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Random Sampling of Mellin Band-Limited Signals

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2024年第2期45卷 136-150页

作者： Bajpeyi, Shivam Patel, Dhiraj Sivananthan, S. Sardar Vallabhbhai Natl Inst Technol Surat Dept Math Surat 395007 Gujarat India Indian Inst Technol Delhi Dept Math New Delhi India

In this paper, we address the random sampling problem for the class of functions in the space of Mellin band-limited functions BT, which are concentrated on a bounded cube. It is established that any Mellin band-limited function can be approximated by an element in a finite-dimensional subspace of BT. Utilizing the notion of covering number and Bernstein's inequality to the sum of independent random variables, we prove that the probabilistic sampling inequality holds for the set of concentrated signals in BT with an overwhelming probability provided the sampling size is large enough.

关键词： Mellin band-limited functions Mellin transform random sampling reproducing kernel space sampling inequality

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Random sampling and reconstruction in reproducing kernel subspace of mixed Lebesgue spaces

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2023年第5期46卷 5119-5138页

作者： Goyal, Prashant Patel, Dhiraj Sivananthan, S. Indian Inst Technol Delhi Dept Math New Delhi 110016 India

In this article, we consider the random sampling in the image space V of an idempotent integral operator on mixed Lebesgue space L-p,L-q ( Rn+1). We assume some decay and regularity conditions on the integral kernel and show that the bounded functions in V can be approximated by an element in a finite- dimensional subspace of V on CR, S = [ - R/2, R/2] n x [ - S/2, S/2]. Consequently, we show that the set of bounded functions concentrated on C-R,C- S is totally bounded and prove with an overwhelming probability that the random sample set uniformly distributed over C-R,C-S is a stable set of sampling for the set of concentrated functions on C-R,C-S. Further, we propose an iterative scheme to reconstruct the concentrated functions from their random measurements.

关键词： mixed Lebesgue space random sampling reconstruction algorithm reproducing kernel space

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A new method of solving the best approximate solution for a nonlinear fractional equation

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2023年第8期100卷 1702-1718页

作者： Du, Hong Yang, Xinyue Chen, Zhong GuangDong Ocean Univ Coll Math & Comp Sci Zhanjiang Guangdong Peoples R China Lanzhou Univ Coll Earth & Environm Sci Lanzhou Gansu Peoples R China Harbin Inst Technol Dept Math Weihai 264209 Shandong Peoples R China

A new method of solving the best approximate solution for nonlinear fractional equations with smooth and nonsmooth solutions in reproducing kernel space is proposed in the paper. The nonlinear equation outlines some important equations, such as fractional diffusion-wave equation, nonlinear Klein-Gordon equation and time-fractional sine-Gordon equation. By constructing orthonormal bases in reproducing kernel space using Legendre orthonormal polynomials and Jacobi fractional orthonormal polynomials, the best approximate solution is obtained by searching the minimum of residue in the sense of || . ||(C). Numerical experiments verify that the method has higher accuracy.

关键词： Nonlinear fractional equation reproducing kernel space the best approximate solution nonsmooth solution

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Random sampling of signals concentrated on compact set in localized reproducing kernel subspace of L^p(Rⁿ)

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2023年第6期49卷 76-76页

作者： Patel, Dhiraj Sivananthan, S. Indian Inst Technol Delhi Dept Math New Delhi 110016 India

The paper is devoted to studying the stability of random sampling in a localized reproducing kernel space. We show that if the sampling set on Omega(compact) discretizes the integral norm of simple functions up to a given error, then the sampling set is stable for the set of functions concentrated on Omega. Moreover, we prove with an overwhelming probability that O(mu(Omega)(log mu(Omega))(3))many random points uniformly distributed over Omega yield a stable set of sampling for functions concentrated on Omega.

关键词： Random sampling reproducing kernel space Sampling inequality Covering number Discretization

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The kernel regularized learning algorithm for solving Laplace equation with Dirichlet boundary

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INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING 2022年第6期20卷 2250031-2250031页

作者： Sheng, Baohuai Zhou, Dapao Wang, Shuhua Zhejiang Yuexiu Univ Dept Econ Stat Shaoxing 312000 Peoples R China Shaoxing Univ Dept Appl Stat Shaoxing 312000 Peoples R China Jingdezhen Ceram Univ Sch Informat Engn Jingdezhen 333403 Peoples R China

In this paper, we give an investigation on the problem of solving Laplace equation with the kernel regularized regression. We provide a Sobolev type space corresponding to the Dirichlet boundary value problem on a compact domain, and with which define a reproducing kernel space (RKS), which is used as the hypothesis space for constructing kernel regularized learning algorithm. We give theory analysis for the convergence of the learning algorithm, bound an upper bound for the error. The discussions show that the learning rate is controlled by a K-functional corresponding to the RKS. As an application we give the learning rate in case that the domain is the unit ball. The simulations show that the algorithm has better fitting performance. The investigations show that the problem of solving an elliptic boundary problem can be attributed to constructing an orthonormal basis with respect to a bilinear form corresponding to the boundary value condition.

关键词： Dirichlet boundary problem reproducing kernel space kernel regularized regression the unit ball the unit sphere K-functional learning rate

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An improved collocation method for solving a fractional integro-differential equation

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COMPUTATIONAL & APPLIED MATHEMATICS 2021年第1期40卷 1-14页

作者： Zhang, Xiaoguang Du, Hong Heilongjiang Univ Sci & Technol Coll Sci Harbin 150022 Heilongjiang Peoples R China GuangDong Ocean Univ Coll Math & Comp Sci Zhanjiang 524000 Guangdong Peoples R China

In the paper, an improved collocation method is proposed for solving a linear fractional integro-differential equation. The method is proposed by improving the residual to vanish and require the residual to the minimum in sense of .C. It can avoid efficiently ill-conditioned of higher degree polynomials. Convergence order is superconvergence and stability analysis is also provided.

关键词： Superconvergence reproducing kernel space epsilon approximate solutions

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THE PIECEWISE reproducing kernel METHOD FOR THE TIME VARIABLE FRACTIONAL ORDER ADVECTION-REACTION-DIFFUSION EQUATIONS

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THERMAL SCIENCE 2021年第2期25卷 1261-1268页

作者： Dai, Dan-Dan Ban, Ting-Ting Wang, Yu-Lan Zhang, Wei Inner Mongolia Univ Technol Dept Math Hohhot Peoples R China Jining Normal Univ Inst Phys Jining Inner Mongolia Peoples R China Jining Normal Univ Inst Econ & Management Jining Inner Mongolia Peoples R China

This paper structures some new reproductive kernel spaces based on Legendre polynomials to solve time variable order fractional advection-reaction-diffusion equations. Some examples are given to show the effectiveness... 详细信息

关键词： advection-reaction-diffusion equation variable fractional derivative piecewise reproducing kernel method reproducing kernel space

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A new multiscale algorithm for solving second order boundary value problems

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APPLIED NUMERICAL MATHEMATICS 2020年 156卷 528-541页

作者： Zheng, Yaqin Lin, Yingzhen Shen, Yang Beijing Inst Technol Sch Appl Sci & Civil Engn Zhuhai 519088 Peoples R China

In this study, a new multiscale algorithm was proposed to solve the boundary value problems of second order differential equations. A multiscale basis consisting of two sets of multiscale functions was constructed in the reproducing kernel space, and the proposed multiscale basis was proved to be orthonormal. The epsilon- approximate solution was defined, and then it was proved to be the optimal solution. In addition, the stability, convergence and complexity of this algorithm were discussed and illustrated in this study. Numerical examples verify the effectiveness and feasibility of the algorithm, and the results show that the proposed intelligent multiscale algorithm has advantages in accuracy and stability compared with other methods. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Boundary value problem Multiscale orthonormal basis reproducing kernel space Convergence

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THE reproducing kernel FOR THE REACTION-DIFFUSION MODEL WITH A TIME VARIABLE FRACTIONAL ORDER

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THERMAL SCIENCE 2020年第4期24卷 2553-2559页

作者： Zhang, Wei Wang, Yulan Wang, Meichun Inner Mongolia Univ Technol Dept Math Hohhot Peoples R China Jining Normal Univ Inst Econ & Management Jining Inner Mongolia Peoples R China

The variable-order fractional calculus has become a useful mathematical framework to describe a complex reaction-diffusion process. It is very hard to solve the problem, and there is almost no analytical method available in open literature. In this article, the reproducing kernel method is proposed for this purpose, and some examples show that the method is of high precision.

关键词： reaction-diffusion model variable fractional derivative reproducing kernel space

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The reproducing kernel Hilbert space method for solving Troesch's problem

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Journal of the Association of Arab Universities for Basic and Applied Sciences 2013年第1期14卷 19-27页

作者： Inc, Mustafa Akgül, Ali Department of Mathematics Science Faculty Firat University 23119 Elaziǧ Turkey Department of Mathematics Education Faculty Dicle University 21280 Diyarbakir Turkey

In this paper, the reproducing kernel Hilbert space method (RKHSM) is applied for solving Troesch's problem. We used numerical examples to illustrate the accuracy and implementation of the method. The analytical result of the equation has been obtained in terms of a convergent series with easily computable components. The results are compared with the ones obtained by the homotopy perturbation method (HPM), the Laplace decomposition method (LDM), the perturbation method (PM), the Adomian decomposition method (ADM), the variational iteration method (VIM), the B-spline method and the nonstandard finite difference scheme (FDS) by using tables and figures. Numerical results show that the present method is effective. © 2013.

关键词： reproducing kernel method reproducing kernel space Series solutions Troesch's problem

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