In this paper, we derive an explicit real-variable inversion formula for the Laplace transform on the half-line ( 0 , ∞ ) by using Laguerre polynomials. The formula is obtained from the Fourier–Laguerre expansion, with the corresponding Laguerre coefficients expressed directly in terms of derivatives of the Laplace transform at the fixed point s = 1 / 2 . This gives a reconstruction formula which is structurally different from the classical Bromwich contour formula, the Post–Widder inversion formula, and other real-variable inversion formulas in the literature. We also establish pointwise convergence estimates for the truncated Laguerre reconstruction under suitable decay assumptions on the Laguerre coefficients. Finally, an explicit example is presented to verify the formula analytically and numerically.

Quantitative convergence of weighted ergodic averages We investigate quantitative conditions on a sequence of weights to ensure pointwise convergence of the corresponding weighted ergodic averages. We provide concrete uniform decay rates in Fourier space that guarantee convergence, construct a counterexample at the boundary and propose applications.

We consider a history-dependent hemivariational inequality in a reflexive Banach space X, stated on the interval of time [ 0 , T ] with T>0 and governed by a time-dependent set of constraints. We use arguments of pseudomonotonicity, Mosco convergence and fixed point in order to provide the existence of a unique solution u ∈ C ( [ 0 , T ] ; X ) of the inequality, together with a pointwise convergence result. Next, under additional assumptions, we provide necessary and sufficient conditions which guarantee the uniform convergence of a sequence of functions { u n } ⊂ C ( [ 0 , T ] ; X ) to the solution u. We then introduce and study two well-posedness concepts for the corresponding inequality. Our results give rise to various applications. To provide an example, we illustrate their use in the study of a mathematical model which describes the equilibrium of a viscoelastic rod in contact with a rigid-deformable obstacle, the so-called foundation.

We introduce a new set-open topology on function spaces namely the semi-compact-open topology. The topology of uniform convergence on semi-compacta lies between the topology of pointwise convergence and the topology of uniform convergence. We show that for the space of quasicontinuous functions the topology of uniform convergence on semi-compacta coincides with the semi-compact-open topology. We also investigate results relating to induced functions, cardinal invariants and Ascoli like properties on the space of quasicontinuous functions when equipped with the topology of uniform convergence on semi-compacta.

We investigate the class of sequences ()wn that can serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. We show that any such sequence must satisfy the bounds 2log( )logn wnn. Our main result establishes a general equivalenceprinciple between one-dimensional and multidimensional trigonometric systems, which allows one to extend certain estimates known for the one-dimensional case to higher dimensions.

Abstract In this article, we capture Phillips type operators based on adjoint Bernoulli polynomials and their connection to other operators. We derive their characteristic function and provide pointwise convergence and estimate errors using various types of modulus of continuity. Then we establish the theorems based on the difference of operators with their decomposed parts. Additionally, we modify these operators in order to preserve e Ax and e 2Ax and give asymptotic formula and a Korovkin-type result for modified operators.

Our work is part of the close link between continuous-time dissipative dynamical systems and optimization algorithms, and more precisely here, in the stochastic setting. We aim to study stochastic convex minimization problems through the lens of stochastic inertial differential inclusions that are driven by the subgradient of a convex objective function. This will provide a general mathematical framework for analyzing the convergence properties of stochastic second-order inertial continuous-time dynamics involving vanishing viscous damping and measurable stochastic subgradient selections. Our chief goal in this paper is to develop a systematic and unified way that transfers the properties recently studied for first-order stochastic differential equations to second-order ones involving even subgradients in lieu of gradients. This program will rely on two tenets: time scaling and averaging, following an approach recently developed in the literature by one of the coauthors in the deterministic case. Under a mild integrability assumption involving the diffusion term and the viscous damping, our first main result shows that almost surely, there is weak convergence of the trajectory toward a minimizer of the objective function and fast convergence of the values and gradients. We also provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex, strongly convex, and (local) Polyak-Łojasiewicz case. Finally, using Tikhonov regularization with a properly tuned vanishing parameter, we can obtain almost sure strong convergence of the trajectory toward the minimum norm solution. Funding: This research was supported by Agence Nationale de la Recherche (ANR) [Projet-ANR-20-CE92-0037].

The behavior of a new modification of operators of the Baskakov–Schurer–Stancu variant is discussed in this study. First, we establish certain necessary moment and central moment estimates. We then demonstrate the weighted approximation result of the suggested operators using a Korovkin-type theorem in weighted spaces. We also give the rate at which these operators converge. Next, we establish theorems of pointwise convergence. Finally, we show several graphical representations to illustrate the accuracy and functionality of the operators.

In complex modeling that may lack explicit parametric structures, understanding the contribution of specific covariates to explaining or predicting a response variable is essential. In particular, it is crucial to examine whether this contribution varies with certain characteristic variables, such as age in psychological studies. We refer to this varying contribution as "heterogeneous variable importance," a concept that allows for evaluating variable relevance across different groups of individuals. To quantify heterogeneous variable importance, we introduce a measure defined as the ratio of two conditional mean squared errors. We then propose a point estimator for this ratio parameter and establish its pointwise and uniform convergence rates. Additionally, we develop procedures for constructing asymptotic confidence intervals and bands, which are guaranteed to achieve their nominal coverage rates. Moreover, the proposed approach demonstrates satisfactory finite-sample performance in simulation studies, and is further illustrated through its application to a real data set.

ABSTRACT The main objective of this article is to construct a new sequence of ‐Chlodowsky–Durrmeyer type operators including shape parameter , where . We compute some needed moment estimates. Also, we study some direct and local approximation properties of the proposed operators. Next, we obtain the order of convergence in terms of the weighted modulus of continuity. Then, in order to check the asymptotic behavior of related operators, we prove Voronovskaya's type theorem. Further, we evaluate pointwise convergence of the proposed operators. To validate our theoretical results, we perform some graphical and numerical experiments that illustrate the precision, effectiveness, and advantages of the proposed approach in practical applications of the related operators.

MANIFOLD LEARNING IN METRIC SPACES.

A sequence of positive linear operators acting on suitable function spaces on convex Borel cones is introduced and studied. Such operators generalize the well-known Bernstein-Chlodovsky operators on $[0,+\infty[$ and, in addition, they unify many of their more recent extensions to other settings. The study is mainly addressed to highlight their pointwise convergence as well as their uniform convergence on compact subsets for particular classes of bounded Borel measurable functions. In some particular cases, by means of such operators, a Weierstrass-type density result is obtained which concerns the approximation of such class of functions in terms of polynomials or, more generally, of elements of some function algebras. In order to achieve the main results some new Korovkin-type theorems are also discussed in the framework of completely regular spaces. In a final section, some examples and applications are discussed as well.

This work reports a novel multi-frame Bundle Adjustment (BA) framework called RKHS-BA. It uses continuous landmark representations that encode RGB-D/LiDAR and semantic observations in a reproducing kernel hilbert space (RKHS). With a correspondence-free pose graph formulation, the proposed system constructs a loss function that achieves more generalized convergence than classical point-wise convergence. We demonstrate its applications in multi-view point cloud registration, sliding-window odometry, and global LiDAR mapping on simulated and real data. It shows highly robust pose estimations in extremely noisy scenes and exhibits strong generalization with various types of semantic inputs.

We develop a monotone, two-scale discretization for a class of integro-differential operators of order 2s , s \in (0,1) . We apply it to develop numerical schemes, and derive pointwise convergence rates for linear and obstacle problems governed by such operators. As applications of the monotonicity, we provide error estimates for free boundaries and a convergent numerical scheme for a concave fully nonlinear, nonlocal, problem.

Abstract In this paper, we first present the second-order Riesz and Nörlund logarithmic summation methods, followed by a class of regular summation techniques based on second-order logarithmic averages. We then examine their applications to the pointwise convergence of Fourier series for continuous and integrable functions.

In this work, we focus on investigating statistical convergence in function spaces equipped with semi-uniform convergence topologies. We provide a comparative analysis of pointwise and uniform statistical convergence under these topological structures and investigate the conditions that allow transitions between these types of convergence.

Pointwise convergence to initial data for some evolution equations on symmetric spaces

To solve convex optimization problems with a noisy gradient input, we analyze the global behavior of subgradient-like flows under stochastic errors. The objective function is composite, being equal to the sum of two convex functions, one being differentiable and the other potentially non-smooth. We then use stochastic differential inclusions where the drift term is minus the subgradient of the objective function, and the diffusion term is either bounded or square-integrable. In this context, under Lipschitz’s continuity of the differentiable term and a growth condition of the non-smooth term, our first main result shows almost sure weak convergence of the trajectory process towards a minimizer of the objective function. Then, using Tikhonov regularization with a properly tuned vanishing parameter, we can obtain almost sure strong convergence of the trajectory towards the minimum norm solution. We find an explicit tuning of this parameter when our objective function satisfies a local error-bound inequality. We also provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex, strongly convex, and Łojasiewicz case.

This paper deals with some behavior of Baskakov-Schurer-Stancu type operators in approximating functions, grounded on non-negative parameter α. Firstly, we establish some needed moment estimations. Next, according to the famous Korovkin theorem we prove weighted approximation result of proposed operators. Also, we provide rate of convergence of these operators and as well as pointwise convergence theorems. Lastly, to demonstrate the efficiency and consistency of the operators, we provide some graphical representations.

We study spaces X for which the space Homp(X) of automorphisms with the topology of point-wise convergence is a topological group. We identify large classes of spaces X for which Homp(X) is or is not a topological group.