Coherent states and projective representation of the linear canonical

... intrinsic and coordinate-free notations differing from the notations used in Refs. 2, 3, or 4. At the end of the paper we rewrite some of the results in the more familiar x-p notations. Following the prescription given in Ref. 11 for the dequantization of these operators, we proceed then to compute ...

... intrinsic and coordinate-free notations differing from the notations used in Refs. 2, 3, or 4. At the end of the paper we rewrite some of the results in the more familiar x-p notations. Following the prescription given in Ref. 11 for the dequantization of these operators, we proceed then to compute ...

Deconvolutions of Gaussian kernels

... series) in the same fashion as the usual exponential function exp(ξ ); ξ may either be a real or complex number. This expansion is referred to as a Lie series of an operator function. In quantum mechanics such operator functions can be defined by the Hamiltonian H, which has to be a continuous (H ma ...

... series) in the same fashion as the usual exponential function exp(ξ ); ξ may either be a real or complex number. This expansion is referred to as a Lie series of an operator function. In quantum mechanics such operator functions can be defined by the Hamiltonian H, which has to be a continuous (H ma ...

Fine Grained Kernel Logging with KLogger

... mechanism, called KLogger schemata, allows kernel developers to share their expertise and insights, thus allowing other researchers to analyze code without having to fully understand its intricacies. The idea behind this design is based on the notion that a high level understanding of a subsystem sh ...

... mechanism, called KLogger schemata, allows kernel developers to share their expertise and insights, thus allowing other researchers to analyze code without having to fully understand its intricacies. The idea behind this design is based on the notion that a high level understanding of a subsystem sh ...

Fine Grained Kernel Logging with KLogger: Experience and Insights

... compilation time. This structured specialization mechanism, called KLogger schemata, allows kernel developers to share their expertise and insights, thus allowing other researchers to analyze code without having to fully understand its intricacies. The idea behind this design is based on the notion ...

... compilation time. This structured specialization mechanism, called KLogger schemata, allows kernel developers to share their expertise and insights, thus allowing other researchers to analyze code without having to fully understand its intricacies. The idea behind this design is based on the notion ...

The path integral representation kernel of evolution operator in

... the path integral representation was constructed for the operator evolution kernel (propagator) of a Merton-Garman Hamiltonian. Meanwhile, while constructing the evolution operator kernel in [6, 7], there were encountered a few shortcomings. In particular, the operator structure of the Merton-Garman ...

... the path integral representation was constructed for the operator evolution kernel (propagator) of a Merton-Garman Hamiltonian. Meanwhile, while constructing the evolution operator kernel in [6, 7], there were encountered a few shortcomings. In particular, the operator structure of the Merton-Garman ...

THE WIGHTMAN AXIOMS AND THE MASS GAP FOR STRONG

... sequence of positive operators uniformly bounded by 1, so that ...

... sequence of positive operators uniformly bounded by 1, so that ...

Self-adjoint operators and solving the Schrödinger equation

... for solving the Schrödinger equation. Specific topics include the spectral theorem and functional calculus for self-adjoint operators, Stone’s Theorem, Laplacians and Fourier transform, Duhamel’s formula and Dyson series, time-evolution of non-interacting quantum systems, as well as the Baker-Campb ...

... for solving the Schrödinger equation. Specific topics include the spectral theorem and functional calculus for self-adjoint operators, Stone’s Theorem, Laplacians and Fourier transform, Duhamel’s formula and Dyson series, time-evolution of non-interacting quantum systems, as well as the Baker-Campb ...

# Reproducing kernel Hilbert space

In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space associated with a kernel that reproduces every function in the space or, equivalently, where every evaluation functional is bounded. The reproducing kernel was first introduced in the 1907 work of Stanisław Zaremba concerning boundary value problems for harmonic and biharmonic functions. James Mercer simultaneously examined functions which satisfy the reproducing property in the theory of integral equations. The idea of the reproducing kernel remained untouched for nearly twenty years until it appeared in the dissertations of Gábor Szegő, Stefan Bergman, and Salomon Bochner. The subject was eventually systematically developed in the early 1950s by Nachman Aronszajn and Stefan Bergman. These spaces have wide applications, including complex analysis, harmonic analysis, and quantum mechanics. Reproducing kernel Hilbert spaces are particularly important in the field of statistical learning theory because of the celebrated Representer theorem which states that every function in an RKHS can be written as a linear combination of the kernel function evaluated at the training points. This is a practically useful result as it effectively simplifies the empirical risk minimization problem from an infinite dimensional to a finite dimensional optimization problem. For ease of understanding, we provide the framework for real-valued Hilbert spaces. The theory can be easily extended to spaces of complex-valued functions and hence include the many important examples of reproducing kernel Hilbert spaces that are spaces of analytic functions.