Related papers: RKHS, Berezin and Odzijewicz-type quantizations on…
We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of…
We present a N-dimensional quantization a la Berezin-Klauder or frame quantization of the complex plane based on overcomplete families of states (coherent states) generated by the N first harmonic oscillator eigenstates. The spectra of…
Generalized flag manifolds endowed with the Bruhat-Poisson bracket are compact Poisson homogeneous spaces, whose decompositions in symplectic leaves coincide with their stratifications in Schubert cells. In this note it is proved that the…
Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of…
We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact K\"ahler manifolds under deformations of the complex structure. We show that the…
In the theory of Toeplitz quantization of algebras, as developed by the second author, coherent states are defined as eigenvectors of a Toeplitz annihilation operator. These coherent states are studied in the case when the algebra is the…
Ambiguities arising in different approaches (canonical, quasiclassical, path integration) to quantization are discussed by an example of the mechanics of a point-like particle in the Riemannian space (the geodesic dynamics). A way to select…
It is often easier to study pseudo-Riemannian manifolds by presenting them as surfaces in some ambient space. We propose an algorithm for construction of explicit isometric embeddings of pseudo-Riemannian manifolds with symmetries into an…
We study equivariant localization formulas for phase space path integrals when the phase space is a multiply connected compact Riemann surface. We consider the Hamiltonian systems to which the localization formulas are applicable and show…
In this article, we completely characterize the Berezin range of Toeplitz operators with harmonic symbols acting on weighted Bergman spaces, illustrating the necessity of the harmonicity condition through examples. We then introduce a new…
Exploiting a notion of Kaehler structure on a stratified space introduced elsewhere we show that, in the Kaehler case, reduction after quantization coincides with quantization after reduction: Key tools developed for that purpose are…
We prove a characterization for the Peetre type $K$-functional on $\mathbb{M}$, a compact two-point homogeneous space, in terms the rate of approximation of a family of multipliers operator defined to this purpose. This extends the well…
These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing…
A closed subset of $\mathbb{R}^q$, definable in some given o-minimal structure, is Lipschitz normally embedded in $\mathbb{R}^q$ if and only if its one-point compactification is Lipschitz normally embedded in the unit sphere ${\bf S}^q$($ =…
The geometrical description of a Hilbert space asociated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a…
We introduce a nonlinear potential theory problem for the Laplacian, the solution of which characterizes the Berezin density $B(z,\cdot)$ for the polynomial Bergman space, where the point $z\in\mathbb{C}$ is fixed. When $z=\infty$, the…
Geometric quantization is an attempt at using the differential-geometric ingredients of classical phase spaces regarded as symplectic manifolds in order to define a corresponding quantum theory. Generally, the process of geometric…
The formulation of Geometric Quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the…
Generalizing previous results for orbifolds, in this paper we describe the compactification of Matrix model on an orientifold which is a quotient space as a Yang-Mills theory living on a quantum space. The information of the…
Plemelj projection operators are introduced for spaces of square integrable functions defined over the boundaries of a class of compact real n-dimensional manifolds lying in C^n. These manifolds posses many properties similar to domains in…