Hilbert Space Of Functions Research Articles

Covariant Integral Quantization of the Semi-Discrete SO(3)-Hypercylinder

Covariant integral quantization with rotational SO(3) symmetry is established for quantum motion on this group manifold. It can also be applied to Gabor signal analysis on this group. The corresponding phase space takes the form of a discrete-continuous hypercylinder. The central tool for implementing this procedure is the Weyl–Gabor operator, a non-unitary operator that operates on the Hilbert space of square-integrable functions on SO(3). This operator serves as the counterpart to the unitary Weyl or displacement operator used in constructing standard Schrödinger–Glauber–Sudarshan coherent states. We unveil a diverse range of properties associated with the quantizations and their corresponding semi-classical phase-space portraits, which are derived from different weight functions on the considered discrete-continuous hypercylinder. Certain classes of these weight functions lead to families of coherent states. Moreover, our approach allows us to define a Wigner distribution, satisfying the standard marginality conditions, along with its related Wigner transform.

Weak convergence of inertial proximal point algorithm for a family of nonexpansive mappings in Hilbert spaces

In this paper, we modified proximal point algorithm with some convex combination technique to approximate a minimizer, equilibrium point and a common fixed point of a family of nonexpansive mappings in Hilbert spaces. We establish a weak convergence theorem under some mild conditions. Moreover, we also provide a numerical example to illustrate the convergence behavior of the proposed iterative method.

Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

N-Best kernel approximation in reproducing kernel Hilbert spaces

By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of n-best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for the corresponding class of Bochner type spaces of stochastic processes. This study thus generalizes the classical result of n-best rational approximation for the Hardy space and a recent result of n-best kernel approximation for the weighted Bergman spaces of the unit disc. The type of approximations have significant applications to signal and image processing and system identification, as well as to numerical solutions of the classical and the stochastic type integral and differential equations.

Hilbert Space Decomposition Properties of Complex Functions and Their Applications

We analyzed the classical problem of decomposing the Hilbert space of holomorphic functions, especially their splitting into the product or sum of domain-separated components. For the Bergman space of analytical functions, we obtained a special decomposition satisfying the assigned growth degree properties. Concerning a general Hilbert space of analytical functions on a connected domain, we studied its α-invariant decomposition and related ergodic consequences. As an interesting consequence, we obtained the decomposition theorem for an ergodic α-mapping on the Bergman space of holomorphic functions.

Inertial reflected Krasnosel’ski-Mann iteration with applications to monotone inclusion problems

This paper studies an inertial reflected method for the approximation of fixed points (assuming existence) of nonexpansive mappings in Hilbert spaces. The proposed method is a combination of Krasnosel'ski-Mann iteration, inertial extrapolation step and reflected step. The aim is to improve and complement various versions of inertial Krasnosel'ski-Mann iterations and the recently proposed reflected Krasnosel'ski-Mann iteration in the literature for approximating fixed points of nonexpansive mappings in Hilbert spaces. Some applications to problems of finding zeros of the sum of monotone operators are given. Finally, numerical simulations including image restoration problems are performed using standard test examples to show the superiority of our proposed method.

Single-Valued Demicontractive Mappings: Half a Century of Developments and Future Prospects

Demicontractive operators form an important class of nonexpansive type mappings whose study led researchers to the creation of some beautiful results in the framework of metric fixed-point theory. This article aims to provide an overview of the most relevant results on the approximation of fixed points of single-valued demicontractive mappings in Hilbert spaces. Subsequently, we exhibit the role of additional properties of demicontractive operators, as well as the main features of the employed iterative algorithms to ensure weak convergence or strong convergence. We also include commentaries on the use of demicontractive mappings to solve some important nonlinear problems with the aim of providing a comprehensive starting point to readers who are attempting to apply demicontractive mappings to concrete applications. We conclude with some brief statements on our view on relevant and promising directions of research on demicontractive mappings in nonlinear settings (metric spaces) and some application challenges.

Higher Order Difference Operators and Associated Relative Reproducing Kernel Hilbert Spaces

We study multiple notions of Hilbert spaces of functions which, via the respective inner products, reproduce function values, or differences of function values. We do this by extending results from the more familiar settings of reproducing kernel Hilbert spaces, RKHSs. Our main results deal with operations on infinite graphs of vertices and edges, and associated Hilbert spaces. For electrical network models, the differences represent voltage differences for pairs of vertices x, y. In these cases, relative RKHSs will depend on choices of conductance functions c, where an appropriate function c is specified as a positive function defined on the edge-set E from G. Our present study of higher order differences, using choices of relative RKHSs, is motivated in part by existing numerical algorithms for discretization of PDEs. Our approach to higher order differences uses both combinatorial operations on graphs, and operator theory for the respective RKHSs. Starting with a graph , we introduce an induced graph such that the vertices in are the edges in E from G, while the edges in are pairs of neighboring edges from G.

Confounder Adjustment in Shape-on-Scalar Regression Model: Corpus Callosum Shape Alterations in Alzheimer’s Disease

Large-scale imaging studies often face challenges stemming from heterogeneity arising from differences in geographic location, instrumental setups, image acquisition protocols, study design, and latent variables that remain undisclosed. While numerous regression models have been developed to elucidate the interplay between imaging responses and relevant covariates, limited attention has been devoted to cases where the imaging responses pertain to the domain of shape. This adds complexity to the problem of imaging heterogeneity, primarily due to the unique properties inherent to shape representations, including nonlinearity, high-dimensionality, and the intricacies of quotient space geometry. To tackle this intricate issue, we propose a novel approach: a shape-on-scalar regression model that incorporates confounder adjustment. In particular, we leverage the square root velocity function to extract elastic shape representations which are embedded within the linear Hilbert space of square integrable functions. Subsequently, we introduce a shape regression model aimed at characterizing the intricate relationship between elastic shapes and covariates of interest, all while effectively managing the challenges posed by imaging heterogeneity. We develop comprehensive procedures for estimating and making inferences about the unknown model parameters. Through real-data analysis, our method demonstrates its superiority in terms of estimation accuracy when compared to existing approaches.

Analysis of the single reference coupled cluster method for electronic structure calculations: the full-coupled cluster equations

The central problem in electronic structure theory is the computation of the eigenvalues of the electronic Hamiltonian—a semi-unbounded, self-adjoint operator acting on an L^2-type Hilbert space of antisymmetric functions. Coupled cluster (CC) methods, which are based on a non-linear parameterisation of the sought-after eigenfunction and result in non-linear systems of equations, are the method of choice for high-accuracy quantum chemical simulations. The existing numerical analysis of coupled cluster methods relies on a local, strong monotonicity property of the CC function that is valid only in a perturbative regime, i.e., when the sought-after ground state CC solution is sufficiently close to zero. In this article, we introduce a new well-posedness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Under the minimal assumption that the sought-after eigenfunction is intermediately normalisable and the associated eigenvalue is isolated and non-degenerate, we prove that the continuous (infinite-dimensional) CC equations are always locally well-posed. Under the same minimal assumptions and provided that the discretisation is fine enough, we prove that the discrete Full-CC equations are locally well-posed, and we derive residual-based error estimates with guaranteed positive constants. Preliminary numerical experiments indicate that the constants that appear in our estimates are a significant improvement over those obtained from the local monotonicity approach.

Modified-operator method for the calculation of band diagrams of crystalline materials

In solid state physics, electronic properties of crystalline materials are often inferred from the spectrum of periodic Schrödinger operators. As a consequence of Bloch’s theorem, the numerical computation of electronic quantities of interest involves computing derivatives or integrals over the Brillouin zone of so-called energy bands, which are piecewise smooth, Lipschitz continuous periodic functions obtained by solving a parametrized elliptic eigenvalue problem on a Hilbert space of periodic functions. Classical discretization strategies for resolving these eigenvalue problems produce approximate energy bands that are either non-periodic or discontinuous, both of which cause difficulty when computing numerical derivatives or employing numerical quadrature. In this article, we study an alternative discretization strategy based on an ad hoc operator modification approach. While specific instances of this approach have been proposed in the physics literature, we introduce here a systematic formulation of this operator modification approach. We derive a priori error estimates for the resulting energy bands and we show that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach. Numerical experiments involving a toy model in 1D, graphene in 2D, and silicon in 3D validate our theoretical results and showcase the efficiency of the operator modification approach.

Composition-differentiation operators acting on certain Hilbert spaces of analytic functions

We study composition-differentiation operators acting on the Bergman and Dirichlet space of the open unit disk. We first characterize the compactness of composition-differentiation operator on weighted Bergman spaces. We shall then prove that for an analytic self-map $\varphi$ on the open unit disk $\mathbb{D}$, the induced composition-differentiation operator is bounded with dense range if and only if $\varphi$ is univalent and the polynomials are dense in the Bergman space on $\Omega:=\varphi(\mathbb{D})$.

A geometrical index for the group Sn and its applications in periodic solutions of differential and differential-difference equations

In this paper, we construct a new index theory, Sn index theory, which is an improvement of the S1 index theory given by Benci in [1]. Also S1 index theory is a powerful tool in the study of the multiplicity of periodic orbits of autonomous differential equations, the definition of index confined the functional space being composed by functions whose mean values must zero. Under the condition that the functional Φ is even, our new index theory can be applied to study the multiplicity of periodic orbits in Hilbert space of functions without the restriction of mean value zero. After the shift of the origin it can be applied to the multiplicity of periodic orbits with a nonzero mean value. Besides, we give in this paper a delay differential system as a complete example to show the calculation of the multiplicity of periodic orbits.

Topological transitivity of translation operators in a non-separable Hilbert space

We consider a Hilbert space of entire analytic functions on a non-separable Hilbert space, associated with a non-separable Fock space. We show that under some conditions operators, like the differentiation operators and translation operators, are topologically transitive in this space.

Modified Mann-Type Algorithm for Two Countable Families of Nonexpansive Mappings and Application to Monotone Inclusion and Image Restoration Problems

In this paper, we introduce and study a modified Mann-type algorithm that combines inertial terms for solving common fixed point problems of two countable families of nonexpansive mappings in Hilbert spaces. Under appropriate assumptions on the sequences of parameters, we establish a strong convergence result for the sequence generated by the proposed method in finding a common fixed point of two countable families of nonexpansive mappings. This method can be applied to solve the monotone inclusion problem. Additionally, we employ a modified Mann-type iterative algorithm to address image restoration problems. Furthermore, we present numerical results across different scenarios to demonstrate the superior efficiency of our algorithm compared to existing algorithms.

On solving split equilibrium problems and fixed point problems of α-nonexpansive multi-valued mappings in Hilbert spaces

On solving split equilibrium problems and fixed point problems of α-nonexpansive multi-valued mappings in Hilbert spaces

Applications of Alughte Transform for Berezin Radius Inequalities

– In functional analysis, linear operators induced by functions are frequently encountered; thesecontain Hankel operators, constitution operators, and Toeplitz operators. The symbol of the resultantoperator is another name for the inciting function. In many instances, a linear operator on a Hilbert spaceℋ results in a function on a subset of a topological space. As a result, we regularly investigate operatorsinduced by functions, and we may also investigate functions induced by operators. The Berezin sign is awonderful representation of an operator-function relationship. F. Berezin proposed the Berezin switch in[8], and it has proven to be a vital tool in operator theory given that it utilizes many essential aspects ofsignificant operators. Many mathematicians and physicists are fascinated by the Berezin symbol of anoperator defined on the functional Hilbert space. The Berezin radius inequality has been extensively studiedin this situation by a number of mathematicians. In this paper, we use the Alughte transform and thegeneralized Alughte transform to develop Berezin radius inequalities for Hilbert space operators. Weadditionally offer fresh Berezin radius inequality results. Huban et al. [15] and Başaran et al. [6] supply theBerezin radius inequality.

Power inequalities for the Berezin radius of operators in functional Hilbert spaces

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Berezin Quantization, Conformal Welding and the Bott–Virasoro Group

Following Nag–Sullivan, we study the representation of the group Diff+(S1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Diff}^+(S^1)$$\\end{document} of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space. This lift provides quantization of conformal welding, gives a new representative of the Bott–Virasoso cocycle class, and leads to a surprising identity for the Takhtajan–Teo energy functional on Diff+(S1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Diff}^+(S^1)$$\\end{document}.

Iterative Approximation of Common Fixed Points for Edge-Preserving Quasi-Nonexpansive Mappings in Hilbert Spaces along with Directed Graph

We present iterative approximation results of an iterative scheme for finding common fixed points of edge-preserving quasi-nonexpansive self-maps in Hilbert spaces along with directed graph. We obtain weak as well as strong convergence of our scheme under various assumptions. That is, we impose several possible mild conditions on the domain, on the mapping, or on the parameters involved in our scheme to prove convergence results. We support numerically our main outcome by giving an example. Eventually, an application is provided for solving a variational inequality problem. Our result are new/generalized some recently announced results of the literature.