Type Integral Operators Research Articles

Finite Sum of Integral Operators from the Fractional Cauchy Spaces to Bloch-Type and Zygmund-Type Spaces

The family of fractional Cauchy transforms, defined on the open unit disc in the complex plane, is of classical and modern interest. Membership of an analytic function in the family is determined by the requirement that the function can be expressed as an integral of a certain kernel against a complex Borel measure on the disc. Such an integral representation imposes a growth condition on the function and its derivatives. This exposes a connection between the families of Cauchy transforms and familiar spaces of analytic functions, such as the Bloch spaces and the Zygmund space. The notion of a composition operator has been a fruitful area of study. More generally, many authors have studied weighted composition operators, the differentiation operator, integral-type operators, and various products of such operators, acting from one normed linear space of analytic functions to another such space. A common theme of such works is to characterize the operator-theoretic notions of boundedness and compactness in terms of the inducing symbols of the operator. We extend these studies to a specific linear transformation which will be defined as the sum of finitely many integral operators. Our conclusions include a complete characterization of boundedness and compactness of the integral sum, acting from the fractional Cauchy spaces to the Bloch-type and Zygmund-type spaces.

A class of processes defined in the white noise space through generalized fractional operators

The generalization of fractional Brownian motion in the white and grey noise spaces has been recently carried over, following the Mandelbrot–Van Ness representation, through Riemann–Liouville type fractional operators. Our aim is to extend this construction by means of more general fractional derivatives and integrals, which we define as Fourier-multiplier operators and then specialize by means of Bernstein functions. More precisely, we introduce a general class of kernel-driven processes which encompasses, as special cases, a number of models in the literature, including fractional Brownian motion, tempered fractional Brownian motion, Ornstein–Uhlenbeck process. The greater generality of our model, with respect to the previous ones, allows a higher flexibility and a wider applicability. We derive here some properties of this class of processes (such as continuity, occupation density, variance asymptotics and persistence) according to the conditions satisfied by the Fourier symbol of the operator or the Bernstein function chosen. On the other hand, these processes are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral type operator, respectively, regardless of the kernel used in their definition. Finally, this kind of construction allows us to define the corresponding noise and to solve a Langevin-type integral equation.

Construction of New Operators by Composition of Integral-Type Operators and Discrete Operators

In this paper, we propose some new positive linear approximation operators, which are obtained from a composition of certain integral type operators with certain discrete operators. It turns out that the new operators can be expressed in discrete form. We provide representations for their coefficients. Furthermore, we study their approximation properties and determine their moment generating functions, which may be useful in finding several other convergence results in different settings.

Composition of integral-type operators and discrete operators involving Laguerre polynomials

In this note we provide a new way to capture operators involving Laguerre polynomials by composition of an integral operator and a discrete operator. The new operator so obtained is a discrete operator. We give three examples by considering composition of Szász-Durrmeyer operator, exponential-type operator related to 2x3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2x^{3/2}$$\\end{document} and the Phillips operator, respectively, with Szász-Mirakyan operators. In all cases we obtain positive linear, discretely defined operators which are based on Laguerre polynomials and approximate functions on the positive real half-axis.

On a new approach in the space of measurable functions

In this paper, we present a new modulus of continuity for locally integrable function spaces which is effected by the natural structure of the L_{p} space. After basic properties of it are expressed, we provide a quantitative type theorem for the rate of convergence of convolution type integral operators and iterates of them. Moreover, we state their global smoothness preservation property including the new modulus of continuity. Finally, the obtained results are performed to the Gauss-Weierstrass operators.

Lp boundedness of Forelli-Rudin type operators on the Hartogs triangle

The Hartogs triangle H={(z1,z2)∈C2:|z1|<|z2|<1}, compared with the unit ball, is a non-smooth pseudoconvex domain which reveals various remarkable phenomena in several complex variables. Motivated by the recent work of Kures and Zhu (2006) [14] over the unit ball setting, in this paper, we completely characterize Lp boundedness of Forelli-Rudin type integral operators on the Hartogs triangle for all 1≤p≤∞.

Maximal estimate and integral operators in Bergman spaces with doubling measure

The boundedness of the maximal operator on the upper half-plane Π + \Pi ^{+} is established. Here Π + \Pi ^+ is equipped with a positive Borel measure d ω ( y ) d x d\omega (y)dx satisfying the doubling property ω ( ( 0 , 2 t ) ) ≤ C ω ( ( 0 , t ) ) \omega ((0,2t))\leq C\omega ((0,t)) . This result is connected to the Carleson embedding theorem, which we use to characterize the boundedness and compactness of the Volterra type integral operators on the Bergman spaces A ω p ( Π + ) A_{\omega }^{p}(\Pi ^{+}) .

Optimal Sobolev embeddings for the Ornstein-Uhlenbeck operator

A comprehensive analysis of Sobolev-type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space is offered. A unified approach is proposed, providing one with criteria for their validity in the class of rearrangement-invariant function norms. Optimal target and domain norms in the relevant inequalities are characterized via a reduction principle to one-dimensional inequalities for a Calderón type integral operator patterned on the Gaussian isoperimetric function. Consequently, the best possible norms in a variety of specific families of spaces, including Lebesgue, Lorentz, Lorentz-Zygmund, Orlicz and Marcinkiewicz spaces, are detected. The reduction principle hinges on a preliminary discussion of the existence and uniqueness of generalized solutions to equations, in the Gauss space, for the Ornstein-Uhlenbeck operator, with a just integrable right-hand side. A decisive role is also played by a pointwise estimate, in rearrangement form, for these solutions.

Generalized Volterra type integral operators on large Bergman spaces

Let ϕ be an analytic self-map of the open unit disk D and g analytic in D. We characterize boundedness and compactness of generalized Volterra type integral operatorsGI(ϕ,g)f(z)=∫0zf′(ϕ(ξ))g(ξ)dξ andGV(ϕ,g)f(z)=∫0zf(ϕ(ξ))g(ξ)dξ, acting between large Bergman spaces Aωp and Aωq for 0<p,q≤∞. To prove our characterizations, which involve Berezin type integral transforms, we use the Littlewood-Paley formula of Constantin and Peláez and establish corresponding embedding theorems, which are also of independent interest. When ϕ(z)=z, our results for GV(ϕ,g) complement the descriptions of Pau and Peláez.

Approximation results for beta Jakimovski-Leviatan type operators via q-analogue

We construct a new version of q-Jakimovski-Leviatan type integral operators and show that set of all continuous functions f defined on [0,?) are uniformly approximated by our new operators. Finally we construct the Stancu type operators and obtain approximation properties in weighted spaces. Moreover, with the aid of modulus of continuity we discuss the rate of convergence, Lipschitz type maximal approximation and some direct theorems.

Integral Baskakov type operator with quadratic order of approximation

Integral Baskakov type operator with quadratic order of approximation

Essential norm of generalized integral type operator from QK(p,q) to Zygmund spaces

Let ? be an analytic self-map on D, n ? N and 1 ? H(D). We consider the essential norm of the generalized integral-type operator Cn ?,g : QK (p,q) ? Z? that is defined as follows (Cn ?,g f) (z) = ?z0 f(n)(?(?))1(?) d?, for all f ? QK (p,q). We give an estimate for the essential norm of the above operator.

Norms of some operators between weighted-type spaces and weighted Lebesgue spaces

&lt;abstract&gt;&lt;p&gt;We calculate the norms of several concrete operators, mostly of some integral-type ones between weighted-type spaces of continuous functions on several domains. We also calculate the norm of an integral-type operator on some subspaces of the weighted Lebesgue spaces.&lt;/p&gt;&lt;/abstract&gt;

Compactness of Fractional Type Integral Operators on Spaces of Homogeneous Type

Compactness of Fractional Type Integral Operators on Spaces of Homogeneous Type

Modified Bernstein–Durrmeyer Type Operators

We constructed a summation–integral type operator based on the latest research in the linear positive operators area. We establish some approximation properties for this new operator. We highlight the qualitative part of the presented operator; we studied uniform convergence, a Voronovskaja-type theorem, and a Grüss–Voronovskaja type result. Our subsequent study focuses on a direct approximation theorem using the Ditzian–Totik modulus of smoothness and the order of approximation for functions belonging to the Lipschitz-type space. For a complete image on the quantitative estimations, we included the convergence rate for differential functions, whose derivatives were of bounded variations. In the last section of the article, we present two graphs illustrating the operator convergence.

Subclass of analytic functions involving Erdély–Kober type integral operator in conic regions and applications to neutrosophic Poisson distribution

In this article we familiarize a new subclass of analytic functions comprising Erdély–Kober type integral operator linked with the Janowski functions. Further, we confer some significant geometric properties like necessary and sufficient condition, growth and distortion bounds convex combination, partial sums and Fekete–Szegő inequality for this newly demarcated class. Further we conferred Fekete–Szegő inequality related with neutrosophic Poisson distribution.

Approximation of GBS Type q-Jakimovski-Leviatan-Beta Integral Operators in Bögel Space

In the present article, we introduce the bivariate variant of Beta integral type operators based on Appell polynomials via q-calculus. We study the local and global type approximation properties for these new operators. Next, we introduce the GBS form for these new operators and then study the degree of approximation by means of modulus of smoothness, mixed modulus of smoothness and Lipschitz class of Bögel continuous functions.

Continuity of $$L_{p}$$ Balls and an Application to Input-Output Systems

In this paper the continuity of the set valued map $p\rightarrow B_{\Omega,\mathcal{X},p}(r),$ $p\in (1,+\infty),$ is proved where $B_{\Omega,\mathcal{X},p}(r)$ is the closed ball of the space $L_{p}\left(\Omega,\Sigma,\mu; \mathcal{X}\right)$ centered at the origin with radius $r,$ $\left(\Omega,\Sigma,\mu\right)$ is a finite and positive measure space, $\mathcal{X}$ is separable Banach space. An application to input-output system described by Urysohn type integral operator is discussed.

Intertwining relations for composition operators and integral-type operators between the Bloch-type spaces

&lt;abstract&gt;&lt;p&gt;In this paper, the compact intertwining relations of integral-type operators and composition operators between the Bloch-type spaces are investigated.&lt;/p&gt;&lt;/abstract&gt;

Path connectedness of Volterra type integral operators on Bergman and Dirichlet type spaces

Path connectedness of Volterra type integral operators on Bergman and Dirichlet type spaces