Range Convergence Monotonicity for Vector
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Vector Measures with Variation in a Banach Function Space
VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE O. BLASCO Departamento de An´alisis Matem´atico, Universidad de Valencia, Doctor Moliner, 50 E-46100 Burjassot (Valencia), Spain E-mail:[email protected] PABLO GREGORI Departament de Matem`atiques, Universitat Jaume I de Castell´o, Campus Riu Sec E-12071 Castell´o de la Plana (Castell´o), Spain E-mail:[email protected] Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the K¨othe-Bochner function space defined as the set of measurable functions f :Ω→ X such that the nonnegative functions fX :Ω→ [0, ∞) are in the lattice E. The notion of E-variation of a measure —which allows to recover the p- variation (for E = Lp), Φ-variation (for E = LΦ) and the general notion introduced by Gresky and Uhl— is introduced. The space of measures of bounded E-variation VE (X) is then studied. It is shown, amongother thingsand with some restriction ∗ ∗ of absolute continuity of the norms, that (E(X)) = VE (X ), that VE (X) can be identified with space of cone absolutely summingoperators from E into X and that E(X)=VE (X) if and only if X has the RNP property. 1. Introduction The concept of variation in the frame of vector measures has been fruitful in several areas of the functional analysis, such as the description of the duality of vector-valued function spaces such as certain K¨othe-Bochner function spaces (Gretsky and Uhl10, Dinculeanu7), the reformulation of operator ideals such as the cone absolutely summing operators (Blasco4), and the Hardy spaces of harmonic function (Blasco2,3). -
Integral Representation of a Linear Functional on Function Spaces
INTEGRAL REPRESENTATION OF LINEAR FUNCTIONALS ON FUNCTION SPACES MEHDI GHASEMI Abstract. Let A be a vector space of real valued functions on a non-empty set X and L : A −! R a linear functional. Given a σ-algebra A, of subsets of X, we present a necessary condition for L to be representable as an integral with respect to a measure µ on X such that elements of A are µ-measurable. This general result then is applied to the case where X carries a topological structure and A is a family of continuous functions and naturally A is the Borel structure of X. As an application, short solutions for the full and truncated K-moment problem are presented. An analogue of Riesz{Markov{Kakutani representation theorem is given where Cc(X) is replaced with whole C(X). Then we consider the case where A only consists of bounded functions and hence is equipped with sup-norm. 1. Introduction A positive linear functional on a function space A ⊆ RX is a linear map L : A −! R which assigns a non-negative real number to every function f 2 A that is globally non-negative over X. The celebrated Riesz{Markov{Kakutani repre- sentation theorem states that every positive functional on the space of continuous compactly supported functions over a locally compact Hausdorff space X, admits an integral representation with respect to a regular Borel measure on X. In symbols R L(f) = X f dµ, for all f 2 Cc(X). Riesz's original result [12] was proved in 1909, for the unit interval [0; 1]. -
The Total Variation Flow∗
THE TOTAL VARIATION FLOW∗ Fuensanta Andreu and Jos´eM. Maz´on† Abstract We summarize in this lectures some of our results about the Min- imizing Total Variation Flow, which have been mainly motivated by problems arising in Image Processing. First, we recall the role played by the Total Variation in Image Processing, in particular the varia- tional formulation of the restoration problem. Next we outline some of the tools we need: functions of bounded variation (Section 2), par- ing between measures and bounded functions (Section 3) and gradient flows in Hilbert spaces (Section 4). Section 5 is devoted to the Neu- mann problem for the Total variation Flow. Finally, in Section 6 we study the Cauchy problem for the Total Variation Flow. 1 The Total Variation Flow in Image Pro- cessing We suppose that our image (or data) ud is a function defined on a bounded and piecewise smooth open set D of IRN - typically a rectangle in IR2. Gen- erally, the degradation of the image occurs during image acquisition and can be modeled by a linear and translation invariant blur and additive noise. The equation relating u, the real image, to ud can be written as ud = Ku + n, (1) where K is a convolution operator with impulse response k, i.e., Ku = k ∗ u, and n is an additive white noise of standard deviation σ. In practice, the noise can be considered as Gaussian. ∗Notes from the course given in the Summer School “Calculus of Variations”, Roma, July 4-8, 2005 †Departamento de An´alisis Matem´atico, Universitat de Valencia, 46100 Burjassot (Va- lencia), Spain, [email protected], [email protected] 1 The problem of recovering u from ud is ill-posed. -
Geometric Integration Theory Contents
Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants . -
Combinatorics in Banach Space Theory Lecture 2
Combinatorics in Banach space theory Lecture 2 Now, we will derive the promised corollaries from Rosenthal's lemma. First, following [Ros70], we will show how this lemma implies Nikod´ym's uniform boundedness principle for bounded vector measures. Before doing this we need to recall some definitions. Definition 2.4. Let F be a set algebra. By a partition of a given set E 2 F we mean Sk a finite collection fE1;:::;Ekg of pairwise disjoint members of F such that j=1Ej = E. We denote Π(E) the set of all partitions of E. Let also X be a Banach space and µ: F ! X be a finitely additive set function (called a vector measure). The variation of µ is a function jµj: F ! [0; 1] given by ( ) X jµj(E) = sup kµ(A)k: π 2 Π(E) : A2π The semivariation of µ is a function kµk: F ! [0; 1] given by ∗ ∗ kµk = sup jx µj(E): x 2 BX∗ : By a straightforward calculation, one may check that the variation jµj is always finitely additive, whereas the semivariation kµk is monotone and finitely subadditive. Of course, we always have kµk 6 jµj. Moreover, since we know how the functionals on the scalar space (R or C) look like, it is easily seen that for scalar-valued measures the notions of variation and semivariation coincide. By saying that a vector measure µ: F ! X is bounded we mean that kµk is finitely valued. This is in turn equivalent to saying that the range of µ is a bounded subset of X. -
On Stochastic Distributions and Currents
NISSUNA UMANA INVESTIGAZIONE SI PUO DIMANDARE VERA SCIENZIA S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI LEONARDO DA VINCI vol. 4 no. 3-4 2016 Mathematics and Mechanics of Complex Systems VINCENZO CAPASSO AND FRANCO FLANDOLI ON STOCHASTIC DISTRIBUTIONS AND CURRENTS msp MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Vol. 4, No. 3-4, 2016 dx.doi.org/10.2140/memocs.2016.4.373 ∩ MM ON STOCHASTIC DISTRIBUTIONS AND CURRENTS VINCENZO CAPASSO AND FRANCO FLANDOLI Dedicated to Lucio Russo, on the occasion of his 70th birthday In many applications, it is of great importance to handle random closed sets of different (even though integer) Hausdorff dimensions, including local infor- mation about initial conditions and growth parameters. Following a standard approach in geometric measure theory, such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are sin- gular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper, how to cope with these difficulties has been suggested by introducing random generalized densities (dis- tributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. For the last one, mean generalized densities are analyzed, and they have been related to densities of the expected values of the relevant measures. Ac- tually, distributions are a subclass of the larger class of currents; in the usual Euclidean space of dimension d, currents of any order k 2 f0; 1;:::; dg or k- currents may be introduced. -
Invariant Radon Measures on Measured Lamination Space
INVARIANT RADON MEASURES ON MEASURED LAMINATION SPACE URSULA HAMENSTADT¨ Abstract. Let S be an oriented surface of genus g ≥ 0 with m ≥ 0 punctures and 3g − 3+ m ≥ 2. We classify all Radon measures on the space of measured geodesic laminations which are invariant under the action of the mapping class group of S. 1. Introduction Let S be an oriented surface of finite type, i.e. S is a closed surface of genus g 0 from which m 0 points, so-called punctures, have been deleted. We assume that≥ 3g 3 + m 1,≥ i.e. that S is not a sphere with at most 3 punctures or a torus without− puncture.≥ In particular, the Euler characteristic of S is negative. Then the Teichm¨uller space (S) of S is the quotient of the space of all complete hyperbolic metrics of finite volumeT on S under the action of the group of diffeomorphisms of S which are isotopic to the identity. The mapping class group MCG(S) of all isotopy classes of orientation preserving diffeomorphisms of S acts properly discontinuously on (S) with quotient the moduli space Mod(S). T A geodesic lamination for a fixed choice of a complete hyperbolic metric of finite volume on S is a compact subset of S foliated into simple geodesics. A measured geodesic lamination is a geodesic lamination together with a transverse translation invariant measure. The space of all measured geodesic laminations on S, equipped with the weak∗-topology,ML is homeomorphic to S6g−7+2m (0, ) where S6g−7+2m is the 6g 7+2m-dimensional sphere. -
Banach Space Properties of V of a Vector Measure
proceedings of the american mathematical society Volume 123, Number 12. December 1995 BANACHSPACE PROPERTIES OF V OF A VECTOR MEASURE GUILLERMO P. CURBERA (Communicated by Dale Alspach) Abstract. We consider the space L'(i>) of real functions which are integrable with respect to a measure v with values in a Banach space X . We study type and cotype for Lx{v). We study conditions on the measure v and the Banach space X that imply that Ll(v) is a Hilbert space, or has the Dunford-Pettis property. We also consider weak convergence in Lx(v). 1. Introduction Given a vector measure v with values in a Banach space X, Lx(v) denotes the space of (classes of) real functions which are integrable with respect to v in the sense of Bartle, Dunford and Schwartz [BDS] and Lewis [L-l]. This space has been studied by Kluvanek and Knowles [KK], Thomas [T] and Okada [O]. It is an order continuous Banach lattice with weak unit. In [C-l, Theorem 8] we have identified the class of spaces Lx(v) , showing that every order continuous Banach lattice with weak unit can be obtained, order isometrically, as L1 of a suitable vector measure. A natural question arises: what is the relation between, on the one hand, the properties of the Banach space X and the measure v, and, on the other hand, the properties of the resulting space Lx(v) . The complexity of the situ- ation is shown by the following example: the measures, defined over Lebesgue measurable sets of [0,1], vx(A) = m(A) £ R, v2(A) = Xa e £'([0, 1]) and v3 — (]"Ar„(t)dt) £ Co, where rn are the Rademacher functions, generate, order isometrically, the same space, namely 7J([0, 1]). -
FLOWS of MEASURES GENERATED by VECTOR FIELDS 1. Introduction Every Sufficiently Smooth and Bounded Vector Field V in the Euclide
FLOWS OF MEASURES GENERATED BY VECTOR FIELDS EMANUELE PAOLINI AND EUGENE STEPANOV Abstract. The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of N. Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ODE. We further show, that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure “flows along” the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense. 1. Introduction Every sufficiently smooth and bounded vector field V in the Euclidean space E = Rn is well-known to generate a canonical flow of a given finite Borel measure t + t µ in E by setting µt := ϕ µ, t R , where ϕ (x) := θ(t), θ standing for the V # ∈ V unique solution to the differential equation (1.1) θ˙(t)= V (θ(t)) satisfying the initial condition θ(0) = x. Such a flow satisfies the continuity equation ∂µ (1.2) t + div v µ =0 ∂t t t + in the weak sense in E R , where vt : E E is some velocity field (of course, non unique for the given×V , but in particular,→ this equation is satisfied in this case with vt := V ). -
Algebras and Orthogonal Vector Measures
proceedings of the american mathematical society Volume 110, Number 3, November 1990 STATES ON IT*-ALGEBRAS AND ORTHOGONAL VECTOR MEASURES JAN HAMHALTER (Communicated by William D. Sudderth) Abstract. We show that every state on a If*-algebra sf without type I2 direct summand is induced by an orthogonal vector measure on sf . This result may find an application in quantum stochastics [1, 7]. Particularly, it allows us to find a simple formula for the transition probability between two states on sf [3, 8], 1. Preliminaries Here we fix some notation and recall basic facts about Hilbert-Schmidt op- erators (for details, see [9]). Let H be a complex Hilbert space and let 3§'HA) denote the set of all linear bounded operators on H . An operator A e 33(H) is said to be a trace class operator if the series ^2aeI(Aea, ea) is absolutely convergent for an orthonormal basis (ea)ae! of //. In this case the sum J2a€¡(Aea, ea) does not depend on the choice of the basis (ea)a€¡ and is called a trace of A (abbreviated Tr,4). Let us denote by CX(H) the set of all trace class operators on //. Then Tr AB = Tr BA whenever the product AB belongs to CX(H). Suppose that A e 33(H). Then A is called a Hilbert-Schmidt operator if IMII2 = {X3„e/II^JI } < 00 for some (and therefore for any) orthonor- mal basis (en)n€, of //. Let us denote by C2(H) the set of all Hilbert- Schmidt operators on //. Thus, C2(H) is a two-sided ideal in 33(H), and CX(H) c C2(H). -
Total Variation Deconvolution Using Split Bregman
Published in Image Processing On Line on 2012{07{30. Submitted on 2012{00{00, accepted on 2012{00{00. ISSN 2105{1232 c 2012 IPOL & the authors CC{BY{NC{SA This article is available online with supplementary materials, software, datasets and online demo at http://dx.doi.org/10.5201/ipol.2012.g-tvdc 2014/07/01 v0.5 IPOL article class Total Variation Deconvolution using Split Bregman Pascal Getreuer Yale University ([email protected]) Abstract Deblurring is the inverse problem of restoring an image that has been blurred and possibly corrupted with noise. Deconvolution refers to the case where the blur to be removed is linear and shift-invariant so it may be expressed as a convolution of the image with a point spread function. Convolution corresponds in the Fourier domain to multiplication, and deconvolution is essentially Fourier division. The challenge is that since the multipliers are often small for high frequencies, direct division is unstable and plagued by noise present in the input image. Effective deconvolution requires a balance between frequency recovery and noise suppression. Total variation (TV) regularization is a successful technique for achieving this balance in de- blurring problems. It was introduced to image denoising by Rudin, Osher, and Fatemi [4] and then applied to deconvolution by Rudin and Osher [5]. In this article, we discuss TV-regularized deconvolution with Gaussian noise and its efficient solution using the split Bregman algorithm of Goldstein and Osher [16]. We show a straightforward extension for Laplace or Poisson noise and develop empirical estimates for the optimal value of the regularization parameter λ. -
Guaranteed Deterministic Bounds on the Total Variation Distance Between Univariate Mixtures
Guaranteed Deterministic Bounds on the Total Variation Distance between Univariate Mixtures Frank Nielsen Ke Sun Sony Computer Science Laboratories, Inc. Data61 Japan Australia [email protected] [email protected] Abstract The total variation distance is a core statistical distance between probability measures that satisfies the metric axioms, with value always falling in [0; 1]. This distance plays a fundamental role in machine learning and signal processing: It is a member of the broader class of f-divergences, and it is related to the probability of error in Bayesian hypothesis testing. Since the total variation distance does not admit closed-form expressions for statistical mixtures (like Gaussian mixture models), one often has to rely in practice on costly numerical integrations or on fast Monte Carlo approximations that however do not guarantee deterministic lower and upper bounds. In this work, we consider two methods for bounding the total variation of univariate mixture models: The first method is based on the information monotonicity property of the total variation to design guaranteed nested deterministic lower bounds. The second method relies on computing the geometric lower and upper envelopes of weighted mixture components to derive deterministic bounds based on density ratio. We demonstrate the tightness of our bounds in a series of experiments on Gaussian, Gamma and Rayleigh mixture models. 1 Introduction 1.1 Total variation and f-divergences Let ( R; ) be a measurable space on the sample space equipped with the Borel σ-algebra [1], and P and QXbe ⊂ twoF probability measures with respective densitiesXp and q with respect to the Lebesgue measure µ.