Let (X) be a Banach space, (G) be a closed subspace of (X), ((Omega,Sigma,mu)) be a (sigma)-finite measure space, (L(mu,X)) be the space of all strongly measurable functions from (Omega) to (X), and (L^{p}(mu,X)) be the space of all Bochner (p-)integrable functions from (Omega) to (X). Discussing the relationship between the pointwise coproximinality of (L(mu, G)) in (L(mu, X)) and the pointwise coproximinality of (L^{p}(mu, G)) in (L^{p}(mu, X)) is the purpose of this paper.
{"title":"Pointwise coproximinality in (L^p(mu, X))","authors":"Eyad Abu-Sirhan","doi":"10.33993/jnaat521-1328","DOIUrl":"https://doi.org/10.33993/jnaat521-1328","url":null,"abstract":"Let (X) be a Banach space, (G) be a closed subspace of (X), ((Omega,Sigma,mu)) be a (sigma)-finite measure space, (L(mu,X)) be the space of all strongly measurable functions from (Omega) to (X), and (L^{p}(mu,X)) be the space of all Bochner (p-)integrable functions from (Omega) to (X). Discussing the relationship between the pointwise coproximinality of (L(mu, G)) in (L(mu, X)) and the pointwise coproximinality of (L^{p}(mu, G)) in (L^{p}(mu, X)) is the purpose of this paper.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122354393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we take the piecewise linear equation system (x-W|x|=b), which is also known by absolute value equation, where (Win {mathbb R}^ {ntimes n}), (bin {mathbb R}^{n}) are given and to undetermined the value of (xin {mathbb R}^{n}). The absolute value equation (AVE) has many applications in various fields of mathematics like bi-matrix games, linear interval systems, linear complementarity problems (LCP) etc. By the equivalence relation of AVE with LCP, some necessary and sufficient conditions proved the existence and unique solvability of the AVE. Some examples are also provided to highlight the current singular value conditions for a unique solution that may revise in the future. (small corrections operated in the pdf file on January 7, 2023)
{"title":"On unique solvability of the piecewise linear equation system","authors":"Shubham Kumar, Deepmala","doi":"10.33993/jnaat512-1271","DOIUrl":"https://doi.org/10.33993/jnaat512-1271","url":null,"abstract":"In this article, we take the piecewise linear equation system (x-W|x|=b), which is also known by absolute value equation, where (Win {mathbb R}^ {ntimes n}), (bin {mathbb R}^{n}) are given and to undetermined the value of (xin {mathbb R}^{n}). The absolute value equation (AVE) has many applications in various fields of mathematics like bi-matrix games, linear interval systems, linear complementarity problems (LCP) etc. By the equivalence relation of AVE with LCP, some necessary and sufficient conditions proved the existence and unique solvability of the AVE. Some examples are also provided to highlight the current singular value conditions for a unique solution that may revise in the future.\u0000(small corrections operated in the pdf file on January 7, 2023)","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130062581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, the author presented some applications of the Laplace, (L^2), and Post-Widder transforms for solving fractional Singular Integral Equations, impulsive differential equation and systems of differential equations. Finally, analytic solution for a non-homogeneous partial differential equation with non-constant coefficients is given. The obtained results reveal that the integral transform method is an effective tool and convenient.
{"title":"Direct methods for singular integral equations and non-homogeneous parabolic PDEs","authors":"A. Aghili","doi":"10.33993/jnaat512-1269","DOIUrl":"https://doi.org/10.33993/jnaat512-1269","url":null,"abstract":"In this article, the author presented some applications of the Laplace, (L^2), and Post-Widder transforms for solving fractional Singular Integral Equations, impulsive differential equation and systems of differential equations. Finally, analytic solution for a non-homogeneous partial differential equation with non-constant coefficients is given. The obtained results reveal that the integral transform method is an effective tool and convenient.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126203679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce the notion of periodic wavelet bi-frames on local fields and establish the theory for the construction of periodic Bessel sequences and periodic wavelet bi-frames on local fields.
本文引入了局部域上周期小波双框架的概念，建立了局部域上周期贝塞尔序列和周期小波双框架的构造理论。
{"title":"Wavelet bi-frames on local fields","authors":"O. Ahmad, Neyaz Ahmad, M. Ahmad","doi":"10.33993/jnaat512-1265","DOIUrl":"https://doi.org/10.33993/jnaat512-1265","url":null,"abstract":"In this paper, we introduce the notion of periodic wavelet bi-frames on local fields and establish the theory for the construction of periodic Bessel sequences and periodic wavelet bi-frames on local fields.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125320400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For the Hermite type quartic spline interpolating on the partition knots and at the midpoint of each subinterval, we consider the estimation of the derivatives on the knots, and the values of these derivatives are obtained by constructing an algorithm of Akima's type. For computing the derivatives on endpoints are also considered alternatives that request optimal properties near the endpoints. The error estimate in the interpolation with this quartic spline is generally obtained in terms of the modulus of continuity. In the case of interpolating smooth functions, the corresponding error estimate reveal the maximal order of approximation O(h^3). A numerical experiment is presented for making the comparison between the Akima's cubic spline and the Akima's variant quartic spline havingdeficiency 2 and natural endpoint conditions.
{"title":"The Akima's fitting method for quartic splines","authors":"A. Bica, D. Curilă (Popescu)","doi":"10.33993/jnaat512-1278","DOIUrl":"https://doi.org/10.33993/jnaat512-1278","url":null,"abstract":"For the Hermite type quartic spline interpolating on the partition knots and at the midpoint of each subinterval, we consider the estimation of the derivatives on the knots, and the values of these derivatives are obtained by constructing an algorithm of Akima's type. For computing the derivatives on endpoints are also considered alternatives that request optimal properties near the endpoints. The error estimate in the interpolation with this quartic spline is generally obtained in terms of the modulus of continuity. In the case of interpolating smooth functions, the corresponding error estimate reveal the maximal order of approximation O(h^3). A numerical experiment is presented for making the comparison between the Akima's cubic spline and the Akima's variant quartic spline havingdeficiency 2 and natural endpoint conditions.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132372903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
High convergence order methods are important in computational mathematics, since they generate sequences converging to a solution of a non-linear equation. The derivation of the order requires Taylor series expansions and the existence of derivatives not appearing on the method. Therefore, these results cannot assure the convergence of the method in those cases when such high order derivatives do not exist. But, the method may converge. In this article, a process is introduced by which the semi-local convergence analysis of a sixth order method is obtained using only information from the operators on the method. Numerical examples are included to complement the theory.
{"title":"On the semi-local convergence of a sixth order method in Banach space","authors":"I. Argyros, Jinny Ann John, Jayakumar Jayaraman","doi":"10.33993/jnaat512-1284","DOIUrl":"https://doi.org/10.33993/jnaat512-1284","url":null,"abstract":"High convergence order methods are important in computational mathematics, since they generate sequences converging to a solution of a non-linear equation. The derivation of the order requires Taylor series expansions and the existence of derivatives not appearing on the method. Therefore, these results cannot assure the convergence of the method in those cases when such high order derivatives do not exist. But, the method may converge.\u0000In this article, a process is introduced by which the semi-local convergence analysis of a sixth order method is obtained using only information from the operators on the method. Numerical examples are included to complement the theory.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"137 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130759047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, the researchers develop a new type of spline function with fractional order which constructs two distinct formulas for the proposed method by using fractional boundary conditions and fractional continuity conditions. These methods are used to solve linear Volterra and Fredholm-integral equations of the second kind. The convergence analysis is studied. Moreover, some numerical examples are provided and compared to illustrate the efficiency and applicability of the proposed methods.
{"title":"On the numerical solution of Volterra and Fredholm integral equations using the fractional spline function method","authors":"Faraidun Hamasalih, Rahel J. Qadir","doi":"10.33993/jnaat512-1272","DOIUrl":"https://doi.org/10.33993/jnaat512-1272","url":null,"abstract":"In this article, the researchers develop a new type of spline function with fractional order which constructs two distinct formulas for the proposed method by using fractional boundary conditions and fractional continuity conditions. These methods are used to solve linear Volterra and Fredholm-integral equations of the second kind. The convergence analysis is studied. Moreover, some numerical examples are provided and compared to illustrate the efficiency and applicability of the proposed methods.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115616705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we introduce a modified fixed point method to process the large and sparse linear complementarity problem (LCP) and formulate an equivalent fixed point equation for the LCP and show the equivalence. Also, we provide convergence conditions when the system matrix is a (P)-matrix and two sufficient convergence conditions when the system matrix is an (H_+)-matrix. To show the efficiency of our proposed method, we illustrate two numerical examples for different parameters.
{"title":"On general fixed point method based on matrix splitting for solving linear complementarity problem","authors":"B. Kumar, Deepmala, Arup K Das","doi":"10.33993/jnaat512-1285","DOIUrl":"https://doi.org/10.33993/jnaat512-1285","url":null,"abstract":"In this article, we introduce a modified fixed point method to process the large and sparse linear complementarity problem (LCP) and formulate an equivalent fixed point equation for the LCP and show the equivalence. Also, we provide convergence conditions when the system matrix is a (P)-matrix and two sufficient convergence conditions when the system matrix is an (H_+)-matrix. To show the efficiency of our proposed method, we illustrate two numerical examples for different parameters.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123787972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Here we expose multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or (mathbb{R}^{N}), (Nin mathbb{N}), by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Frechet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the arctangent function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.
{"title":"General multivariate arctangent function activated neural network approximations","authors":"G. Anastassiou","doi":"10.33993/jnaat511-1262","DOIUrl":"https://doi.org/10.33993/jnaat511-1262","url":null,"abstract":"Here we expose multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or (mathbb{R}^{N}), (Nin mathbb{N}), by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Frechet derivatives. \u0000Our multivariate operators are defined by using a multidimensional density function induced by the arctangent function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130709812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a previous paper the author presented a Kantorovich modification of Baskakov operators which reproduce affine functions and he provided an upper estimate for the rate of convergence in polynomial weighted spaces. In this paper, for the same family of operators, a strong inverse inequality is given for the case of approximation in norm.
{"title":"Baskakov-Kantorovich operators reproducing affine functions: inverse results","authors":"J. Bustamante","doi":"10.33993/jnaat511-1264","DOIUrl":"https://doi.org/10.33993/jnaat511-1264","url":null,"abstract":"In a previous paper the author presented a Kantorovich modification of Baskakov operators which reproduce affine functions and he provided an upper estimate for the rate of convergence in polynomial weighted spaces.\u0000In this paper, for the same family of operators, a strong inverse inequality is given for the case of approximation in norm.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121337208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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