Hilbert Space Of Functions Research Articles

The Iterative Method for Generalized Equilibrium Problems and a Finite Family of Lipschitzian Mappings in Hilbert Spaces

In this research, we introduced the S‐mapping generated by a finite family of contractive mappings, Lipschitzian mappings and finite real numbers using the results of Kangtunyakarn (2013). Then, we prove the strong convergence theorem for fixed point sets of finite family of contraction and Lipschitzian mapping and solution sets of the modified generalized equilibrium problem introduced by Suwannaut and Kangtunyakarn (2014). Finally, numerical examples are provided to illustrate our main theorem.

Weak stability of Ishikawa iterations for strongly demicontractive mappings in Hilbert spaces

In this article, we establish a weak stability theorem for Ishikawa iterations in Hilbert spaces. Moreover, a strongly demicontractive mapping is presented to illustrate that the Ishikawa iteration of T is weakly T-stable but not T-stable. Our results are new and extend several known results.

New modified hybrid algorithm for pseudo-contractive mappings in Hilbert spaces

New modified hybrid algorithm for pseudo-contractive mappings in Hilbert spaces

Multiplicative property of series used in the Nevanlinna-Pick problem

In the paper we obtained substantially new sufficient condition for negativity of coefficients of power series inverse to series with positive ones. It has been proved that element-wise product of power series retains this property. In particular, it gives rise to generalization of the classical Hardy theorem about power series. These results are generalized for cases of series with multiple variables. Such results are useful in Nevanlinna – Pick theory. For example, if function k(x, y) can be represented as power series Pn≥0 an(x¯y)n, an > 0, and reciprocal function 1/k(x, y) can be represented as power series Pn≥0 bn(x¯y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna – Pick property. The reproducing kernel 1/(1 − x¯y) of the classical Hardy space H2(D) is a prime example for our theorems.

Multiplier tests and subhomogeneity of multiplier algebras

Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space.

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.

Композиции независимых случайных операторов и связанные с ними дифференциальные уравнения

Iterations of independent random linear operators in the Hilbert space of square integrable functions on a finite dimensional Euclidean space are studied. Random operator under consideration take values in the algebra of operators which is generated by an operators of a shift on a vector of Euclidean space of the argument of a function or the argument of its Fourier image, operators of orthogonal mapping and operators of contraction of argument space. We obtain the conditions sufficient to convergence of a sequence of mean values of compositions of operator valued processes with values in the considered algebra of linear operators to the semigroup describing the diffusion in finite dimensional Euclidean space. Generators of limit semigroups are described.

Fixed Point Theorems for Pemetaan Nonexpansive, Nonspreading, and Hybrid Mappings in Hilbert Space.

A Firmly nonexpansive mapping is a map which is closely related to equilibrium problems Nonexpansive, nonspreading, and hybrid mappings are nonlinear mappings in Hilbert spaces. In this paper, we show the existence of a fixed point of nonexpansive, nonspreading, and hybrid mappings in Hilbert space using The Banach limit.

$$\mathbb{K}$$-homogeneous tuple of operators on bounded symmetric domains

Let Ω be an irreducible bounded symmetric domain of rank r in ℂd. Let $$\mathbb{K}$$ be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group $$\mathbb{K}$$ consisting of linear transformations acts naturally on any d-tuple T = (T1, …, Td) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is $$\mathbb{K}$$ -homogeneous. In this paper, we obtain a model for a certain class of $$\mathbb{K}$$ -homogeneous d-tuple T as the operators of multiplication by the coordinate functions z1, …, zd on a reproducing kernel Hilbert space of holomorphic functions defined on Ω. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B1(Ω). For an irreducible bounded symmetric domain Ω of rank 2, an explicit description of the operator $$\sum\nolimits_{i = 1}^d {T_i^\ast {T_i}} $$ is given. In general, based on this formula, we make a conjecture giving the form of this operator.

Simultaneous confidence bands for the functional mean of convex curves

Simultaneous confidence bands for the mean of convex curves in a functional, design-based approach are presented. Specifically, a constrained functional Horvitz-Thompson estimator is proposed to address the constrained nature of the data. Indeed, a naive direct application of the functional Horvitz-Thompson estimator on the smoothed trajectories may lead to non-convex estimates. As a consequence, convex functions are defined as solutions of differential equations. Thus, data are embedded in the Hilbert space of square integrable functions, where ‘standard’ functional data analysis methodologies can be coherently interpreted. An approximation of the variance of the estimator is derived using the delta method. The coverage level of the estimated bands is evaluated through a simulation study. The proposed method is employed on a real dataset concerning the concentration of tourists daily expenditure in the Majella National Park, Central Italy.

Ocean Surface Radial Velocity Imaging in the AT‐INSAR Velocity Bunching Model: A Functional Approach

AbstractImages obtained by airborne along‐track interferometric synthetic aperture radar (AT‐INSAR), provide tools to obtain ocean surface properties, such as the radial velocities of sea surface elevations. The estimation of these radial velocities from AT‐INSAR images is the interest of this work. We assume the Velocity Bunching Model relating the complex AT‐INSAR image at a point in the image plane, with the radial velocity of a scatterer point in the sea surface. The relation is by means of a nonlinear integral operator. Consequently, the estimation of radial velocities amounts to the solution of nonlinear and oscillatory integral equations. Our proposal is to solve the latter by Newton's methods on Hilbert function spaces, the so‐called optimize then discretize approach. The effectiveness is illustrated by means of numerical simulations evaluating the reconstruction of the radial velocities and the AT‐INSAR image. We show that this continuous version is accurate, and faster than the classical discretize then optimize version. A physical comparison is also carried out with the interferometric velocities.

A note on topological aspects in dynamic games of resource extraction and economic growth theory

We show that right-continuous monotone strategies used in Markov perfect equilibria for economic growth models and related dynamic games can be recognised as members of the Hilbert space of square integrable functions of the state variable. We provide an application of this result to a bequest game and point out that this result also holds for the class of left-continuous monotone functions. The result is fundamental for using the Schauder fixed point theorem. Furthermore, it considerably simplifies the classical approach, where such strategies are represented by non-negative measures on the state space.

Best kernel approximation in Bergman spaces

Let H be a reproducing kernel Hilbert space of analytic functions on the unit disk D. The best kernel approximation problem for H is the following: given any positive integer n and any function f∈H find the best norm approximation of f by a linear combination of no more than n kernel functions K(z,zk), 1≤k≤n. The purpose of this paper is to prove the existence of best kernel approximation for weighted Bergman spaces with standard weights.

Best approximation mappings in Hilbert spaces

The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional spaces was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of a BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspaces to nonempty closed convex sets and from $${\mathbb {R}}^{n}$$ to general Hilbert spaces. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al. proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in $${\mathbb {R}}^{n}$$ . We generalize their result from $${\mathbb {R}}^{n}$$ to general Hilbert spaces and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAMs with circumcenter mappings in Hilbert spaces and we report on Cegielski’s results on relationships with strongly quasinonexpansive mappings.

Weak and Strong Convergence Theorems for Common Attractive Points of Widely More Generalized Hybrid Mappings in Hilbert Spaces

In this work, we study iterative methods for the approximation of common attractive points of two widely more generalized hybrid mappings in Hilbert spaces and obtain weak and strong convergence theorems without assuming the closedness for the domain. A numerical example supporting our main result is also presented. As a consequence, our main results can be applied to solving a common fixed point problem.

Optimal approximants and orthogonal polynomials in several variables II: Families of polynomials in the unit ball

We obtain closed expressions for weighted orthogonal polynomials and optimal approximants associated with the function f ( z ) = 1 − 1 2 ( z 1 + z 2 ) f(z)=1-\frac {1}{\sqrt {2}}(z_1+z_2) and a scale of Hilbert function spaces in the unit 2 2 -ball having reproducing kernel ( 1 − ⟨ z , w ⟩ ) − γ (1-\langle z,w\rangle )^{-\gamma } , γ > 0 \gamma >0 . Our arguments are elementary but do not rely on reduction to the one-dimensional case.

Iterative algorithm with self-adaptive step size for approximating the common solution of variational inequality and fixed point problems

In this paper, we propose and study new inertial viscosity Tseng's extragradient algorithms with self-adaptive step size to solve the variational inequality problem (VIP) and the fixed point problem (FPP) in Hilbert spaces. Our proposed methods involve a projection onto a half-space and self-adaptive step size. We prove that the sequence generated by our proposed methods converges strongly to a common solution of the VIP and FPP of an infinite family of strict pseudo-contractive mappings in Hilbert spaces under some mild assumptions when the underlying operator is monotone and Lipschitz continuous. Furthermore, we apply our results to find a common solution of VIP and zero-point problem (ZPP) for an infinite family of maximal monotone operators. Finally, we provide some numerical experiments of the proposed methods in comparison with other existing methods in the literature.

Iterated nonexpansive mappings in Hilbert spaces

In [T. Dominguez Benavides and E. Llorens-Fuster, Iterated nonexpansive mappings, J. Fixed Point Theory Appl. 20 (2018), no. 3, Paper No. 104, 18 pp.], the authors raised the question about the existence of a fixed point free continuous INEA mapping T defined on a closed convex and bounded subset (or on a weakly compact convex subset) of a Banach space with normal structure. Our main goal is to give the affirmative answer to this problem in the very special case of a Hilbert space.

The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

The generalized viscosity implicit rules of nonexpansive asymptotically mappings in Hilbert spaces are considered. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequences of parameters. An application of it in the convex minimization problem is considered. The results presented in this paper improve and extend some recent corresponding results in the literature.

Multi-Step Inertial Regularized Methods for Hierarchical Variational Inequality Problems Involving Generalized Lipschitzian Mappings

In this paper, we construct two multi-step inertial regularized methods for hierarchical inequality problems involving generalized Lipschitzian and hemicontinuous mappings in Hilbert spaces. Then we present two strong convergence theorems and some numerical experiments to show the effectiveness and feasibility of our new iterative methods.