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We introduce weighted amalgam-type spaces and analyze their relations with some known spaces. Integrability results for the Fourier transform of a function with the derivative from one of those spaces are proved. The obtained results are applied to the integrability of trigonometric series with the sequence of coefficients of bounded variation.

New relations between the Banach algebras of absolutely convergent Fourier integrals of complex-valued measures of Wiener and various issues of trigonometric Fourier series (see classical monographs by A. Zygmund [1] and N. K. Bary [2]) are described. Those bilateral interrelations allow one to derive new properties of the Fourier series from the known properties of the Wiener algebras, as well as new results to be obtained for those algebras from the known properties of Fourier series. For example, criteria, i.e. simultaneously necessary and sufficient conditions, are obtained for any trigonometric series to be a Fourier series, or the Fourier series of a function of bounded variation, and so forth. Approximation properties of various linear summability methods of Fourier series (comparison, approximation of function classes and single functions) and summability almost everywhere (often with the set indication) are considered. The presented material was reported by the author on 12.02.2021 at the Zoom-seminar on the theory of real variable functions at the Moscow State University.

We propose a scheme for the solution of a mixed problem for a parabolic differential equation with coefficients that are generalized derivatives of functions of bounded variation. We seek the solution of this problem by the method of reduction. According to this method, the solution of the proposed problem is reduced to the solution of two problems: (i) a quasistationary boundary-value problem with input boundary conditions and (ii) a mixed problem with trivial boundary conditions. The first of these problems is solved by introducing the quasiderivative. For the solution of the second problem, we use the Fourier method and the expansion in eigenfunctions of a certain boundary-value problem for a quasidifferential equation of the second-order. The obtained results can be used, in particular, for the investigation of the processes of heat transfer in multilayer plates, hollow cylinders, and spheres.

This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schrodinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space L 2 ( R 2 ) . This result is sharp in view of the fact that the flow map cannot be C 3 continuous below L 2 ( R 2 ) . Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space U S 2 and its dual V S 2 of square bounded variation functions. We also prove that the initial-value problem is locally well-posed in H s ( R 2 ) , s > 0 . Our results extend to the finite depth version of the Dysthe equation.

We consider the problem of reconstructing a compactly supported function from samples of its Fourier transform taken along a spiral. We determine the Nyquist sampling rate in terms of the density of the spiral and show that, below this rate, spirals suffer from an approximate form of aliasing. This sets a limit to the amount of undersampling that compressible signals admit when sampled along spirals. More precisely, we derive a lower bound on the condition number for the reconstruction of functions of bounded variation, and for functions that are sparse in the Haar wavelet basis.

We study the trigonometric series $$\sum_{n=1}^\infty \lambda_n \cos nx$$ and $$\sum_{n=1}^\infty \lambda_n \sin nx$$ with $$\{\lambda_n\}$$ being a sequence of bounded variation. Let $$\psi$$ denote the sum of such a series. We obtain necessary and sufficient conditions for the validity of the weighted Fourier inequality $$\left(\intop_0^\pi\mathopen|\psi(x)|^q \omega(x)\,dx\right)^{1/q} \le C\!\left(\sum_{n=1}^\infty u_n\left(\sum_{k=n}^\infty\mathopen|\lambda_{k}-\lambda_{k+1}|\right)^p \right)^{1/p}$$ , $$0

Let sequences , satisfy the relations , , , as , and let and . We redefine the function as on the interval by polygonal arcs in such a way that the function remains continuous and vanishes on a neighbourhood of the ends of the interval. Also let the function and the pair of sequences , be connected by the equiconvergence condition. Then for the classical Lagrange–Jacobi interpolation processes to approximate uniformly with respect to on it is sufficient that have bounded variation on . In particular, if the sequences and are bounded, then for the classical Lagrange–Jacobi interpolation processes to approximate uniformly with respect to on it is sufficient that the variation of be bounded on , .

In this study, the authors propose a fractional derivative-based image decomposition and denoising model which decomposes the image into the cartoon component (the component formed by homogeneous regions and with sharp boundaries) and the texture (or noise) component. The cartoon component is modelled by a function of the fractional-order total bounded variation, while the texture component is modelled by an oscillatory function, bounded in the negative Sobolev space norm. The authors give the corresponding minimisation functional, after some transformations, and then the resulting fractional-order partial differential equation can be solved using the Fourier transform. By symmetry and asymmetry of the fractional-order derivative, some generalisations and variants of the proposed model are also introduced. Finally, the authors implement the algorithm by the fractional-order finite difference in the frequency-domain. The experimental results demonstrate that the proposed models make objective and visual improvements compared with other standard approaches in the task of decomposition and denoising.

A conditioned equivalence is proved for a certain weighted Sobolev space to the space of Riemann integrable functions. An equivalence representing a new result that not only asserts the sufficiency (but non-necessity) nature of bounded variation of functions for their Riemann integrability, but also reveals a potential for some novel computational findings.

We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes account of area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied. Concerning the total mean and Gauss curvature, the classical counterexample by Schwarz-Peano to the definition of area is also analyzed., Comment: 25 pages

Henstock--Kurzweil integral, a nonabsolute integral, is a natural extension of the Riemann integral that was studied independently by Ralph Henstock and Jaroslav Kurzweil. This paper will introduce the Henstock--Kurzweil--Stieltjes integral of $\mathcal{C}[a,b]$-valued functions defined on a closed interval $[f,g]\subseteq\mathcal{C}[a,b]$, where $\mathcal{C}[a,b]$ is the space of all continuous real-valued functions defined on $[a,b]\subseteq\mathbb{R}$. Some simple properties of this integral will be formulated including the Cauchy criterion and an existence theorem will be provided.

This paper addresses an optimal impulsive control problem whose trajectories are functions of bounded variation and impulsive controls are regular vector measures. This problem is characterized by two main features. First, the dynamical control system to be considered may not possess the so-called well-posedness property. Second, the constraints on the one-sided limits of states are presented. Such constraints are interpreted as multipoint state constraints. For this problem, we derive global optimality conditions based on using of compound Lyapunov type functions which possess strongly monotone properties with respect to the control system. As a motivating case, a model of advertising expenses optimization for mutually complementary products is considered. For this model, we propose four variants of resolving sets of Lyapunov type functions and explain the technique of applying the optimality conditions.

In this paper, we study the existence of solutions for evolution problems of the form \begin{document}$ -\frac{du}{dr}(t) \in A(t)u(t) + F(t, u(t))+f(t, u(t)) $\end{document} , where, for each \begin{document}$ t $\end{document} , \begin{document}$ A(t) : D(A(t)) \to 2 ^H $\end{document} is a maximal monotone operator in a Hilbert space \begin{document}$ H $\end{document} with continuous, Lipschitz or absolutely continuous variation in time. The perturbation \begin{document}$ f $\end{document} is separately integrable on \begin{document}$ [0, T] $\end{document} and separately Lipschitz on \begin{document}$ H $\end{document} , while \begin{document}$ F $\end{document} is scalarly measurable and separately scalarly upper semicontinuous on \begin{document}$ H $\end{document} , with convex and weakly compact values. Several new applications are provided.

An impulse nonlinear problem admitting discontinuous solutions that are functions of bounded variation is studied. This problem models the deformation of a discontinuous string (chains of strings fastened together by springs) with elastic supports in the form of linear and nonlinear springs (for example, springs with different turns, whose deformations do not obey Hooke’s law). The model is described by a second-order differential equation with derivatives in special measures and Dirichlet boundary conditions. Existence theorems are proved, and conditions for the existence of nonnegative solutions are obtained.

We provide three different characterizations of the space \begin{document}$ BV(O, \gamma) $\end{document} of the functions of bounded variation with respect to a centred non-degenerate Gaussian measure \begin{document}$ \gamma $\end{document} on open domains \begin{document}$ O $\end{document} in Wiener spaces. Throughout these different characterizations we deduce a sufficient condition in order to belong to \begin{document}$ BV(O, \gamma) $\end{document} by means of the Ornstein-Uhlenbeck semigroup and we provide an explicit formula for one-dimensional sections of functions of bounded variation. Finally, we apply our techniques to Fomin differentiable probability measures \begin{document}$ \nu $\end{document} on a Hilbert space \begin{document}$ X $\end{document} , and we infer a characterization of the space \begin{document}$ BV(O, \nu) $\end{document} of the functions of bounded variation with respect to \begin{document}$ \nu $\end{document} on open domains \begin{document}$ O\subseteq X $\end{document} .

This paper proposes a gradient extremum seeking method to address locally quadratic static maps in the presence of time-varying delays. Accommodating nonconstant delays has a strong impact in the predictor construction in terms of the associated transport partial differential equation (PDE) with variable convection speeds beyond the restrictions imposed on the delay regarding its arbitrary duration but bounded variation. A novel predictor design using perturbation-based estimates of the unknown Gradient and Hessian of the map must be introduced to handle this variable nature of the delays, which can arise both in the input and output channels of the nonlinear map to be optimized. Local exponential stability and convergence to a small neighborhood of the unknown extremum point are guaranteed. This technical result is assured by using backstepping transformation and averaging theory in infinite dimensions. Implementation aspects of the presented predictor for variable delays as well as the extension to state-dependent delays are also discussed. At last, we introduce the first results in the topic of extremum seeking control for cascades of transport PDEs that are interconnected through boundary conditions. Such a PDE-PDE cascade is useful to represent simultaneous time- and state-dependent delays. A simulation example illustrates the effectiveness of the proposed predictor-based extremum seeking approach for time-delay compensation.

In a concise description of mathematical methods, Henri Lebesgue underlined the importance of definitions and axioms (see [47]).

We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and “interpolates” the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) L∞-estimates and L2-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for p and hp methods for singular problems, and answer some open questions posed in some recent literature. Ministry of Education (MOE) The research of the first author was supported by the National Natural Science Foundation of China (Nos. 11801120 and 11771107), the Fundamental Research Funds for the Central Universities (Grant No. HIT.NSRIF.2020081), the Natural Science Foundation of Heilongjiang Province (Nos. LH2020A004 and LH2021A011), and the Guangdong Basic and Applied Basic Research Foundation (No.2020B1515310006). The research of the second author is partially supported by Singapore MOE AcRF Tier 2 Grant: MOE2018-T2-1-059 and Tier 1 Grant: RG15/21. The research of the third author was supported by the National Natural Science Foundation of China (Nos. 11971131, U1637208, 61873071, 51476047).

In this chapter, we discuss a unique method to time-frequency analysis which gives a centralized way to represent discrete and continuous time-frequency. This serves as a straightforward way to include all possible (countable) discrete and continuous time scales in one model. We consider the Mexican hat wavelet which is one of the basic wavelet functions formulated by the second derivative of Gaussian function to define the Mexican hat wavelet transform (MHWT). Further, the theory of MHWT is implemented to obtain the Mexican hat wavelet Stieltjes transform (MHWST) of a bounded variation function. Some convenient properties of MHWST are also presented. Further, a standard method is introduced for representing functions of class B(m, n). Besides, an integral transform is constructed with the help of the Fourier summation kernel. This construction results in a flexible way to present some conditions that are necessary and sufficient for a function of class B(m, n) to be Mexican hat wavelet and MHWST.

In this paper, we establish the exponential BV stability of general systems of discretized scalar conservation laws with positive speed. The focus is on numerical approximation of such systems using a wide class of slope limiter schemes built from the upwind monotone flux. The proof is based on a Lyapunov analysis taken from the continuous theory (Coron et al. in J Differ Equ 262(1):1–30, 2017) and a careful use of Harten formalism.

We study the one-dimensional version of the Rudin–Osher–Fatemi (ROF) denoising model and some related TV-minimization problems. A new proof of the equivalence between the ROF model and the so-called taut string algorithm is presented, and a fundamental estimate on the denoised signal in terms of the corrupted signal is derived. Based on duality and the projection theorem in Hilbert space, the proof of the taut string interpretation is strictly elementary with the existence and uniqueness of solutions (in the continuous setting) to both models following as by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends to zero, are recalled and efficient proofs provided. The taut string interpretation plays an essential role in the proof of the fundamental estimate. This estimate implies, among other things, the strong convergence (in the space of functions of bounded variation) of the denoised signal to the corrupted signal as the regularization parameter vanishes. It can also be used to prove semi-group properties of the denoising model. Finally, it is indicated how the methods developed can be applied to related problems such as the fused lasso model, isotonic regression and signal restoration with higher-order total variation regularization.

For a Lebesgue integrable complex-valued function $f$ defined on $\mathbb R^+:=[0,\infty )$ let $\hat f$ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that $\hat f(y)\to 0$ as $y\to \infty $. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of $L^1(\mathbb R^+)$ there is a definite rate at which the Walsh-Fourier transform tends to zero. We determine this rate for functions of bounded variation on $\mathbb R^+$. We also determine such rate of Walsh-Fourier transform for functions of bounded variation in the sense of Vitali defined on $(\mathbb R^+)^N$, $N\in \mathbb N$.

We consider the strong rate of convergence of the Euler–Maruyama approximation for stochastic differential equations with possibly discontinuous drift and Holder continuous diffusion coefficient. In particular, we show that the rates obtained in some recent papers can be improved under an additional assumption that the diffusion coefficient is of bounded variation.

We are interested in the evolution of a compressible fluid under its self-generated gravitational field. Assuming here Gowdy symmetry, we investigate the algebraic structure of the Euler equations satisfied by the mass density and velocity field. We exhibit several interaction functionals that provide us with a uniform control on weak solutions in suitable Sobolev norms or in bounded variation. These functionals allow us to study the local regularity and nonlinear stability properties of weakly regular fluid flows governed by the Euler-Gowdy system. In particular for the Gowdy equations, we prove that a spurious matter field arises under weak convergence, and we establish the nonlinear stability of weak solutions., Comment: 37 pages. v2: fix typos to match version published in 2019

The purpose of this paper is twofold. Firstly I present an optimal regularity result for minimizers of a \begin{document}$ 1D $\end{document} convex functional involving the BV-norm, under Neumann boundary condition. This functional is a simplified version of models occuring in Image Processing. Secondly I investigate the existence of minimizers for the same functional under Dirichlet boundary condition. Surprisingly, this turns out to be a delicate issue, which is still widely open.

Using an explicit catching-up algorithm, we prove the existence of absolutely continuous as well as bounded variation continuous solutions to a second-order perturbed Moreau’s sweeping process with the normal cone of a subsmooth moving set, which depends both on the time and on the state.

This is the first attempt to generalize the problem of $$L^1$$ convergence of trigonometric series to the non-periodic case. We extend one of the most general results and then show the way how to derive its prototype from the obtained extension.

We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda~Jr. We show that these two notions coincide, if the measure is doubling and supports a $1$-Poincar\'e inequality. In doing so, we also prove that if the measure is doubling and supports a $1$-Poincar\'e inequality, then the metric space supports a \emph{Semmes family of curves} structure., Comment: 20 pages

Summary Calculation of electromagnetic fields in layered models of the medium leads to the need to calculate the direct and inverse Fourier-Bessel (Hankel) transformations. The calculation of such improper integrals is complicated by the oscillating nature of the integrand. As a rule, it is the product of a smooth function of bounded variation by a Bessel function of the first kind. This circumstance allows the first function to be approximated in such a way that the integral of its product by the Bessel function is calculated analytically. We have analyzed two calculation algorithms. One of them is based on the expansion in terms of eigenfunctions of the Hankel integral operator, the second is based on approximation by a linear combination of exponentials. Both approaches make it possible to present the results of calculating integrals in an analytical form that allows their further mathematical analysis.

We consider the regular Lagrangian flow X associated to a bounded divergence-free vector field b with bounded variation. We prove a Lusin-Lipschitz regularity result for X and we show that the Lipschitz constant grows at most linearly in time. As a consequence we deduce that both geometric and analytical mixing have a lower bound of order $t^{-1}$ as $t \to \infty$.

In this chapter we define the Lebesgue measure and Lebesgue integral on time scales. We show the difference between the classical Lebesgue integral and the time scale Lebesgue integral. We introduce the absolutely continuous functions and deduct some of their properties. In this chapter we define functions of bounded variation and give some of their properties. We introduce Lp spaces, Sobolev spaces, and generalized derivatives. As their applications, we investigate the weak solutions and Euler solutions of dynamic systems. We prove an analogue of the Gronwall type inequality. We define \(\varDelta \times \mathcal {B}\)-measurable set-valued functions and deduct some of their basic properties.

We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces., 26 pages, 1 figure

We study Riemann-Lebesgue integrability for interval-valued multifunctions relative to an interval-valued set multifunction. Some classic properties of the RL integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the purpose of illustration, an example is given in case of fractal image coding for image compression, and for edge detection algorithm. In these contexts, the image modelization as an interval valued multifunction is crucial since allows to take into account the presence of quantization errors (such as the so-called round-off error) in the discretization process of a real world analogue visual signal into a digital discrete one.

This paper is concerned with an optimal impulsive control problem under intermediate state constraints. The control system of this problem is bilinear with respect to states of bounded variation and impulsive controls given by vector Borel measures. We propose necessary and sufficient optimality conditions based on special functions of the Lyapunov type. These functions possess the property of strong and weak monotonicity with respect to the impulsive control system.

We study fine potential theory and in particular partitions of unity in quasiopen sets in the case $p=1$. Using these, we develop an analog of the discrete convolution technique in quasiopen (instead of open) sets. We apply this technique to show that every function of bounded variation (BV function) can be approximated in the BV and $L^{\infty}$ norms by BV functions whose jump sets are of finite Hausdorff measure. Our results seem to be new even in Euclidean spaces but we work in a more general complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality., Comment: arXiv admin note: text overlap with arXiv:1806.04647, arXiv:1811.07713

In a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we study strict subsets, i.e. sets whose variational capacity with respect to a larger reference set is finite, in the case $p=1$. Relying on the concept of fine topology, we give a characterization of those strict subsets that are also sets of finite perimeter, and then we apply this to the study of condensers as well as BV capacities. We also apply the theory to prove a pointwise approximation result for functions of bounded variation., Comment: arXiv admin note: text overlap with arXiv:1812.11087

A new class of functions is introduced closely related to that of functions with bounded Tonelli variation and to the real Hardy space. For this class, conditions for integrability of the Fourier transform are established.

The Riemann integral can be considered an evolution of Cauchy’s integral, in that certain functions that are not integrable according to Cauchy become integrable in Riemann’s theory. At the same time, alas, in the new framework integration is no longer the inverse operation to differentiation. Thus the fundamental theorem of calculus, in the version for continuous maps proved by Cauchy, loses its status of calculus’ highest pinnacle and becomes a mere special case of a much bigger picture. The existence of continuous maps with no derivative, of integrable functions whose integral map is not differentiable and the ensuing demise of the fundamental theorem of Cauchy’s integral calculus, persuaded many mathematicians, most notably Lebesgue, to investigate the relationship between integrals and primitives. In particular Lebesgue observed that the issues with integral calculus arise when the derivative f is not bounded. Lebesgue showed that for a function f to be summable the corresponding primitive F must have bounded variation. The idea of functions with bounded variation had in the meantime been elaborated by Jordan for others reasons. Yet, Lebesgue stopped short of saying $$|F(x)-F(a)|=\int _{[a, x]} |f(t)|\ell (dt)$$ for every \(x\in [a,b]\), because in that case the difference of the two sides would be a monotone map with bounded variation and zero derivative \(\ell \)-almost everywhere on [a, x]. He proved that among functions with bounded variation, the only ones satisfying $$F(x)-F(a)=\int _{[a, x]}f(t)\ell (dt)$$ are the absolutely continuous functions, as defined by Giuseppe Vitali. This Chapter is therefore devoted to the study of functions with bounded variation and of absolutely continuous functions.

We prove various extensions of the Coifman–Rubio de Francia–Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call ℓ r ( ℓ s ) {\ell ^{r}(\ell ^{s})} -boundedness, which implies R \mathcal {R} -boundedness in many cases. The proofs are based on new Littlewood–Paley–Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors.

‎Let $C[0,T]$ denote the space of continuous real-valued functions on $[0,T]$‎. ‎In this paper we introduce two Banach algebras‎: ‎one of them is defined on $C[0,T]$ and the other is a space of equivalence classes of measures over paths of bounded variation on $[0,T]$‎. ‎We establish an isometric isomorphism between them and evaluate analytic Feynman integrals of the functions in the Banach algebras‎, ‎which play significant roles in the Feynman integration theories and quantum mechanics‎.

In this paper, we investigate the initial-boundary value problem of compressible Euler equations including friction and heating that model the transonic Fanno–Rayleigh flows through symmetric variable area nozzles. In particular, the case of contracting nozzles is considered. A new version of a generalized Glimm scheme (GGS) is presented for establishing the global existence of entropy solutions with bounded variation. Modified Riemann and boundary Riemann solutions are applied to design this GGS, which is constructed using the contraction matrices acting on the homogeneous Riemann (or boundary-Riemann) solutions. The extended Glimm–Goodman’s type of wave interaction estimates are investigated to determine the stability of the scheme and the positivity of gas velocity that results in the existence of the weak solution. The limit of approximation solutions serves as an entropy solution. Moreover, a quantitative relation between the shape of the nozzle, friction, and heat is proposed for the global existence result in the contracting nozzle. Numerical simulations of the contraction-expansion and expansion-contraction nozzles are presented to validate the scheme.

Extremal functions are exhibited in Poincare trace inequalities for functions of bounded variation in the unit ball $${\mathbb {B}}^n$$ of the n-dimensional Euclidean space $${{\mathbb {R}}}^n$$ . Trial functions are subject to either a vanishing mean value condition, or a vanishing median condition in the whole of $${\mathbb {B}}^n$$ , instead of just on $$\partial {\mathbb {B}}^n$$ , as customary. The extremals in question take a different form, depending on the constraint imposed. In particular, under the median constraint, unusually shaped extremal functions appear. A key step in our approach is a characterization of the sharp constant in the relevant trace inequalities in any admissible domain $$\Omega \subset {{\mathbb {R}}}^n$$ , in terms of an isoperimetric inequality for subsets of $$\Omega $$ .

In this note we give a characterization of Nikolskii–Besov classes of functions of fractional smoothness (see [1–3]) by means of a nonlinear integration by parts formula in the form of a certain nonlinear inequality. This characterization is motivated by the recent papers [4–6] on distributions of polynomials in Gaussian random variables, where it has been shown that the distribution densities of nonconstant polynomials in Gaussian random variables belong to Nikolskii–Besov classes. Our main result is a generalization of the classical description of the class BV of functions of bounded variation in terms of integration by parts.

This paper is devoted to the study of evolution problems of the form $-\frac {du}{dr}(t) \in A(t)u(t) + f(t, u(t))$ in a new setting, where, for each t, A(t) : D(A(t)) → 2 H is a maximal monotone operator in a Hilbert space H and the mapping t↦A(t) has continuous bounded or Lipschitz variation on [0, T], in the sense of Vladimirov’s pseudo-distance. The measure dr gives an upper bound of that variation. The perturbation f is separately integrable on [0, T] and separately Lipschitz on H. Several versions and new applications are presented.

Total variation (TV) based Models have been widely used in image restoration problems. However, these models are always accompanied by staircase effect due to the property of bounded variation (BV) space. In this paper, we present two high order variational models based on total generalized variation (TGV) with two common and important non-quadratic fidelity data terms for blurred images corrupted by impulsive and Poisson noises. Since the direct extension of alternative direction method of multipliers (ADMM) to solve three-block convex minimization problems is not necessarily convergent, we develop an efficient algorithm called Prediction–Correction ADMM to solve our models and also show the convergence of the proposed method. Moreover, we extend our models to deal with color images restoration. Numerical experiments demonstrate that the proposed high order models can reduce staircase effect while preserving edges and outperform classical TV based models in SNR and SSIM values.

We estimate the error of Gauss-Jacobi quadrature rule applied to a function f, which is supposed locally absolutely continuous in some Besov type spaces, or of bounded variation on [-1,1]. In the first case the error bound concerns the weighted main part phi-modulus of smoothness of f introduced by Z. Ditzian and V. Totik, while in the second case we deal with a Stieltjes integral with respect to f. The stated estimates generalize several error bounds from literature and, in both the cases, they assure the same convergence rate of the error of best polynomial approximation in weighted L-1 space. (C) 2017 IMACS. Published by Elsevier B.V. All rights reserved.

International audience; In this paper, we study the well−posedness (in the sense of existence and uniqueness of a solution) of a discontinuous sweeping process involving prox-regular sets in Hilbert spaces. The variation of the moving set is controlled by a positive Radon measure and the perturbation is assumed to satisfy a Lipschitz property. The existence of a solution with bounded variation is achieved thanks to the Moreau’s catching-up algorithm adapted to this kind of problem. Various properties and estimates of jumps of the solution are also provided. We give sufficient conditions to ensure the uniform prox-regularity when the moving set is described by inequality constraints. As an application, we consider a nonlinear differential complementarity system which is a combination of an ordinary differential equation with a nonlinear complementarily condition. Such problems appear in many areas such as nonsmooth mechanics, nonregular electrical circuits and control systems.

We establish a local martingaleMassociate withf(X,Y) under some restrictions onf, whereYis a process of bounded variation (on compact intervals) and eitherXis a jump diffusion (a special case being a Lévy process) orXis some general (càdlàg metric-space valued) Markov process. In the latter case,fis restricted to the formf(x,y)=∑k=1Kξk(x)ηk(y). This local martingale unifies both Dynkin's formula for Markov processes and the Lebesgue–Stieltjes integration (change of variable) formula for (right-continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed, then this local martingale becomes anL2-martingale. Convergence of the product of this Martingale with some deterministic function ( of time ) to 0 both inL2and almost sure is also considered and sufficient conditions for functions for which this happens are identified.

Let R is a Riemann surface, glued from finitely or countably many domains in the extended complex plane Open image in new window so that the following conditions are satisfied: each point in R projects onto a point w = prW in one on the glued domains, each point in R has a neighbourhood which is a univalent disk, or multivalent disk with the unique ramification point at the centre of disk. We study elementary properties of functions of bounded variation and sets of finite perimeter in an open set Q ⊂ R {W ∈ R: W is a ramification point or prW = ∞}. Further, by using Ziemer’s technique, we obtain the main result $$C\left( {{F_{0,}}{F_1},G} \right) \cdot M\left( {{F_{0,}}{F_1},G} \right) = 1$$ . Here G is an open set with the compact closure on R, F0 and F1 are disjoint compact sets in the closure of G, C(F0, F1, G) is the conformal capacity of the condenser (F0, F1, G), M(F0, F1, G) is the conformal module of the family of all curves that separate F0 from F1 in G.

We give a simple proof of a well-known theorem of Gal and of the recent related results of Aistleitner, Berkes and Seip [1] regarding the size of GCD sums. In fact, our method obtains the asymptotically sharp constant in Gal's theorem, which is new. Our approach also gives a transparent explanation of the relationship between the maximal size of the Riemann zeta function on vertical lines and bounds on GCD sums; a point which was previously unclear. Furthermore we obtain sharp bounds on the spectral norm of GCD matrices which settles a question raised in [2] . We use bounds for the spectral norm to show that series formed out of dilates of periodic functions of bounded variation converge almost everywhere if the coefficients of the series are in L 2 ( log ⁡ log ⁡ 1 / L ) γ , with γ > 2 . This was previously known with γ > 4 , and is known to fail for γ 2 . We also develop a sharp Carleson–Hunt-type theorem for functions of bounded variations which settles another question raised in [1] . Finally we obtain almost sure bounds for partial sums of dilates of periodic functions of bounded variations improving [1] .