Search results for "Hilbert space"
showing 10 items of 234 documents
Hilbert Space Embeddings for Gelfand–Shilov and Pilipović Spaces
2017
We consider quasi-Banach spaces that lie between a Gelfand–Shilov space, or more generally, Pilipovi´c space, \(\mathcal{H}\), and its dual, \(\mathcal{H}^\prime\) . We prove that for such quasi-Banach space \(\mathcal{B}\), there are convenient Hilbert spaces, \(\mathcal{H}_{k}, k=1,2\), with normalized Hermite functions as orthonormal bases and such that \(\mathcal{B}\) lies between \(\mathcal{H}_1\; \mathrm{and}\;\mathcal{H}_2\), and the latter spaces lie between \(\mathcal{H}\; \mathrm{and}\;\mathcal{H}^\prime\).
Weak commutation relations of unbounded operators and applications
2011
Four possible definitions of the commutation relation $[S,T]=\Id$ of two closable unbounded operators $S,T$ are compared. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the operators act. Some consequences on the existence of eigenvectors of two number-like operators are derived and the partial O*-algebra generated by $S,T$ is studied. Some applications are also considered.
Quantum like modelling of decision making: quantifying uncertainty with the aid of the Heisenberg-Robertson inequality
2018
This paper contributes to quantum-like modeling of decision making (DM) under uncertainty through application of Heisenberg’s uncertainty principle (in the form of the Robertson inequality). In this paper we apply this instrument to quantify uncertainty in DM performed by quantum-like agents. As an example, we apply the Heisenberg uncertainty principle to the determination of mutual interrelation of uncertainties for “incompatible questions” used to be asked in political opinion pools. We also consider the problem of representation of decision problems, e.g., in the form of questions, by Hermitian operators, commuting and noncommuting, corresponding to compatible and incompatible questions …
Upport vector machines for nonlinear kernel ARMA system identification.
2006
Nonlinear system identification based on support vector machines (SVM) has been usually addressed by means of the standard SVM regression (SVR), which can be seen as an implicit nonlinear autoregressive and moving average (ARMA) model in some reproducing kernel Hilbert space (RKHS). The proposal of this letter is twofold. First, the explicit consideration of an ARMA model in an RKHS (SVM-ARMA 2k) is proposed. We show that stating the ARMA equations in an RKHS leads to solving the regularized normal equations in that RKHS, in terms of the autocorrelation and cross correlation of the (nonlinearly) transformed input and output discrete time processes. Second, a general class of SVM-based syste…
Optimizing Kernel Ridge Regression for Remote Sensing Problems
2018
Kernel methods have been very successful in remote sensing problems because of their ability to deal with high dimensional non-linear data. However, they are computationally expensive to train when a large amount of samples are used. In this context, while the amount of available remote sensing data has constantly increased, the size of training sets in kernel methods is usually restricted to few thousand samples. In this work, we modified the kernel ridge regression (KRR) training procedure to deal with large scale datasets. In addition, the basis functions in the reproducing kernel Hilbert space are defined as parameters to be also optimized during the training process. This extends the n…
Intertwining operators between different Hilbert spaces: connection with frames
2009
In this paper we generalize a strategy recently proposed by the author concerning intertwining operators. In particular we discuss the possibility of extending our previous results in such a way to construct (almost) isospectral self-adjoint operators living in different Hilbert spaces. Many examples are discussed in details. Many of them arise from the theory of frames in Hilbert spaces, others from the so-called g-frames.
A concise review on pseudo-bosons, pseudo-fermions and their relatives
2017
We review some basic definitions and few facts recently established for $\D$-pseudo bosons and for pseudo-fermions. We also discuss an extended version of these latter, based on biorthogonal bases, which lives in a finite dimensional Hilbert space. Some examples are described in details.
Some remarks on unconditionally convergent multipliers
2017
We present some results concerning the representation of unconditionally convergent multipliers, including a reformulation of a conjecture of Balazs and Stoeva.
A new mathematical tool for an exact treatment of open quantum system dynamics
2005
A new method to obtain an operatorial exact solution of a wide class of Markovian master equations is presented. Its key point is the existence of a constant of motion partitioning the Hilbert space into finite-dimensional subspaces. The consequent possibility of representing the reduced density operator as a block diagonal matrix is shown. Each “block operator” evolves under the action of a non-unitary operator explicitly derived. Our mathematical approach is illustrated applying it to simple physical systems.
Orbits of bounded bijective operators and Gabor frames
2020
This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of $L^2(\mathbb{R})$, which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete Gabor frames, with some ordering over $\mathbb{Z}$, which are orbits of bounded operators on $L^2(\mathbb{R})$. Two classes of overcomplete Gabor frames which cannot be ordered over $\mathbb{Z}$ and represented by orbits of operators in $GL(L^2(\mathbb{R}))$ are given. Some results about opera…