Pointwise Estimates Research Articles

A note on Lusin-type approximation of Sobolev functions on Gaussian spaces

We establish new approximation results in the sense of Lusin for Sobolev functions f with |∇f|∈Llog⁡L on infinite-dimensional spaces equipped with Gaussian measures. The proof relies on some new pointwise estimate for the approximations based on the corresponding semigroup which can be of independent interest.

The pointwise estimates of solutions for the 3D incompressible viscoelastic fluids

In this paper, we consider the Cauchy problem of the incompressible viscoelastic hydrodynamic model in dimension three. Firstly, we introduce the appropriate variable transformation, and study the Greens function of the linearized system for the transformed equations.Then, according to the pointwise estimation method of Greens function, combining the expressions of the solutions, we analyze the influence of Riesz operator and obtain the pointwise time-space estimates of the solutions.

О применении метода коллокации в пространствах p-суммируемых функций к интегральному уравнению Фредгольма второго рода

This work is devoted to the study of the numerical solution by the spline collocation method of the Fredholm equation of the second kind. For numerical solutions of such problems, the classical collocation method using polynomials is not always realizable in spaces of p-summable functions for numerical solutions of such problems. It is not always possible to obtain characteristics and estimates of errors of such approximations even in the case of its implementation. In this regard, in recent years, in practice, approximations are built using finite-difference methods. The purpose of this study is to obtain estimates of the error of the obtained approximate solution in the spaces indicated above. In addition, several statements were obtained about a pointwise estimate of this error at collocation nodes in terms of the kernel norm in specially constructed spaces of functions summable over the second variable. To obtain the main results, third-order collocation splines are used, as well as integral and averaged modules of smoothness. In this case, the results obtained can become a starting point for working with collocation splines of higher orders. In the case of the third order, the exact constants involved in the estimates are obtained. These results can be extended to the case of linear, parabolic collocation splines, as well as splines of order higher than the third.

Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms

This article concerns the Cauchy problem of conservation laws with nonlocal dissipation-type terms in R^3. By using Green's function and the time-frequency decomposition method, we study global classical solutions and their long time behavior including pointwise estimates for large initial data, for solutions near the nontrivial equilibrium state.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/129/abstr.html

Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups

Let be a Schrödinger operator on the Heisenberg group , where is the sublaplacian on and the nonnegative potential V belongs to the reverse Hölder class with . Here is the homogeneous dimension of . Assume that is the heat semigroup generated by. The Lusin area integral and the Littlewood–Paley–Stein function associated with the Schrödinger operator are defined, respectively, bywhereandWhere is a parameter. In this article, the author shows that there is a relationship between and the operator and for any , the following inequality holds true:Based on this inequality and known results for the Lusin area integral , the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function on . In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator on . By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator acting on the Morrey spaces for an appropriate choice of . It can be shown that the same conclusions hold for the operator on generalized Morrey spaces as well.

Optimal point-wise error estimate of two conservative fourth-order compact finite difference schemes for the nonlinear Dirac equation

In this paper, we propose and analyze two conservative fourth-order compact finite difference schemes for the (1+1) dimensional nonlinear Dirac equation with periodic boundary conditions. Based on matrix knowledge, we convert the point-wise forms of the proposed compact schemes into equivalent vector forms and analyze their conservative and convergence properties. We prove that the proposed schemes preserve the total mass and energy in the discrete level and the convergence rate of the schemes, without any restrictions on the grid ratio, are at the order of O(h4+τ2) in l∞-norm, where h and τ are spatial and temporal steps, respectively. The error analysis techniques include the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. The numerical experiments are carried out to confirm our theoretical analysis. The research method in this paper can be easily extended to higher order compact schemes or other types of wave equations.

Rational Approximation on Exponential Meshes

Error estimates of pointwise approximation, that are not possible by polynomials, are obtained by simple rational operators based on exponential-type meshes, improving previous results. Rational curves deduced from such operators are analyzed by Discrete Fourier Transform and a CAGD modeling technique for Shepard-type curves by truncated DFT and the PIA algorithm is developed.

Global existence and decay in multi-component reaction–diffusion–advection systems with different velocities: oscillations in time and frequency

It is well-known that quadratic or cubic nonlinearities in reaction-diffusion-advection systems can lead to growth of solutions with small, localized initial data and even finite time blow-up. It was recently proved, however, that, if the components of two nonlinearly coupled reaction-diffusion-advection equations propagate with different velocities, then quadratic or cubic mixed-terms, i.e.~nonlinear terms with nontrivial contributions from both components, do not affect global existence and Gaussian decay of small, localized initial data. The proof relied on pointwise estimates to capture the difference in velocities. In this paper we present an alternative method, which is better applicable to multiple components. Our method involves a nonlinear iteration scheme that employs $L^1$-$L^p$ estimates in Fourier space and exploits oscillations in time and frequency, which arise due to differences in transport. Under the assumption that each component exhibits different velocities, we establish global existence and decay for small, algebraically localized initial data in multi-component reaction-diffusion-advection systems allowing for cubic mixed-terms and nonlinear terms of Burgers' type.

Semigroup Maximal Functions, Riesz Transforms, and Morrey Spaces Associated with Schrödinger Operators on the Heisenberg Groups

Let L = − Δ ℍ n + V be a Schrödinger operator on the Heisenberg group ℍ n , where Δ ℍ n is the sub-Laplacian on ℍ n and the nonnegative potential V belongs to the reverse Hölder class B q with q ∈ Q / 2 , ∞ . Here, Q = 2 n + 2 is the homogeneous dimension of ℍ n . Assume that e − t L t > 0 is the heat semigroup generated by L . The semigroup maximal function related to the Schrödinger operator L is defined by T L ∗ f u ≔ sup t > 0 e − t L f u . The Riesz transform associated with the operator L is defined by R L = ∇ ℍ n L − 1 / 2 , and the dual Riesz transform is defined by R L ∗ = L − 1 / 2 ∇ ℍ n , where ∇ ℍ n is the gradient operator on ℍ n . In this paper, the author first introduces a class of Morrey spaces associated with the Schrödinger operator L on ℍ n . Then, by using some pointwise estimates of the kernels related to the nonnegative potential, the author establishes the boundedness properties of these operators T L ∗ , R L , and R L ∗ acting on the Morrey spaces. In addition, it is shown that the Riesz transform R L = ∇ ℍ n L − 1 / 2 is of weak-type 1 , 1 . It can be shown that the same conclusions are also true for these operators on generalized Morrey spaces.

Approximation of functions by a new type of Gamma operators

AbstractIn this article, a new type of Gamma operators which preserves 1 and x2 has been presented and tested. In order to show the approximation properties of the newly defined operator, Voronovksya type theorems, weighted approximation, rate of convergence, and pointwise estimates have been presented. Additionally, numerical examples implemented by MATLAB have been provided to show its approximation properties by numerically.

Sharp estimates of the Cesàro kernels for weighted orthogonal polynomial expansions in several variables

We study the Cesàro means of the orthogonal polynomial expansions (OPEs) with respect to the weight function ∏i=1d|xi|2κi on the unit sphere Sd−1⊂Rd for all parameters κ1,⋯,κd>−12. We obtain sharp pointwise estimates for the corresponding Cesàro kernels, which were previously known when all parameters are nonnegative. We settle the problem for the case when min1≤j≤d⁡κj<0. Our estimates allow us to establish sharp results on Cesàro summability with less restriction on the parameters. We also establish similar results for the corresponding weighted OPEs on the unit ball and on the simplex.

Series Expansion and Fourth-Order Global Padé Approximation for a Rough Heston Solution

The rough Heston model has recently been shown to be extremely consistent with the observed empirical data in the financial market. However, the shortcoming of the model is that the conventional numerical method to compute option prices under it requires great computational effort due to the presence of the fractional Riccati equation in its characteristic function. In this study, we contribute by providing an efficient method while still retaining the quality of the solution under varying Hurst parameter for the fractional Riccati equations in two ways. First, we show that under the Laplace–Adomian-decomposition method, the infinite series expansion of the fractional Riccati equation’s solution corresponds to the existing expansion method from previous work for at least up to the fifth order. Then, we show that the fourth-order Padé approximants can be used to construct an extremely accurate global approximation to the fractional Riccati equation in an unexpected way. The pointwise approximation error of the global Padé approximation to the fractional Riccati equation is also provided. Unlike the existing work of third-order global Padé approximation to the fractional Riccati equation, our work extends the availability of Hurst parameter range without incurring huge errors. Finally, numerical comparisons were conducted to verify that our methods are indeed accurate and better than the existing method for computing both the fractional Riccati equation’s solution and option prices under the rough Heston model.

Formation of Shocks for 2D Isentropic Compressible Euler

AbstractWe consider the 2D isentropic compressible Euler equations, with pressure law p(ρ) = (1γ)ργ, with γ &gt; 1. We provide an elementary constructive proof of shock formation from smooth initial data of finite energy, with no vacuum regions, and with nontrivial vorticity. We prove that for initial data which has minimum slope −1ε, for ε &gt; 0 taken sufficiently small relative to the amplitude, there exist smooth solutions to the Euler equations which form a shock in time . The blowup time and location can be explicitly computed and solutions at the blowup time are of cusp‐type, with Hölder C13 regularity.Our objective is the construction of solutions with inherent vorticity at the shock. As such, rather than perturbing from an irrotational regime, we instead construct solutions with dynamics dominated by purely azimuthal wave motion. We consider homogenous solutions to the Euler equations and use Riemann‐type variables to obtain a system of forced transport equations. Using a transformation to modulated self‐similar variables and pointwise estimates for the ensuing system of transport equations, we show the global stability, in self‐similar time, of a smooth blowup profile. © 2020 Wiley Periodicals LLC.

Artificial boundary conditions for linearized stationary incompressible viscous flow around rotating and translating body

AbstractWe consider the linearized and nonlinear stationary incompressible flow around rotating and translating body in the exterior domain , where is open and bounded, with Lipschitz boundary. We derive the pointwise estimates for the pressure in both cases. Moreover, we consider the linearized problem in a truncation domain of the exterior domain under certain artificial boundary conditions on the truncating boundary , and then compare this solution with the solution in the exterior domain to get the truncation error estimate.

Point-Wise Wavelet Estimation in the Convolution Structure Density Model

By using a kernel method, Lepski and Willer establish adaptive and optimal $$L^p$$ risk estimations in the convolution structure density model in 2017 and 2019. They assume their density functions to be in a Nikol’skii space. Motivated by their work, we first use a linear wavelet estimator to obtain a point-wise optimal estimation in the same model. We allow our densities to be in a local and anisotropic Holder space. Then a data driven method is used to obtain an adaptive and near-optimal estimation. Finally, we show the logarithmic factor necessary to get the adaptivity.

Gradient potential estimates for elliptic obstacle problems

In this paper, we consider the solutions of the non-homogeneous quasilinear elliptic obstacle problems involving measure data. At first we prove the existence of approximable solutions to these problems. Then we establish pointwise and oscillation estimates for the gradients of solutions via the nonlinear Wolff potentials, and these yield results on C1,α-regularity of solutions.

Bayesian inference in based-kernel regression: comparison of count data of condition factor of fish in pond systems

ABSTRACT The discrete kernel-based regression approach generally provides pointwise estimates of count data that do not account for uncertainty about both parameters and resulting estimates. This work aims to provide probabilistic kernel estimates of count regression function by using Bayesian approach and then allows for a readily quantification of uncertainty. Bayesian approach enables to incorporate prior knowledge of parameters used in discrete kernel-based regression. An application was proposed on count data of condition factor of fish (K) provided from an experimental project that analyzed various pond management strategies. The probabilistic distribution of estimates were contrasted by discrete kernels, as a support to theoretical results on the performance of kernels. More practically, Bayesian credibility intervals of K-estimates were evaluated to compare pond management strategies. Thus, similarities were found between performances of semi-intensive and coupled fishponds, with formulated feed, in comparison with extensive fishponds, without formulated feed. In particular, the fish development was less predictable in extensive fishpond, dependent on natural resources, than in the two other fishponds, supplied in formulated feed.

Bounds for expected payoff on two stocks

Under certain conditions, we establish explicit upper bounds for an investor's expected payoff on two stocks. The proof of these bounds is based on a certain optimal stopping problem and a pointwise estimate for solutions of a second-order elliptic equation.

On Rational Approximation of Markov Functions by Partial Sums of Fourier Series on a Chebyshev–Markov System

Approximations on the closed interval $$[-1,1]$$ of functions that are combinations of classical Markov functions by partial sums of Fourier series on a system of Chebyshev–Markov rational fractions are considered. Pointwise and uniform estimates for approximations are established. For the case in which the derivative of the measure is weakly equivalent to a power function, an asymptotic expression for the majorant of uniform approximations and an optimal parameter value ensuring the greatest rate of approximation by the method used in the paper are found. In the case of the even multiplicity of the poles of the approximating function, the asymptotic estimate is sharp. Examples of approximations of concrete functions are given.

Pointwise approximation of functions by matrix operators of their Fourier series with r-differences of the entries

We extend the results of Xh.Z. Krasniqi [Acta Comment. Univ. Tartu. Math., 17:89–101, 2013] and the authors [Acta Comment. Univ. Tartu. Math., 13:11–24, 2019] to the case where in the measures of estimations r-differences of the entries are used.