Functions Of Bounded Variation Research Articles

Forward integration of bounded variation coefficients with respect to Hölder continuous processes

In this article, we study the forward integral, in the Russo and Vallois sense, with respect to Hölder continuous stochastic processes Y with exponent bigger than 1∕2. Here, the integrands have the form f(Y), where f is a bounded variation function. As a consequence of our results, we show that this integral agrees with the generalized Stieltjes integral given by Zähle and that, in the case that Y is fractional Brownian motion, this forward integral is equal to the divergence operator plus a trace term, which is related to the local time of Y. Moreover, the definition of the forward integral allows us to obtain a representation of the solutions to forward stochastic differential equations with a possibly discontinuous coefficient and, as a consequence of our analysis, to figure out some explicit solutions.

On Sharp Rate of Convergence for Discretization of Integrals Driven by Fractional Brownian Motions and Related Processes with Discontinuous Integrands

We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order H>12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H>\\frac{1}{2}$$\\end{document} with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to n1-2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{1-2H}$$\\end{document}, which is twice as good as the best known results in the case of discontinuous integrands and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply a change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.

BIG IN REVERSE MATHEMATICS: THE UNCOUNTABILITY OF THE REALS

AbstractThe uncountability of$\mathbb {R}$is one of its most basic properties, known far outside of mathematics. Cantor’s 1874 proof of the uncountability of$\mathbb {R}$even appears in the very first paper on set theory, i.e., a historical milestone. In this paper, we study the uncountability of${\mathbb R}$in Kohlenbach’shigher-orderReverse Mathematics (RM for short), in the guise of the following principle:$$\begin{align*}\mathit{for \ a \ countable \ set } \ A\subset \mathbb{R}, \mathit{\ there \ exists } \ y\in \mathbb{R}\setminus A. \end{align*}$$An important conceptual observation is that the usual definition of countable set—based on injections or bijections to${\mathbb N}$—does not seem suitable for the RM-study of mainstream mathematics; we also propose a suitable (equivalent over strong systems) alternative definition of countable set, namelyunion over${\mathbb N}$of finite sets; the latter is known from the literature and closer to how countable sets occur ‘in the wild’. We identify a considerable number of theorems that are equivalent to the centred theorem based on our alternative definition. Perhaps surprisingly, our equivalent theorems involve most basic properties of the Riemann integral, regulated or bounded variation functions, Blumberg’s theorem, and Volterra’s early work circa 1881. Our equivalences are alsorobust, promoting the uncountability of${\mathbb R}$to the status of ‘big’ system in RM.

Oscillations of BV measures on unbounded nested fractals

Motivated by recent developments in the theory of bounded variation functions on unbounded nested fractals, this paper studies the exact asymptotics of functionals related to the total variation measure associated with unions of $n$-cells. The oscillatory behavior observed implies the non-uniqueness of BV measures in this setting.

Traveling waves for a Fisher-type reaction–diffusion equation with a flux in divergence form

Analysis of the speed of propagation in parabolic operators is frequently carried out considering the minimal speed at which its traveling waves (TWs) move. This value depends on the solution concept being considered. We analyze an extensive class of Fisher-type reaction–diffusion equations with flows in divergence form. We work with regular flows, which may not meet the standard elliptical conditions, but without other types of singularities. We show that the range of speeds at which classic TWs move is an interval unbounded to the right. Contrary to classic examples, the infimum may not be reached. When the flow is elliptic or over-elliptic, the minimum speed of propagation is achieved. The classic TW speed threshold is complemented by another value by analyzing an extension of the first-order boundary value problem to which the classic case is reduced. This singular minimum speed can be justified as a viscous limit of classic minimal speeds in elliptic or over-elliptic flows. We construct a singular profile for each speed between the minimum singular speed and the speeds at which classic TWs move. Under additional assumptions, the constructed profile can be justified as that of a TW of the starting equation in the framework of bounded variation functions. We also show that saturated fronts verifying the Rankine–Hugoniot condition can appear for strictly lower speeds even in the framework of bounded variation functions.

The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation

AbstractIn the classical Fourier analysis, the representation of the double Fourier transform as the integral off(x,y)exp(−i⟨(x,y),(s1,s2)⟩f(x,y)\exp(-i\langle(x,y),(s_{1},s_{2})\rangleis usually defined from the Lebesgue integral. Using an improper Kurzweil–Henstock integral, we obtain a similar representation of the Fourier transform of non-Lebesgue integrable functions onR2\mathbb{R}^{2}. We prove the Riemann–Lebesgue lemma and the pointwise continuity for the classical Fourier transform on a subspace of non-Lebesgue integrable functions which is characterized by the bounded variation functions in the sense of Hardy–Krause. Moreover, we generalize some properties of the classical Fourier transform defined onLp⁢(R2)L^{p}(\mathbb{R}^{2}), where1<p≤21<p\leq 2, yielding a generalization of the results obtained by E. Hewitt and K. A. Ross. With our integral, we define the space of integrable functionsKP⁢(R2)\mathrm{KP}(\mathbb{R}^{2})which contains a subspace whose completion is isometrically isomorphic to the space of integrable distributions on the plane as defined by E. Talvila [The continuous primitive integral in the plane,Real Anal. Exchange45(2020), 2, 283–326]. A question arises about the dual space of the new spaceKP⁢(R2)\mathrm{KP}(\mathbb{R}^{2}).

Fractional Differential Equations in Electronic Information Models

Abstract The article first uses the fractional derivative to define a new fractional bounded variation function space. This method constructs the corresponding electronic information image model denoising mask by setting a smaller fractional integration order. The experimental results show that the image denoising algorithm based on fractional integration can not only improve the signal-to-noise ratio of the image compared with the traditional denoising method, but also can better retain the details of the edge and texture of the electronic information image.

Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation

<p style='text-indent:20px;'>We consider optimal control of an elliptic two-point boundary value problem governed by functions of bounded variation (BV). The cost functional is composed of a tracking term for the state and the BV-seminorm of the control. We use the mixed formulation for the state equation together with the variational discretization approach, where we use the classical lowest order Raviart-Thomas finite elements for the state equation. Consequently the variational discrete control is a piecewise constant function over the finite element grid. We prove error estimates for the variational discretization approach in combination with the mixed formulation of the state equation and confirm our analytical findings with numerical experiments.</p>

BOUNDED VARIATION ON THE SIERPIŃSKI GASKET

Under certain continuity conditions, we estimate upper and lower box dimensions of the graph of a function defined on the Sierpiński gasket. We also give an upper bound for Hausdorff dimension and box dimension of the graph of a function having finite energy. Further, we introduce two sets of definitions of bounded variation for a function defined on the Sierpiński gasket. We show that fractal dimension of the graph of a continuous function of bounded variation is [Formula: see text] We also prove that the class of all bounded variation functions is closed under arithmetic operations. Furthermore, we show that every function of bounded variation is continuous almost everywhere in the sense of [Formula: see text]-dimensional Hausdorff measure.

A General 3D Non-Stationary 6G Channel Model With Time-Space Consistency

This paper proposes a novel general sixth-generation (6G) irregular-shaped geometry-based stochastic model (IS-GBSM) for millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) wireless communication systems. The proposed IS-GBSM can model spherical wavefront propagation, high delay resolution, and three-dimensional (3D) space-time-frequency non-stationarity of channels with time-space consistency, and thus can be applied to integrated communication and sensing systems. To capture the 3D non-stationarity with channel consistency, a new hybrid method combining geometric and parametric methods is developed. Based on the geometric method, i.e., visibility region (VR), the smooth and consistent appearance and disappearance of clusters during array-time evolution are modeled. With the help of the parametric method, a frequency-dependent path gain is introduced, and a visible factor based on uniform continuity theorem and bounded variation function is further proposed to capture the soft cluster power handover during cluster array-time evolution. Key channel statistics of the proposed IS-GBSM are derived and investigated. Simulation results demonstrate that the space-time-frequency non-stationarity is modeled and time-space consistency is captured. Finally, the utility of the proposed IS-GBSM is verified by the excellent agreement between simulation results and measurement.

2-Topological reflexivity of sets of isometries between bounded variation function spaces

2-Topological reflexivity of sets of isometries between bounded variation function spaces

THE EFFECT OF THE WEYL FRACTIONAL INTEGRAL ON FUNCTIONS

This paper mainly discusses the influence of the Weyl fractional integrals on continuous functions and proves that the Weyl fractional integrals can retain good properties of many functions. For example, a bounded variation function is still a bounded variation function after the Weyl fractional integral. Continuous functions that satisfy the Holder condition after the Weyl fractional integral still satisfy the Holder condition, furthermore, there is a linear relationship between the order of the Holder conditions of the two functions. At the end of this paper, the classical Weierstrass function is used as an example to prove the above conclusion.

On BV functions and essentially bounded divergence-measure fields in metric spaces

By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation (BV) in terms of suitable vector fields on a complete and separable metric measure space $(\mathbb{X},d,\mu)$ equipped with a non-negative Radon measure $\mu$ finite on bounded sets. Then, we extend the concept of divergence-measure vector fields $\mathcal{DM}^p(\mathbb{X})$ for any $p\in\[1,\infty]$ and, by simply requiring in addition that the metric space is locally compact, we determine an appropriate class of domains for which it is possible to obtain a Gauss–Green formula in terms of the normal trace of a $\mathcal{DM}^\infty(\mathbb{X})$ vector field. This differential machinery is also the natural framework to specialize our analysis for $\mathsf{RCD}(K,\infty)$ spaces, where we exploit the underlying geometry to determine the Leibniz rules for $\mathcal{DM}^\infty(\mathbb{X})$ and ultimately to extend our discussion on the Gauss–Green formulas.

CONSTRUCTION OF SOME SPECIAL CONTINUOUS FUNCTIONS AND ANALYSIS OF THEIR FRACTIONAL INTEGRALS

In this paper, we mainly construct two continuous functions defined on [Formula: see text] which have only one unbounded variation point, and show that both Box and Hausdorff dimensions of these functions are 1. We prove that their Riemann–Liouville fractional integrals are bounded variation function. We also show that these functions repair inaccuracies of the example in [N. Liu and K. Yao, Fractal dimension of a special continuous function, Fractals 26(4) (2018) 1850048; Y. S. Liang, Definition and classification of one-dimensional continuous functions with unbounded variation, Fractals 25(5) (2017) 1750048]. In the proofs, we mainly use Leibniz’s alternating series test and the properties of logarithmic function.

Adaptive estimation of multivariate piecewise polynomials and bounded variation functions by optimal decision trees

Proposed by Donoho (Ann. Statist. 25 (1997) 1870–1911), Dyadic CART is a nonparametric regression method which computes a globally optimal dyadic decision tree and fits piecewise constant functions in two dimensions. In this article, we define and study Dyadic CART and a closely related estimator, namely Optimal Regression Tree (ORT), in the context of estimating piecewise smooth functions in general dimensions in the fixed design setup. More precisely, these optimal decision tree estimators fit piecewise polynomials of any given degree. Like Dyadic CART in two dimensions, we reason that these estimators can also be computed in polynomial time in the sample size N via dynamic programming. We prove oracle inequalities for the finite sample risk of Dyadic CART and ORT, which imply tight risk bounds for several function classes of interest. First, they imply that the finite sample risk of ORT of order r≥0 is always bounded by CklogN N whenever the regression function is piecewise polynomial of degree r on some reasonably regular axis aligned rectangular partition of the domain with at most k rectangles. Beyond the univariate case, such guarantees are scarcely available in the literature for computationally efficient estimators. Second, our oracle inequalities uncover minimax rate optimality and adaptivity of the Dyadic CART estimator for function spaces with bounded variation. We consider two function spaces of recent interest where multivariate total variation denoising and univariate trend filtering are the state of the art methods. We show that Dyadic CART enjoys certain advantages over these estimators while still maintaining all their known guarantees.

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

With a view toward fractal spaces, by using a Korevaar–Schoen space approach, we introduce the class of bounded variation (BV) functions in the general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry–Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $$L^1$$ Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in $$L^1$$ . The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

Bounded variation functions and some properties on metric spaces

Bounded variation functions of a single variable were first introduced by Camille Jordan (1881). Bounded variation functions or \U0001d435\U0001d449 functions is a function with the total variation is finite. The variation in a function aims to measure how much increase and decline occure in the function. In this paper, we generalize this function in the metric space. Let (\U0001d44b, \U0001d451) be a complete metric space and \U0001d6fe: \U0001d446 → ℝ is a function, where \U0001d446 is a closed and bounded subset of \U0001d44b. The function \U0001d6fe is of bounded variation on \U0001d446 if \U0001d449(\U0001d6fe, \U0001d446) is finite, i.e \U0001d449(\U0001d6fe, \U0001d446) < ∞. The result of this paper is the definition and properties algebraic of bounded variation function on metric spaces. Some properties of this function is multiplication with scalar, sum of two functions, and product of two functions. The function of bounded variation to \U0001d446 are also bounded variation to each subspace from \U0001d446. If \U0001d6fe is a bounded variation function on \U0001d446 then \U0001d6fe is bounded on \U0001d446. We also consider the related topic it is absolute continuity. If a function \U0001d6fe is absolutely continuous on \U0001d446 then \U0001d6fe is of bounded variation on \U0001d446.

Primitive function of the Dunford integral

In this paper, Dunford integrals and their primitive functions are discussed. We discuss its properties related to absolutely continuous, strictly absolutely continuous, bounded variation function, strictly bounded variation function and their generalizations. The result is obtained that for each function which integrated Dunford, then the primitive function is a continuous, absolutely continuous, and bounded variation function. Furthermore, its generalized absolutely continuous and generalized bounded variation function.

Rough traces of BV functions in metric measure spaces

Following a Maz'ya-type approach, we adapt the theory of rough traces of functions of bounded variation (BV) to the context of doubling metric measure spaces supporting a Poincaré inequality. This eventually allows for an integration by parts formula involving the rough trace of such functions. We then compare our analysis with the study done in a recent work by Lahti and Shanmugalingam, where traces of BV functions are studied by means of the more classical Lebesgue-point characterization, and we determine the conditions under which the two notions coincide.

Reflexivity of sets of isometries on bounded variation function spaces

For arbitrary subsets X and Y of the real line with at least two points, let BV(X) (resp. BV(Y)) be the Banach space of all functions of bounded variation on X (resp. Y) endowed with the natural norm , where and denote the supremum norm and the total variation of a function, respectively. We show that the set of all surjective linear isometries from BV(X) onto BV(Y) is topologically reflexive. Among the consequences, it is also proved that the set of all isometric reflections, and the set of all generalized bi-circular projections on BV(X) are topologically reflexive.