Space Of Integrable Functions Research Articles

The integral analog of a series with a two-point sum range

We consider an improper integral corresponding to the series with a two-point sum range which was constructed by Kornilov in the space of integrable functions. We verify that the sum range of the integral is equal to the set of all constant functions.

EXTREMAL EQUILIBRIA FOR DISSIPATIVE PARABOLIC EQUATIONS IN LOCALLY UNIFORM SPACES

We consider a reaction diffusion equation ut = Δu + f(x, u) in ℝN with initial data in the locally uniform space [Formula: see text], q ∈ [1, ∞), and with dissipative nonlinearities satisfying s f(x, s) ≤ C(x)s2 + D(x) |s|, where [Formula: see text] and [Formula: see text] for certain [Formula: see text]. We construct a global attractor [Formula: see text] and show that [Formula: see text] is actually contained in an ordered interval [φm, φM], where [Formula: see text] is a pair of stationary solutions, minimal and maximal respectively, that satisfy φm ≤ lim inft→∞ u(t; u0) ≤ lim supt→∞ u(t; u0) ≤ φM uniformly for u0 in bounded subsets of [Formula: see text]. A sufficient condition concerning the existence of minimal positive steady state, asymptotically stable from below, is given. Certain sufficient conditions are also discussed ensuring the solutions to be asymptotically small as |x| → ∞. In this case the solutions are shown to enter, asymptotically, Lebesgue spaces of integrable functions in ℝN, the attractor attracts in the uniform convergence topology in ℝN and is a bounded subset of W2,r(ℝN) for some r > N/2. Uniqueness and asymptotic stability of positive solutions are also discussed. Applications to some model problems, including some from mathematical biology are given.

H 1-estimates of Littlewood-Paley and Lusin functions for Jacobi analysis

For α ≥ β ≥ −1/2 let $$ \Delta (x) = (2shx)^{2\alpha + 1} (2chx)^{2\beta + 1} $$ denote the weight function on R+ and L1(Δ) the space of integrable functions on R+ with respect to Δ(x)dx, equipped with a convolution structure. For a suitable ϕ ∈ L1(Δ), we put $$ \varphi _t (x) = t^{ - 1} \Delta (x)^{ - 1} \Delta (x/t)\varphi (x/t) $$ for t > 0 and define the radial maximal operator Mϕ as usual manner. We introduce a real Hardy space H1(Δ) as the set of all locally integrable functions f on R+ whose radial maximal function Mϕ(f) belongs to L1(Δ). In this paper we obtain a relation between H1(Δ) and H1(R). Indeed, we characterize H1(Δ) in terms of weighted H1 Hardy spaces on R via the Abel transform of f. As applications of H1(Δ) and its characterization, we shall consider (H1(Δ),L1(Δ))-boundedness of some operators associated to the Poisson kernel for Jacobi analysis: the Poisson maximal operator MP, the Littlewood-Paley g-function and the Lusin area function S. They are bounded on Lp(Δ) for p > 1, but not true for p = 1. Instead, MP, g and a modified Sa, γ are bounded from H1(Δ) to L1(Δ).

Regularity of solutions to stochastic Volterra equations of convolution type

In this paper we consider two classes of linear Volterra equations: of convolution type on ℝ+ and integro-differential with infinite delay on d-dimensional torus, both driven by spatially homogeneous Wiener process. First, we study the existence of solutions to these equations in the space of tempered distributions and then derive conditions under which the solutions take values in Sobolev spaces. We give necessary and sufficient conditions providing regularity of solutions to equations considered. The harmonic analysis techniques and stochastic integration in function spaces are used. The paper is a short review of regularity results for stochastic Volterra equations of convolution type.

Bayesian Nonparametric Functional Data Analysis Through Density Estimation.

In many modern experimental settings, observations are obtained in the form of functions, and interest focuses on inferences on a collection of such functions. We propose a hierarchical model that allows us to simultaneously estimate multiple curves nonparametrically by using dependent Dirichlet Process mixtures of Gaussians to characterize the joint distribution of predictors and outcomes. Function estimates are then induced through the conditional distribution of the outcome given the predictors. The resulting approach allows for flexible estimation and clustering, while borrowing information across curves. We also show that the function estimates we obtain are consistent on the space of integrable functions. As an illustration, we consider an application to the analysis of Conductivity and Temperature at Depth data in the north Atlantic.

Linear systems with locally integrable trajectories

The locally integrable function space L 1 loc is made a module over the ring of proper rational functions. Using this module structure, the notion of relative dimension is defined naturally for every linear subspace in ( L 1 loc ) q . It is shown that the property of having finite relative dimension together with the property of having sufficiently many smooth trajectories and the evident property of differentiation-invariance characterize the weak solutions sets of linear constant coefficient differential systems among all linear subspaces.

Sure wins, separating probabilities and the representation of linear functionals

We discuss conditions under which a convex cone $\K\subset \R^{\Omega}$ admits a probability $m$ such that $\sup_{k\in \K} m(k)\leq0$. Based on these, we also characterize linear functionals that admit the representation as finitely additive expectations. A version of Riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions

Ordered representations of spaces of integrable functions

Let X be a Banach space, (Ω,Σ) a measurable space and let m : Σ → X be a (countably additive) vector measure. Consider the corresponding space of integrable functions L1(m). In this paper we analyze the set of (countably additive) vector measures n satisfying that L1(n) = L1(m). In order to do this we define a (quasi) order relation on this set to obtain under adequate requirements the simplest representation of the space L1(m) associated to downward directed subsets of the set of all the representations.

Radon–Nikodým derivatives for vector measures belonging to Köthe function spaces

Let m, n be a couple of vector measures with values on a Banach space. We develop a separation argument which provides a characterization of when the Radon–Nikodým derivative of n with respect to m—in the sense of the Bartle–Dunford–Schwartz integral—exists and belongs to a particular sublattice Z ( μ ) of the space of integrable functions L 1 ( m ) . We show that this theorem is in fact a particular feature of our separation argument, which can be applied to prove other results in both the vector measure and the function space settings.

Orlicz Spaces of Integrable Functions with Respect to Vector- Valued Measures

Orlicz Spaces of Integrable Functions with Respect to Vector- Valued Measures

Multiplication operators on spaces of integrable functions with respect to a vector measure

We study continuity and other properties related to some kind of compactness of multiplication operators between different spaces of pth power integrable scalar functions with respect to a vector measure.

On the weighted dyadic diaphony of digital (t, s)‐sequences

AbstractIn this note, we present results on the weighted dyadic diaphony of digital (t, s)‐sequences over ℤ2. These results are closely linked to the worst‐case error for quasi‐Monte Carlo integration in certain spaces of functions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Hypercyclic, mixing, and chaoticC0-semigroups induced by semiflows

AbstractWe characterize whenC0-semigroups induced by semiflows are hypercyclic, topologically mixing, or chaotic both on spaces of integrable functions and on spaces of continuous functions. Furthermore, we give characterizations of transitivity for weighted composition operators on these spaces.

Optimal Domains for L0-valued Operators Via Stochastic Measures

We study extension of operators T: E→ L0([0, 1]), where E is an F–function space and L0([0, 1]) the space of measurable functions with the topology of convergence in measure, to domains larger than E, and we study the properties of such domains. The main tool is the integration of scalar functions with respect to stochastic measures and the corresponding spaces of integrable functions.

On the solutions of the nonlinear Liouville hierarchy

We investigate the initial-value problem of the nonlinear Liouville hierarchy. For the general form of the interaction potential we construct a solution in terms of an expansion over particle clusters whose evolution is described by the corresponding-order cumulant of evolution operators of a system of finitely many particles. For the initial data from the space of integrable functions the existence of a strong solution of the Cauchy problem is proved.

ANALYTIC FUNCTIONS OF k SINGLE INTEGRALS

In this paper we establish the existence theorem of the operator-valued function space integral for where F is of the form (1.1). Our result is a generalization for Johnson and Skoug result in [8].

Optimal domain for the Hardy operator

We study the optimal domain for the Hardy operator considered with values in a rearrangement invariant space. In particular, this domain can be represented as the space of integrable functions with respect to a vector measure defined on a δ-ring. A precise description is given for the case of the minimal Lorentz spaces.

Integrable Boehmians, Fourier transforms, and Poisson’s summation formula

Boehmians are classes of generalized functions whose construction is algebraic. The first construction appeared in a paper that was published in 1981 [6]. In [8], P. Mikusinski constructs a space of Boehmians, βL1(R), in which each element has a Fourier transform. Mikusinski shows that the Fourier transform of a Boehmian satisfies some basic properties, and he also proves an inversion theorem. However, the range of the Fourier transform is not investigated. Also, Mikusinski states that βL1(R) contains some elements which are not Schwartz distributions, but no examples are given. We will address these problems in this paper. In this note, we will construct a space of Boehmians β`(R). The space of integrable functions on the real line can be identified with a proper subspace of β`(R). Each element of β`(R) has a Fourier transform which is a continuous function and satisfies a growth condition at infinity. Conditions are given which ensure that a given function is the Fourier transform of an element of β`(R).

Evaluation Formulas for Conditional Function Space Integrals I

In this article, we establish several explicit conditional function space integration formulas for functionals defined on a very general function space C a,b [0,T]. The formulas we obtain are rather simple and don't involve function space integrals. In particular we obtain a formula for the conditional function space integral of each of the functionals exp{∫0 T x(t)db(t)}, exp{−[∫0 T x(t) db(t)]2}, and exp{− ∫0 T x2 (t) db(t)} which arise naturally in quantum mechanics.

Classical solutions of quasilinear parabolic systems on two dimensional domains

Using results on abstract evolutions equations and recently obtained results on elliptic operators with discontinuous coefficients including mixed boundary conditions we prove that quasilinear parabolic systems admit a local, classical solution in the space of p–integrable functions, for some p greater than 1, over a bounded two dimensional space domain. The treatment of such equations in a space of integrable functions enables us to define the normal component of the current across the boundary of any Lipschitz subset. As applications we have in mind systems of reaction diffusion equations, e.g. van Roosbroeck’s system.