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Random sampling and reconstruction of concentrated signals in a reproducing kernel space

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2021年 54卷 273-302页

作者： Li, Yaxu Sun, Qiyu Xian, Jun Hangzhou Normal Univ Dept Math Hangzhou 311121 Zhejiang Peoples R China Univ Cent Florida Dept Math Orlando FL 32816 USA Sun Yat Sen Univ Sch Math Guangzhou 510275 Guangdong Peoples R China Sun Yat Sen Univ Guangdong Prov Key Lab Computat Sci Guangzhou 510275 Guangdong Peoples R China

In this paper, we consider (random) sampling of signals concentrated on a bounded Corkscrew domain Omega of a metric measure space, and reconstructing concentrated signals approximately from their (un)corrupted sampling data taken on a sampling set contained in Omega. We establish a weighted stability of bi-Lipschitz type for a (random) sampling scheme on the set of concentrated signals in a reproducing kernel space. The weighted stability of bi-Lipschitz type provides a weak robustness to the sampling scheme, however due to the nonconvexity of the set of concentrated signals, it does not imply the unique signal reconstruction. From (un)corrupted samples taken on a finite sampling set contained in Omega, we propose an algorithm to find approximations to signals concentrated on a bounded Corkscrew domain Omega. Random sampling is a sampling scheme where sampling positions are randomly taken according to a probability distribution. Next we show that, with high probability, signals concentrated on a bounded Corkscrew domain Omega can be reconstructed approximately from their uncorrupted (or randomly corrupted) samples taken at i.i.d. random positions drawn on Omega, provided that the sampling size is at least of the order mu(Omega) ln(mu(Omega)), where mu(Omega) is the measure of the concentrated domain Omega. Finally, we demonstrate the performance of proposed approximations to the original concentrated signals when the sampling procedure is taken either with large density or randomly with large size. (C) 2021 Elsevier Inc. All rights reserved.

关键词： Random sampling and reconstruction reproducing kernel space Corkscrew domain

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Solving the generalized regularized long wave equation on the basis of a reproducing kernel space

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2011年第14期235卷 4003-4014页

作者： Mohammadi, Maryam Mokhtari, Reza Isfahan Univ Technol Dept Math Sci Esfahan *** Iran

On the basis of a reproducing kernel space, an iterative algorithm for solving the generalized regularized long wave equation is presented. The analytical solution in the reproducing kernel space is shown in a series form and the approximate solution u(n) is constructed by truncating the series to n terms. The convergence of u(n) to the analytical solution is also proved. Results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such evolution equations. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Generalized regularized long wave equation reproducing kernel space

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Random average sampling and reconstruction in a reproducing kernel space

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2024年第4期47卷 2702-2716页

作者： Li, Yaxu Xian, Jun Hangzhou Normal Univ Sch Math Hangzhou Zhejiang Peoples R China Sun Yat Sen Univ Sch Math Guangzhou Guangdong Peoples R China Sun Yat Sen Univ Sch Math Guangzhou 510275 Guangdong Peoples R China

Suppose that signals of interest reside in a reproducing kernel space defined on a metric measure space. We consider the scenario that the sampling positions are distributed on a bounded domain omega$$ \Omega $$ of a metric measure space, and the sampling data are local averages of the original signals in a reproducing kernel space. For signals concentrated on omega$$ \Omega $$ in that reproducing kernel space, we study the stability of this sampling procedure by establishing a weighted sampling inequality of bi-Lipschitz type. This type of stability implies a weak version of conventional sampling inequality. We propose an iterative algorithm that reconstruct these concentrated signals from finite sampling data. The reconstruction error is characterized through the concentration ratio and the Hausdorff distance between the set of sampling positions and omega$$ \Omega $$. We also consider the random sampling scheme where the sampling positions are i.i.d. randomly drawn on omega$$ \Omega $$, and the sampling data are local averages of concentrated signals. We demonstrate that these concentrated signals can be approximated from the random sampling data with high probability when the sampling size is at least of the order mu(omega)ln(mu(omega))$$ \mu \left(\Omega \right)\ln \left(\mu \left(\Omega \right)\right) $$ with mu(omega)$$ \mu \left(\Omega \right) $$ being the measure of omega$$ \Omega $$.

关键词： average sampling random sampling and reconstruction reproducing kernel space

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Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2011年第9期16卷 3639-3645页

作者： Jiang, Wei Lin, Yingzhen Harbin Inst Technol Dept Math Weihai 264209 Shandong Peoples R China

The reproducing kernel theorem is used to solve the time-fractional telegraph equation with Robin boundary value conditions. The time-fractional derivative is considered in the Caputo sense. We discuss and derive the exact solution in the form of series with easily computable terms in the reproducing kernel space. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Telegraph equation Captuo fractional derivative reproducing kernel space

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Sampling and quasi-optimal approximation for signals in a reproducing kernel space of homogeneous type

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017年 319卷 296-307页

作者： Jiang, Yingchun Wang, Suping Guilin Univ Elect Technol Sch Math & Computat Sci Guilin 541004 Peoples R China Tianjin Univ Sch Math Tianjin 300354 Peoples R China

This paper considers the reconstruction of signals in a reproducing kernel space of homoge neous type from finite samples. First, a pre-reconstruction operator based on finite samples and probability measure is proposed and its bounded property is studied. Secondly, the stability and an iterative algorithm with exponential convergence are established for sampling and recovering signals in a subspace of homogeneous reproducing kernel space. Then, we show that the proposed algorithm also provides a quasi-optimal approximation to signals in a reproducing kernel space of homogeneous type. Finally, some numerical simulations are given to reconstruct signals on an interval. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Average sampling Quasi-optimal approximation Homogeneous type reproducing kernel space Iterative algorithm

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The exact solution for solving a class nonlinear operator equations in the reproducing kernel space

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APPLIED MATHEMATICS AND COMPUTATION 2003年第2-3期143卷 393-399页

作者： Li, CL Cui, MG Harbin Inst Technol Dept Math Harbin 150001 Peoples R China

In this paper, the author will introduce the Sobolev space W-2(1)(Omega(n)) firstly and then give the reproducing properties of the space W-2(1) (Omega(n)). As these results of those, a class of the nonlinear operator equation Sigma(i=1)(n) Pi(j=1)(mi) (A(ij)u) = f is transformed to the n-dimensional linear operator equation Au = f. Finally, we also give the exact solution of this nonlinear operator equation. (C) 2002 Published by Elsevier Science Inc.

关键词： operator equation reproducing kernel space nonlinear operator

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An efficient numerical algorithm for solving nonlinear Volterra integral equations in the reproducing kernel space

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2023年第4期69卷 3131-3149页

作者： Dai, Xuefei Niu, Jing Xu, Yanxin Harbin Normal Univ Harbin 150025 Heilongjiang Peoples R China

The main purpose of this paper is to approximate the solution of the nonlinear Volterra integral equation numerically in the reproducing kernel space. Consequently, in the study, combining Quasi-Newton's method and the least-square method, we develop a new method for solving this kind of equation. This technique transforms the non-linear Volterra integral equation into a linear algebraic system of equations, which can be solved by using the least-square method breezily. At the same time, to ensure the preciseness of the method, we strictly analyze the existence and uniqueness of e-approximate solution and its convergence. Finally, we illustrate the accuracy and reliability of this method by giving some examples.

关键词： Nonlinear Volterra integral equation reproducing kernel space Least-square method Quasi-Newton's method

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Representation of the exact solution and a stability analysis on the Fredholm integral equation of the first kind in reproducing kernel space

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APPLIED MATHEMATICS AND COMPUTATION 2006年第2期182卷 1608-1614页

作者： Du, Hong Cui, Minggen Harbin Inst Technol Dept Math Shandong 264209 Peoples R China Harbin Inst Technol Dept Math Harbin 150001 Peoples R China Heilongjiang Inst Sci & Technol Dept Math & Mech Harbin 150027 Hei Long Jiang Peoples R China

In this paper, a new method is given in order to solve an ill-posed problem on Fredholm integral equation of the first kind. The representation of the exact solution is given and the stability of the solution on Fredholm integral equation of the first kind is discussed in the reproducing kernel space. By the discussions, a conclusion is obtained the stability problem is a well-posed problem in the reproducing kernel space, namely, the measurement errors of the experimental data can not result in unbounded errors of the exact solution. The numerical experiment shows that the new method given in the paper is valid. (c) 2006 Elsevier Inc. All rights reserved.

关键词： Fredholm integral equation ill-posed problem reproducing kernel space

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Solving nonlinear pseudoparabolic equations with nonlocal boundary conditions in reproducing kernel space

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NUMERICAL ALGORITHMS 2009年第2期52卷 173-186页

作者： Lin Yingzhen Zhou Yongfang Harbin Inst Technol WEIHAI Dept Math Wei Hai 264209 Shan Dong Peoples R China Heilongjiang Inst Sci & Technol Dept Math & Mech Harbin 150027 Heilongjiang Peoples R China

In this paper, a function space is constructed, in which an arbitrary function satisfies the nonlocal boundary conditions of a nonlinear pseudoparabolic equation. A very simple numerical algorithm for the approximations of the nonlinear pseudoparabolic equation with nonlocal boundary conditions based on the function space is provided. A numerical example is given to illustrate the applicability and efficiency of the algorithm.

关键词： Pseudoparabolic equations Nonlinear equations Nonlocal boundary conditions reproducing kernel space

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A Solution of an Infinite System of Quadratic Equations in reproducing kernel space

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COMPLEX ANALYSIS AND OPERATOR THEORY 2007年第4期1卷 571-579页

作者： Lin, Yinzhen Chung, Philung Cui, Minggen Harbin Inst Technol Wei Hai Dept Math Wei Hai Shan Dong 264209 Peoples R China Kangwon Natl Univ Dept Math Chunchon 200701 Kangwon South Korea

We introduce several operators in certain reproducing kernel Hilbert spaces, and we use them in the solution of a general family of quadratic equations in an infinite number of variables. We further describe an approx... 详细信息

关键词： Infinite system of quadratic equations reproducing kernel space adjoint operator

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