,gamma)$. Finally, given a topological space $X$ with Borel measure $mu$, a locally compact group $G$ and a representation $B$ of $G$ in the group of all homeomorphisms of $X$, we construct a Borel measure $mu_s$ on $X$ which is invariant under $B(G)$.

Teoria spettrale in spazi di Hilbert

In questa tesi ci si occuperà di presentare alcuni aspetti salienti della teoria spettrale per gli operatori limitati negli spazi di Hilbert. Nel primo capitolo verranno presentate alcune nozioni fondamentali di analisi funzionale, necessarie per lo studio degli operatori. Il secondo capitolo si occupa invece di analizzare la teoria spettrale per operatori compatti. In particolare, verrà presentato il Teorema Spettrale per Operatori Normali Compatti e il Teorema dell'Alternativa di Fredholm. In seguito verrà applicata tale teoria alla risolubilità del problema di Dirichlet. Nel terzo capitolo verrà esteso quanto ottenuto per gli operatori compatti ad operatori limitati autoaggiunti e per gli operatori normali limitati, passando attraverso le famiglie spettrali.

Hilbert transform

The Hilbert transform is an important tool in both pure and applied mathematics. It is largely used in the field of signal processing. Lately has been used in mathematical finance as the fast Hilbert transform method is an efficient and accurate algorithm for pricing discretely monitored barrier and Bermudan style options. The purpose of this report is to show the basic properties of the Hilbert transform and to check the domain of definition of this operator.

Basi ortonormali negli spazi di Hilbert

Questo elaborato presenta gli elementi di base della Teoria degli Spazi di Hilbert, con particolare attenzione al Teorema della Proiezione sui convessi e ai sistemi ortonormali completi.

The Poincaré polynomial for the Hilbert scheme of points of C^2

Questo lavoro prende in esame lo schema di Hilbert di punti di C^2, il quale viene descritto assieme ad alcune sue proprietà, ad esempio la sua struttura hyper-kahleriana. Lo scopo della tesi è lo studio del polinomio di Poincaré di tale schema di Hilbert: ciò che si ottiene è una espressione del tipo serie di potenze, la quale è un caso particolare di una formula molto più generale, nota con il nome di formula di Goettsche.

Knowledge Creation as a Square Dance on the Hilbert Cube

This paper presents a micro-model of knowledge creation and transfer in a small group of people. Our model incorporates two key aspects of the cooperative process of knowledge creation: (i) heterogeneity of people in their state of knowledge is essential for successful cooperation in the joint creation of new ideas, while (ii) the very process of cooperative knowledge creation a¤ects the heterogeneity of people through the accumulation of knowledge in common. The model features myopic agents in a pure externality model of interaction. In the two person case, we show that the equilibrium process tends to result in the accumulation of too much knowledge in common compared to the most productive state. Unlike the two-person case, in the four person case we show that the equilibrium process of knowledge creation may converge to the most productive state. Equilibrium paths are found analytically, and they are a discontinuous function of initial heterogeneity.

First-order equivalent to Einstein-Hilbert Lagrangian

A first-order Lagrangian L ∇ variationally equivalent to the second-order Einstein- Hilbert Lagrangian is introduced. Such a Lagrangian depends on a symmetric linear connection, but the dependence is covariant under diffeomorphisms. The variational problem defined by L ∇ is proved to be regular and its Hamiltonian formulation is studied, including its covariant Hamiltonian attached to ∇ .