Class Of Integrable Functions Research Articles

A class of integral functionals with a $W^{1,1}_0$-minimum

In this paper, we study the minimization of the integral functionals of the type $$ \int\_{\Omega}\frac{|\nabla v|^2}{\big\[1+B|v|\big]^{2}} + \frac12\int\_{\Omega} a(x)|v|^2 - \frac1q\int\_{\Omega} \rho(x)|v|^q, $$ where $0\<B$, $0\<a\_0\leq a(x)\in L^1\_{\mathrm{loc}}(\Omega)$, $0\not\equiv\rho^+(x)\in L^{{\frac{2}{2-q}}}(\Omega)$, and $0\<q<2$. The degeneracy of the principal part of the functional implies that it is not coercive in $W\_{0}^{1,2}(\Omega)$ and pushes to set and solve the minimization problem in $W\_{0}^{1,1}(\Omega)$ instead of the space $W\_{0}^{1,2}(\Omega)$. In addition, when the coefficients $a(x)$ and $\rho(x)$ are merely in $L^1(\Omega)$, but satisfy for some $Q>0$ the condition $|\rho(x)|\leq Q a(x)$, we show the existence of a minimum of the functional which belongs to $W\_{0}^{1,2}(\Omega){\cap L^\infty(\Omega)}\setminus{0}$.

Lineability and integrability in the sense of Riemann, Lebesgue, Denjoy, and Khintchine

In this paper, we continue the ongoing research on lineability related questions. On this occasion, we shall consider (among others) the classes of integrable functions (in the sense of Riemann, Lebesgue, Denjoy, and Khintchine), improving some already known results and expanding the study of lineability to other famous integrable classes never considered before.

Universal functions with respect to the double Walsh system for classes of integrable functions

The paper addresses questions on existence and structure of universal functions for spaces L1(E), E ⊂ [0, 1)2, with respect to the doubleWalsh system in the sense of signs of Fourier coefficients. It is shown that for each e > 0 one can find a measurable set Ee ⊂ [0, 1)2 with measure |Ee| > 1−e, such that by a proper modification of any integrable function f ∈ L1[0, 1)2 outside Ee one can get an integrable function f ∈ L1[0, 1)2 which is universal for L1(Ee) with respect to the double Walsh system in the sense of signs of Fourier coefficients.

Quasi-Quadrature Solution of Integral Equations Fredholm of the Second Kind in the Class of Integrable Functions

The analog of the quadrature solution of the equation of Fredholm of the second kind is considered. Fundamental difference from classical quadrature formulas is as follows. On segments of the chosen grid not values of functions, but their integral average values are used. Computing examples show expediency of such approach in appropriate cases.

Norm convergence of logarithmic means on unbounded Vilenkin groups

In this paper we prove that, in the case of some unbounded Vilenkin groups, the Riesz logarithmic means converges in the norm of the spaces X(G) for every f∈X(G), where by X(G) we denote either the class of continuous functions with supremum norm or the class of integrable functions.

The Controllability of Parabolic Systems with Delay and Distributed Parameters on the Graph

The optimal boundary control problem for a differential system with delay and distributed parameters on the graph is considered. As a state-space systems and the space of boundary control and observation, the classes of integrable functions is used. The conditions of the unique solvability of the problem of optimal control are obtained. And finally, the existence conditions of a unique control action and the controllability system are obtained.

Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions

In this paper we study we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The characteristic feature of this control object is the fact that the skew-symmetric part of matrix-valued control $A(x)$ belongs to $L^2$-space (rather than $L^\infty)$. In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP, under rather general assumptions on the class of admissible controls, is well-posed and admits a nonempty set of solutions [9]. However, the optimal solutions to such problem may have a singular character. We show that some of optimal solutions can be attainable by solutions of special optimal control problems in perforated domains with fictitious boundary controls on the holes.

A Class of Special Empirical Processes of Independence

In this paper we investigate the asymptotic properties of one class of empirical processes for certain classes of integrable functions.

Об одном классе интегральных функционалов с неизвестной областью интегрирования

Об одном классе интегральных функционалов с неизвестной областью интегрирования

Higher-order total variations for functions of several variables and their application in the theory of ill-posed problems

In the space of functions of two variables with Hardy-Krause property, new notions of higher-order total variations and Banach spaces of functions of two variables with bounded higher variations are introduced. The connection of these spaces with the Sobolev spaces W 1 , m ∈ ℕ, is studied. In the Sobolev spaces, a wide class of integral functionals with the weak regularization properties and the H-property is defined. It is proved that the application of these functionals in the Tikhonov variational scheme generates for m ≥ 3 the convergence of approximate solutions with respect to the total variation of order m − 3. The results are naturally extended to the case of functions of N variables.

Vector integration

This paper deals with the theory of integration of scalar functions with respect to a measure with values in a, not necessarily locally convex, topological vector space. It focuses on the extension of such integrals from bounded measurable functions to the class of integrable functions, proving adequate convergence theorems, and establishing usable integrability criteria. The aim is to give an account that is both general, comprising all previously known formulations, and sufficiently simple, perhaps to be of use even to those who are interested only in the case of Banach space.

ON THE COMPLETENESS, SEPARABILITY AND DENSITY THEOREMS FOR GENERALIZED LEBESGUE-BOCHNER SPACE $L^p(E,(X_\vartheta,\|.\|))$

Recently, S. Lahrech and al (see \cite{r0}) have introduced a special class of integrable functions denoted $L^p(E,(X_{\vartheta},\|.\|))$ which contains the usual Lebesgue-Bochner space $L^p(E,(X,\|.\|))$, where $(X,\vartheta)$ is a topological vector space, and $\|.\|$ is a norm defined on $X$. Many properties of $L^p(E,(X_{\vartheta},\|.\|))$ have been established in \cite{r0}. The purpose of this paper is to continue the study of the generalized Lebesgue-Bochner space $L^p(E,(X_{\vartheta},\|.\|))$ in the case where the unit ball ${\cal B}_1(X)$ is closed in $(X,\vartheta)$ and sequentially complete under the topology $\vartheta$. Under the above conditions, we establish some results related to the separability and completeness of $L^p(E,(X_{\vartheta},\|.\|))$. Moreover, we prove that the class ${\cal C}(E,(X_\vartheta,\|.\|))$ of continuous vector functions from $E$ into $X_{\vartheta}$ and bounded with respect to the topology generated by the norm $\|.\|$ is dense in $L^p(E,(X_{\vartheta},\|.\|))$.

On the removability of a singularity for minimizers of an integral functional

We obtain a Serrin-type condition for the removability of the isolated singularity for the minimizers of a class of integral functionals of the type: where the integrands satisfy polynomial growth conditions.

Optimal L 1-Control in Coefficients for Dirichlet Elliptic Problems: W-Optimal Solutions

In this paper, we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The coefficients may degenerate and, therefore, the problems may exhibit the so-called Lavrentieff phenomenon and non-uniqueness of weak solutions. We consider the solvability of this problem in the class of W-variational solutions. Using a concept of variational convergence of constrained minimization problems in variable spaces, we prove the existence of W-solutions to the optimal control problem and provide the way for their approximation. We emphasize that control problems of this type are important in material and topology optimization as well as in damage or life-cycle optimization.

A generalized diffusion frame for parsimonious representation of functions on data defined manifolds

One of the now standard techniques in semi-supervised learning is to think of a high dimensional data as a subset of a low dimensional manifold embedded in a high dimensional ambient space, and to use projections of the data on eigenspaces of a diffusion map. This paper is motivated by a recent work of Coifman and Maggioni on diffusion wavelets to accomplish such projections approximately using iterates of the heat kernel. In greater generality, we consider a quasi-metric measure space X (in place of the manifold), and a very general operator T defined on the class of integrable functions on X (in place of the diffusion map). We develop a representation of functions on X in terms of linear combinations of iterates of T. Our construction obviates the need to compute the eigenvalues and eigenfunctions of the operator. In addition, the local smoothness of a function f is characterized by the local norm behavior of the terms in our representation of f. This property is similar to that of the classical wavelet representations. Although the operator T utilizes the values of the target function on the entire space, this ability results in automatic "feature detection", leading to a parsimonious representation of the target function. In the case when X is a smooth compact manifold (without boundary), our theory allows T to be any operator that commutes with the heat operator, subject to certain conditions on its eigenvalues. In particular, T can be chosen to be the heat operator itself, or a Green's operator corresponding to a suitable pseudo-differential operator.

Analytic structure of three-mass triangle coefficients

``Three-mass triangles'' are a class of integral functions appearing in one-loop gauge theory amplitudes. We discuss how the complex analytic properties and singularity structures of these amplitudes can be combined with generalised unitarity techniques to produce compact expressions for three-mass triangle coefficients. We present formulae for the N=1 contributions to the n-point NMHV amplitude.

Hölder Continuity up to the Boundary of Minimizers for Some Integral Functionals with Degenerate Integrands

We study qualitative properties of minimizers for a class of integral functionals, defined in a weighted space. In particular we obtain Hölder regularity up to the boundary for the minimizers of an integral functional of high order by using an interior local regularity result and a modified Moser method with special test function.

On the existence and uniqueness of minimizers for a class of integral functionals

We study the solvability of the minimization problem $$\mathop {\min }\limits_{\eta \in \mathcal{K}_\alpha } \int_0^T {\alpha (t)\left[ {f\left( {|\eta '(t)|} \right) + g\left( {\eta (t)} \right)} \right]} \,dt,$$ where \(\mathcal{K}_\alpha \) is a subset of ACloc[0, T] depending on the weight function α. Neither the convexity nor the superlinearity of f are required. The main application concerns the existence and uniqueness of minimizers to integral functionals on convex domains \(\Omega \subset \mathbb{R}^N ,\) defined in the class of functions in \(W_0 ^{1,1} \left( \Omega \right)\) depending only on the distance from the boundary of Ω. As a corollary, when Ω is a ball we obtain the existence of radially symmetric solutions to nonconvex and noncoercive functionals.

Strong summability for the Marcinkiewicz means in the integral metric and related questions

AbstractThe inequality of strong summability for the Marcinkiewicz means of multiply Fourier series is proved. The inequalities of strong summability with gaps for the different classes of integrable functions are established. The Bernstein inequality for the fractional derivative of analytic polynomials is proved.

On a Class of Functionals Whose Local Minima are Global

In this note we introduce a suitable class of functionals, including the class of integral functionals, and prove that any (strict) local minimum of a functional of this class, defined on a decomposable space, is a (strict) global minimum. So, the recent result obtained by Giner in [1] is specified and extended.