Space Of Integrable Functions Research Articles

Onesided Korovkin approximation

In this paper we study in detail some characterizations of Korovkin closures and we also introduce the notions of onesided upper and lower Korovkin closures. We provide some complete characterizations of these new closures which separate the roles of approximating functions in a Korovkin system. We also present some new characterizations of the classical Korovkin closure in spaces of integrable functions. Again we can introduce and characterize the upper and lower Korovkin closures. Finally, we provide some examples which justify the interest in these new closures.

TWO-DIMENSIONAL INTEGRAL TRANSFORMATIONS WITH KUMMER FUNCTION
 AND HYPERGEOMETRIC GAUSSIAN FUNCTION IN KERNELS AS SPECIAL CASES
 OF TWO-DIMENSIONAL INTEGRAL G-TRANSFORMATION

Two two-dimensional integral transformations with confluent hypergeometric Kummer function and Gauss hypergeometric function in kernels are considered. Applying the Mellin transformation technique, we show that they are special cases of a two-dimensional G-transformation. Based on the theory of the G-transformation, the properties of the considered integral transformations in the weight spaces of integrable functions in the domain  are investigated. The results of the study generalize the results obtained earlier for the corresponding one-dimensional analogues.

Comparison theorems for evolution inclusions with maximal monotone operators. $L^2$-theory

An evolution inclusion with time-dependent family of maximal monotone operators in considered in a separable Hilbert space. If the elements with minimum norm of the family of maximal monotone operators satisfy certain growth conditions, then the domains of definition of this family are closed convex sets. Hence the sweeping process is well defined, whose values are the normal cones of the domains of definition of maximal monotone operators. It is shown that if the sweeping process has a solution for each single-valued perturbation from the space of integrable functions, then the evolution inclusion with the maximal monotone operators and single-valued perturbations from the space of integrable functions is also solvable. Quite general conditions in terms of the properties of the family of maximal monotone operators that ensure the existence of solutions for the sweeping process are presented. All results obtained and the approach presented are new. They are used to prove an existence theorem for evolution inclusions with multivalued perturbations, whose values are closed nonconvex sets. Bibliography: 19 titles.

Some inverse problem remarks of a continuous-in-time financial model in L^1([t_I,Theta_{max}

In the paper we are going to introduce an operator that is involved in the inverse problem of the continuous-in-time financial model. This framework is designed to be used in the finance for any organization and, in particular, for local communities. It allows to set out annual and multiyear budgets, with describing loan, reimbursement and interest payment schemes. We discuss this inverse problem in the space of integrable functions over R having a compact support. The concept of ill-posedness is examined in this space in order to obtain interesting and useful solutions. Then, we will give some remarks for not functionality of the model for a given Repayment Pattern Density γ, when this operator is not invertible in the space. Additionally, this inverse problem is illustrated in order to prove its ability to be used in a financial strategy.

Integrable and absolutely continuous vector-valued functions

The theory of integration for functions with values in a topological vector space is a bit of a messy subject. There are many different choices for the integral, most distinct from one another in general, and with no compelling reason to always adopt one definition or another as the “right” one. No attempt is made here to resolve this question of which integral is the “right” integral, but we overview a few of the common notions and prove a couple of useful properties for the notion of integrability by seminorm. For this notion of integrability, the completeness of the space of integrable functions with values in a complete locally convex space is established. This space is then easily seen to be isomorphic to the completion of the projective tensor product of the usual L1 space of scalar functions with the vector space. Absolute continuity for this notion of integrability is also considered. Here, it is shown that the familiar properties of differentiation for absolutely continuous scalar functions holds in the vector-valued case.

Integrable Solutions for Gripenberg-Type Equations with m-Product of Fractional Operators and Applications to Initial Value Problems

In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with m-product of fractional operators on a half-line R+=[0,∞). We prove the existence of solutions in some weighted spaces of integrable functions, i.e., the so-called L1N-solutions. Because such a space is not a Banach algebra with respect to the pointwise product, we cannot follow the idea of the proof for continuous solutions, and we prefer a fixed point approach concerning the measure of noncompactness to obtain our results. Appropriate measures for this space and some of its subspaces are introduced. We also study the problem of uniqueness of solutions. To achieve our goal, we utilize a generalized Hölder inequality on the noted spaces. Finally, to validate our results, we study the solvability problem for some particularly interesting cases and initial value problems.

Principal eigenvalue, maximum principles and linear stability for nonlocal diffusion equations in metric measure spaces

We study principal eigenvalues and maximum principles for stationary nonlocal operators in spaces of integrable functions defined on general metric measure spaces under minimal assumptions on the kernels. Several characterizations of the principal eigenvalue are given as well as several conditions guaranteeing existence. Characterization of the (strong) maximum principle is also given. For evolution problems we prove the strong maximum principle and characterize stability in terms of the sign of the principal eigenvalue. We recover, extend and improve all previously known results, obtained for smooth open sets in euclidean space under continuity assumptions on the data.

On Geometry of the Unit Ball of Paley–Wiener Space Over Two Symmetric Intervals

Abstract Let $PW_S^1$ be the space of integrable functions on ${\mathbb {R}}$ whose Fourier transform vanishes outside $S$, where $S = [-\sigma ,-\rho ]\cup [\rho ,\sigma ]$, $0<\rho <\sigma $. In the case $\rho>\sigma /2$, we present a complete description of the set of extreme and the set of exposed points of the unit ball of $PW_S^1$ . The structure of these sets becomes more complicated when $\rho <\sigma /2$.

A fixed point theorem in the space of integrable functions and applications

A fixed point theorem in the space of integrable functions and applications

Toeplitz Operators with Bounded Harmonic Symbols

A Cauchy problem associated to a fractional integro-differential de-fined by a Caputo type fractional derivative is studied. It is proved thearcwise connectedness of the solution set and that the set of selectionscorresponding to the solutions of the problem considered is a retractof the space of integrable functions on a given interval.

Isometric factorization of vector measures and applications to spaces of integrable functions

Let X be a Banach space, Σ be a σ-algebra, and m:Σ→X be a (countably additive) vector measure. It is a well known consequence of the Davis-Figiel-Johnson-Pełczyński factorization procedure that there exist a reflexive Banach space Y, a vector measure m˜:Σ→Y and an injective operator J:Y→X such that m factors as m=J∘m˜. We elaborate some theory of factoring vector measures and their integration operators with the help of the isometric version of the Davis-Figiel-Johnson-Pełczyński factorization procedure. Along this way, we sharpen a result of Okada and Ricker that if the integration operator on L1(m) is weakly compact, then L1(m) is equal, up to equivalence of norms, to some L1(m˜) where Y is reflexive; here we prove that the above equality can be taken to be isometric.

Multivariate tempered stable random fields

Multivariate tempered stable random measures (ISRMs) are constructed and their corresponding space of integrable functions is characterized in terms of a quasi-norm utilizing the so-called Rosinski measure of a tempered stable law. In the special case of exponential tempered ISRMs operator-fractional tempered stable random fields are presented by a moving-average and a harmonizable representation, respectively.

Relationship Between the Analytic Generalized Fourier–Feynman Transform and the Function Space Integral

In this paper we investigated a relationship between the analytic generalized Fourier–Feynman transform associated with Gaussian process and the function space integral for exponential type functionals on the function space $$C_{a,b}[0,T]$$ . The function space $$C_{a,b}[0,T]$$ can be induced by a generalized Brownian motion process. The Gaussian processes used in this paper are neither centered nor stationary.

Certain fundamental properties of generalized natural transform in generalized spaces

This paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386–1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed.

THE UNIQUE EXISTENCE OF SOLUTION IN THE q-INTEGRABLE SPACE FOR THE NONLINEAR q-FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we study the solution theory of the nonlinear [Formula: see text]-fractional differential equation of Caputo type [Formula: see text] with given initial values [Formula: see text] where [Formula: see text] is the order, [Formula: see text] and [Formula: see text] is the scale index. For [Formula: see text], by assuming that function [Formula: see text] is bounded and satisfies the Lipschitz condition on variable [Formula: see text], we prove that this problem admits a unique solution in the [Formula: see text]-integrable function space [Formula: see text] and this solution is absolutely stable in the [Formula: see text]-norm. This unique existence condition allows that [Formula: see text] is singular at [Formula: see text] and discontinuous for [Formula: see text]. Finally, a successive approximation method is presented to find out the analytic approximation solution of this problem.

Extension of Lipschitz-type operators on Banach function spaces

We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and other measure-theoretic notions are introduced. We analyze Lipschitz-type inequalities in two fundamental cases. The first concerns almost everywhere pointwise inequalities, while the second considers dominations involving integrals. These Lipschitz-type inequalities provide a suitable frame to work with operators that take values on Banach function spaces. In the last part of the paper we use some interpolation procedures to extend our study to interpolated Banach function spaces.

Interior estimates for solutions of linear elliptic inequalities

We study the wedge of solutions of the inequality , where is a linear elliptic operator of order acting on functions of variables. We establish interior estimates of the form for the elements of this wedge, where is a compact subdomain of , is the Sobolev space, , is the Lebesgue space of integrable functions, and the constant is independent of .

Subregular recourse in nonlinear multistage stochastic optimization

We consider nonlinear multistage stochastic optimization problems in the spaces of integrable functions. We allow for nonlinear dynamics and general objective functionals, including dynamic risk measures. We study causal operators describing the dynamics of the system and derive the Clarke subdifferential for a penalty function involving such operators. Then we introduce the concept of subregular recourse in nonlinear multistage stochastic optimization and establish subregularity of the resulting systems in two formulations: with built-in nonanticipativity and with explicit nonanticipativity constraints. Finally, we derive optimality conditions for both formulations and study their relations.

Characterizing Existence of Minimizers and Optimality to Nonconvex Quadratic Integrals

Quadratic functions play an important role in applied mathematics. In this paper, we consider the problem of minimizing the integral of a (not necessarily convex) quadratic function in a bounded subset of nonnegative integrable functions defined on a finite-dimensional space that is not compact with respect to any (locally convex) topology in the space of integrable functions. We establish a complete description about the existence or nonexistence of solution in terms of the (strict) copositivity of the matrix involved in the integrand. In addition, we characterize optimality via the Hamiltonian function.

On Pointwise Product Vector Measure Duality

This article is devoted to the study of pointwise product vector measure duality. The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.