Operators In Spaces Of Functions Research Articles

Conformable Gehring inequalities and conformable higher integrability

In this paper, we prove some reverse conformable inequalities with weights and employ them to prove some conformable inequalities of Gehring type. Moreover, we prove some interpolation theorems which are powerful tools in\ the study of operators in function spaces. Our results develop a technique based on the applications of a refinement of conformable inequalities.

Fixed point equations with abstract Volterra operators on spaces of functions of several variables

Several notions of abstract Volterra operators on spaces of functions of one variable are well known. In this paper, we introduce various notions of abstract Volterra operators in spaces of functions of several variables. Some fixed point equations with such abstract Volterra operators are also studied. The basic ingredient in the theory of step by step contraction is the notion of G-contraction. The relevance of step by step contraction principle is illustrated by applications in the theory of Darboux-Ionescu problem.

Bilevel Parameter Learning for Nonlocal Image Denoising Models

We propose a bilevel optimization approach for the estimation of parameters in nonlocal image denoising models. The parameters we consider are both the fidelity weight and weights within the kernel of the nonlocal operator. In both cases we investigate the differentiability of the solution operator in function spaces and derive a first order optimality system that characterizes local minima. For the numerical solution of the problems, we use a second-order trust-region algorithm in combination with a finite element discretization of the nonlocal denoising models and we introduce a computational strategy for the solution of the resulting dense linear systems. Several experiments illustrate the applicability and effectiveness of our approach.

Операторы Хаусдорфа специального вида в пространствах типа BMO и пространствах Гёльдера-Липшица

We show that some BMO-type spaces are invariant with respect to Hausdorff operators of a special kind (weighted Hardy–Cesaro operators). Moreover, we establish the necessary and sufficient conditions for the boundedness of such operators in spaces of functions of generalized bounded variation. We study the invariance of Holder–Lipschitz spaces with respect to these operators.

Maximal regularity of the Stokes system with Navier boundary condition in general unbounded domains

Consider the instationary Stokes system in general unbounded domains $\Omega \subset \mathbb{R}^n$, $n \geq 2$, with boundary of uniform class $C^3$, and Navier slip or Robin boundary condition. The main result of this article is the maximal regularity of the Stokes operator in function spaces of the type $\tilde{L}^q$ defined as $L^q \cap L^2$ when $q \geq 2$, but as $L^q + L^2$ when $1 < q < 2$, adapted to the unboundedness of the domain.

General Form of Generalized Invertible Operators in Banach Spaces

We prove theorems on the general form of generalized invertible operators in Banach spaces in the case where the operators are topologically Noetherian or topologically Fredholm. These theorems generalize the well-known Nikol’skii theorem on the general form of Fredholm operators and the Atkinson theorem on the general form of Noetherian operators in function spaces.

A unifying approach for the study of linear sampling type operators in function spaces

AbstractSampling series play a crucial role in Signal and Image Processing. In this note, we illustrate a general method to construct Sampling series and to determine their convergence in Orlicz spaces. (© 2015 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Time-harmonic acoustic wave scattering in an ocean with depth-dependent sound speed

Time-harmonic acoustic wave propagation in an inhomogeneous ocean with depth-dependent sound speed can be modeled by the Helmholtz equation in an infinite, three-dimensional (3D) waveguide of finite height. Using variational theory in Sobolev spaces we prove well posedness of the corresponding scattering problem from a bounded inhomogeneity inside such an ocean. To this end, we introduce an exterior Dirichlet-to-Neumann operator for depth-dependent sound speed and prove boundedness, coercivity, and holomorphic dependence of this operator in function spaces adapted to our weak solution theory. Analytic Fredholm theory then yields existence and uniqueness of solution for the scattering problem for all but a countable sequence of frequencies. The latter result generalizes corresponding theory for waveguide scattering with constant sound speed and easily extends to various related scattering problems, e.g. to scattering from impenetrable obstacles.

Maps with separable dynamics and the spectral properties of the operators generated by them

A map of a space into itself generates weighted shift operators in function spaces on . The spectral properties of such operators are intimately connected with the dynamics of . It was known previously that the spectrum of an operator depends only on the set of invariant ergodic measures for . Conditions for the right invertibility of the operators are obtained when is a spectral value. The main result states that right invertibility is only possible when a nontrivial attractor exists. Bibliography: 29 titles.

Newton--Picard Preconditioners for Time-Periodic Parabolic Optimal Control Problems

We prove existence and uniqueness of solutions of an optimization problem with time-periodic parabolic partial differential equation constraints and show that the solution inherits high smoothness properties from the given data. We use the theory of semigroups in conjunction with spectral decompositions of their generators in order to derive detailed representation formulas for shooting operators in function space and their adjoints. A spectral truncation approach delivers a self-adjoint indefinite Newton--Picard preconditioner for the saddle-point system of optimality conditions in function space. We show that this preconditioner leads to convergence in a function space fixed-point iteration. Moreover, we discuss that this preconditioner can be approximated well by a two-grid approach. We address some implementation issues and present numerical results for three-dimensional instationary problems with more than 100,000,000 degrees of freedom.

Convolution operators in A−∞ for convex domains

We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ n and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.

Abstract Interpolation in Vector-Valued de Branges–Rovnyak Spaces

Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and problems in function theory, vol 146, pp 83-69, Naukova Dumka, Keiv, 1987) for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with an operator-valued Schur class function $S$. A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The interpolation problem studied here contains as particular cases (1) the vector-valued version of the interpolation problem with operator argument considered recently in Ball et al. (Proc Am Math Soc 139(2), 609-618, 2011) (for the nondegenerate and scalar-valued case) and (2) a boundary interpolation problem in $\mathcal{H}(K_S)$. In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.

Locally Lipschitz composition operators in spaces of functions of bounded variation

We give a necessary and sufficient condition on a function \({f:\mathbb{R}\to\mathbb{R}}\) under which the composition operator (Nemytskij operator) F defined by \({Fh=f\circ h}\) acts in the spaces \({BV_\varphi[a,b], HBV[a,b], {\rm and} \,RV_\varphi[a,b]}\) and satisfies a local Lipschitz condition. While the proof of sufficiency consists in a straightforward calculation, the proof of necessity builds on nontrivial arguments like Helly’s selection principle or the Arzelà–Ascoli compactness criterion.

Sequences of composition operators in spaces of functions of bounded Φ-variation

The main results of the paper are contained in Theorems 1 and 2. Theorem 1 presents necessary and sufficient conditions for a sequence of functions hn: 〈c, d〉 → 〈a, b〉, n = 1, 2, ..., to have bounded sequences of Ψ-variations {VΨ (〈c, d〉; f ◦ hn)}n=1∞ evaluated for the compositions of an arbitrary function f: 〈a, b〉 → ℝ with finite Φ-variation and the functions hn. In Theorem 2, the same is done for a sequence of functions hn: ℝ → ℝ, n = 1, 2, ..., and the sequence of Ψ-variations {VΨ(〈a, b〉; hn ◦ f)}n=1∞.

Gantmacher‐Kreĭn theorem for 2 nonnegative operators in spaces of functions

The existence of the second (according to the module) eigenvalueλ2of a completely continuous nonnegative operatorAis proved under the conditions thatAacts in the spaceLp(Ω)orC(Ω)and its exterior squareA∧Ais also nonnegative. For the case when the operatorsAandA∧Aare indecomposable, the simplicity of the first and second eigenvalues is proved, and the interrelation between the indices of imprimitivity ofAandA∧Ais examined. For the case whenAandA∧Aare primitive, the difference (according to the module) ofλ1andλ2from each other and from other eigenvalues is proved.

Boundary regularity of admissible operators

In strictly pseudoconvex domains with smooth boundary, we prove a commutator relationship between admissible integral operators, as introduced by Lieb and Range, and smooth vector fields which are tangential at boundary points. This makes it possible to gain estimates for admissible operators in function spaces which involve tangential derivatives. Examples are given under with circumstances these can be transformed into genuine Sobolev- and Ck-estimates.

Poisson transformation for one-sheeted hyperboloids

This paper is concerned with the study of the Poisson transformation and with finding the complete asymptotic series at infinity for this transformation (not just its dominant term) for an important subclass of the class of semisimple symmetric spaces with non-compact : the real hyperbolic spaces (one-sheeted hyperboloids) . The expansions are obtained for arbitrary (not necessarily -finite) functions. The operators involved in the coefficients of the expansions and acting on functions at the boundary are described. The reducibility and the irreducibility of eigensubspaces of the Laplace-Beltrami operator in function spaces on the hyperboloid are studied. The structure of representations acting in spaces of eigenfunctions is described. Descriptions of the kernel and the closure of the range of the Poisson transformation are presented.

Semismooth Newton Methods for Operator Equations in Function Spaces

We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes nonlinear complementarity problem (NCP)-function-based reformulations of infinite-dimensional nonlinear complementarity problems and thus covers a very comprehensive class of applications. Our results generalize semismoothness and $\alpha$-order semismoothness from finite-dimensional spaces to a Banach space setting. For this purpose, a new infinite-dimensional generalized differential is used that is motivated by Qi's finite-dimensional C-subdifferential [Research Report AMR96/5, School of Mathematics, University of New South Wales, Australia, 1996]. We apply these semismoothness results to develop a Newton-like method for nonsmooth operator equations and prove its local q-superlinear convergence to regular solutions. If the underlying operator is $\alpha$-order semismooth, convergence of q-order $1+\alpha$ is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrative examples and by applications to NCPs and a constrained optimal control problem.

Finite-Dimensional Approximations of Operators in the Hilbert Spaces of Functions on Locally Compact Abelian Groups

A new approach to the approximation of operators in the Hilbert space of functions on a locally compact Abelian (LCA) group is developed. This approach is based on sampling the symbols of such operators. To choose the points for sampling, we use the approximations of LCA groups by finite groups, which were introduced and investigated by Gordon. In the case of the group R n , the constructed approximations include the finite-dimensional approximations of the coordinate and linear momentum operators, suggested by Schwinger. The finite-dimensional approximations of the Schrodinger operator based on Schwinger's approximations were considered by Digernes, Varadarajan, and Varadhan in Rev. Math. Phys. 6 (4) (1994), 621–648 where the convergence of eigenvectors and eigenvalues of the approximating operators to those of the Schrodinger operator was proved in the case of a positive potential increasing at infinity. Here this result is extended to the case of Schrodinger-type operators in the Hilbert space of functions on LCA groups. We consider the approximations of p-adic Schrodinger operators as an example. For the investigation of the constructed approximations, the methods of nonstandard analysis are used.

Pseudodifferential Operators in Function Spaces with Exponential Weights

This paper is a continuation of [17]. We study weighted function spaces of type Bapq(u) and Fapq(u) on the Euclidean space ℝn, where u is a weight function of at most exponential growth. In particular, u(x) = exp(±|x|) is an admissible weight. We consider symbols which belong to the Hormander class Su1,δ, where u ∈ ℝ and 0 ≤ δ ≤ 1. We give sufficient conditions for the boundedness of the corresponding pseudodifferential operators in the above function spaces. As a main tool, we use molecular decompositions of these spaces. Furthermore, we prove that the spaces Bapq (u) and Fapq(u) have the lift property.