Given (fin C[-1,1]) and (n) points (nodes) in ([-1,1]), the Hermite-Fejer interpolation (HFI) polynomial is the polynomial of degree at most (2n-1) which agrees with (f) and has zero derivative at each of the nodes. In 1916, L. Fejer showed that if the nodes are chosen to be the zeros of (T_{n}(x)), the (n)th Chebyshev polynomial of the first kind, then the HFI polynomials converge uniformly to (f) as (nrightarrowinfty). Later, D.L. Berman established the rather surprising result that this convergence property is no longer true for all (f) if the Chebyshev nodes are augmented by including the endpoints (-1) and (1) as additional nodes. This behaviour has become known as Berman's phenomenon. The aim of this paper is to investigate Berman's phenomenon in the setting of ((0,1,2)) HFI, where the interpolation polynomial agrees with (f) and has vanishing first and second derivatives at each node. The principal result provides simple necessary and sufficient conditions, in terms of the (one-sided) derivatives of (f) at (pm 1), for pointwise and uniform convergence of ((0,1,2)) HFI on the augmented Chebyshev nodes if (fin C^{4}[-1,1]), and confirms that Berman's phenomenon occurs for ((0,1,2)) HFI.
{"title":"On Berman's phenomenon for (0,1,2) Hermite-Fejér interpolation","authors":"Graeme J. Byrne, S. Smith","doi":"10.33993/jnaat481-1163","DOIUrl":"https://doi.org/10.33993/jnaat481-1163","url":null,"abstract":"Given (fin C[-1,1]) and (n) points (nodes) in ([-1,1]), the Hermite-Fejer interpolation (HFI) polynomial is the polynomial of degree at most (2n-1) which agrees with (f) and has zero derivative at each of the nodes. In 1916, L. Fejer showed that if the nodes are chosen to be the zeros of (T_{n}(x)), the (n)th Chebyshev polynomial of the first kind, then the HFI polynomials converge uniformly to (f) as (nrightarrowinfty). Later, D.L. Berman established the rather surprising result that this convergence property is no longer true for all (f) if the Chebyshev nodes are augmented by including the endpoints (-1) and (1) as additional nodes. This behaviour has become known as Berman's phenomenon. The aim of this paper is to investigate Berman's phenomenon in the setting of ((0,1,2)) HFI, where the interpolation polynomial agrees with (f) and has vanishing first and second derivatives at each node. The principal result provides simple necessary and sufficient conditions, in terms of the (one-sided) derivatives of (f) at (pm 1), for pointwise and uniform convergence of ((0,1,2)) HFI on the augmented Chebyshev nodes if (fin C^{4}[-1,1]), and confirms that Berman's phenomenon occurs for ((0,1,2)) HFI.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127356213","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 consider the approximation of functions by sublinear positive operators with applications to a big variety of Max-Product operators under Caputo fractional differentiability. Our study is based on our general fractional results about positive sublinear operators. We produce Jackson type inequalities under simple initial conditions. So our approach is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of fractional derivative of the function underapproximation.
{"title":"Caputo fractional approximation by sublinear operators","authors":"G. Anastassiou","doi":"10.33993/jnaat472-1135","DOIUrl":"https://doi.org/10.33993/jnaat472-1135","url":null,"abstract":"Here we consider the approximation of functions by sublinear positive operators with applications to a big variety of Max-Product operators under Caputo fractional differentiability. \u0000Our study is based on our general fractional results about positive sublinear operators. We produce Jackson type inequalities under simple initial conditions. So our approach is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of fractional derivative of the function underapproximation.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121930464","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 deal with the study and implementation of an infeasible interior point method for convex quadratic problems (CQP). The algorithm uses a Newton step and suitable proximity measure for approximately tracing the central path and guarantees that after one feasibility step, the new iterate is feasible and suciently close to the central path. For its complexity analysis, we reconsider the analysis used by the authors for linear optimisation (LO) and linear complementarity problems (LCP). We show that the algorithm has the best known iteration bound, namely (n log (n+1)). Finally, to measure the numerical performance of this algorithm, it was tested on convex quadratic and linear problems.
{"title":"An infeasible interior point methods for convex quadratic problems","authors":"H. Roumili, N. Boudjellal","doi":"10.33993/jnaat472-1147","DOIUrl":"https://doi.org/10.33993/jnaat472-1147","url":null,"abstract":"In this paper, we deal with the study and implementation of an infeasible interior point method for convex quadratic problems (CQP). The algorithm uses a Newton step and suitable proximity measure for approximately tracing the central path and guarantees that after one feasibility step, the new iterate is feasible and suciently close to the central path. For its complexity analysis, we reconsider the analysis used by the authors for linear optimisation (LO) and linear complementarity problems (LCP). \u0000We show that the algorithm has the best known iteration bound, namely (n log (n+1)). \u0000Finally, to measure the numerical performance of this algorithm, it was tested on convex quadratic and linear problems.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129236425","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}
We present a local convergence analysis of an eighth-order Aitken-Newton method for approximating a locally unique solution of a nonlinear equation. Earlier studies have shown convergence of these methods under hypotheses up to the eighth derivative of the function although only the first derivative appears in the method. In this study, we expand the applicability of these methods using only hypotheses up to the first derivative of the function. This way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.
{"title":"Ball convergence for an Aitken-Newton method","authors":"I. Argyros, M. Kansal, V. Kanwar","doi":"10.33993/jnaat472-1082","DOIUrl":"https://doi.org/10.33993/jnaat472-1082","url":null,"abstract":"We present a local convergence analysis of an eighth-order Aitken-Newton method for approximating a locally unique solution of a nonlinear equation. Earlier studies have shown convergence of these methods under hypotheses up to the eighth derivative of the function although only the first derivative appears in the method. In this study, we expand the applicability of these methods using only hypotheses up to the first derivative of the function. This way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130877048","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 study the continuation of harmonic functions in the ball to the entire harmonic functions in space (mathbb{R}^n), (ngeq 3). The generalized order introduced by M.N. Seremeta has been used to characterize the growth of such functions. Moreover, the generalized order, generalized lower order and generalized type have been characterized in terms of harmonic polynomial approximation errors. Our results apply satisfactorily for slow growth.
{"title":"Generalized growth and approximation errors of entire harmonic functions in (R^n), (n geq 3)","authors":"Devendra Kumar","doi":"10.33993/jnaat472-1166","DOIUrl":"https://doi.org/10.33993/jnaat472-1166","url":null,"abstract":"In this paper we study the continuation of harmonic functions in the ball to the entire harmonic functions in space (mathbb{R}^n), (ngeq 3). \u0000The generalized order introduced by M.N. Seremeta has been used to characterize the growth of such functions. Moreover, the generalized order, generalized lower order and generalized type have been characterized in terms of harmonic polynomial approximation errors. \u0000Our results apply satisfactorily for slow growth.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129901410","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 deal with a Stancu type extension of the Cheney and Sharma operators. We consider a recurrence relation to get moments of the operators and give a local approximation result via suitable K-functional. Moreover, we show that each operator preserves the Lipschitz constant and order of a given Lipschitz continuous function.
{"title":"A Stancu type extension of Cheney and Sharma operators","authors":"T. Bostanci, Gülen Başcanbaz-Tunca","doi":"10.33993/jnaat472-1133","DOIUrl":"https://doi.org/10.33993/jnaat472-1133","url":null,"abstract":"In this paper we deal with a Stancu type extension of the Cheney and Sharma operators. \u0000We consider a recurrence relation to get moments of the operators and give a local approximation result via suitable K-functional. Moreover, we show that each operator preserves the Lipschitz constant and order of a given Lipschitz continuous function.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132863812","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, for the univariate Bernstein-Kantorovich-Choquet, Szasz-Kantorovich-Choquet, Baskakov-Kantorovich-Choquet and Bernstein-Durrmeyer-Choquet operators written in terms of the Choquet integrals with respect to monotone and submodular set functions, we study the preservation of the monotonicity and convexity of the approximated functions and the monotonicity of some approximation sequences.
{"title":"Shape preserving properties and monotonicity properties of the sequences of Choquet type integral operators","authors":"S. Gal","doi":"10.33993/jnaat472-1154","DOIUrl":"https://doi.org/10.33993/jnaat472-1154","url":null,"abstract":"In this paper, for the univariate Bernstein-Kantorovich-Choquet, Szasz-Kantorovich-Choquet, Baskakov-Kantorovich-Choquet and Bernstein-Durrmeyer-Choquet operators written in terms of the Choquet integrals with respect to monotone and submodular set functions, we study the preservation of the monotonicity and convexity of the approximated functions and the monotonicity of some approximation sequences.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130301800","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 study approximation methods for solving bi-criteria optimization problems. Initial problem is approximated by a new one which has the components of the objective and the constraints are replaced by their approximation functions. Components of the objective function are first and second order approximated and constraints are first order approximated. Conditions such that efficient solution of the approximate problem will remain efficient for initial problem and reciprocally are studied. Numerical examples are developed to emphasize the importance of these conditions.
{"title":"Approximations of objective function and constraints in bi-criteria optimization problems","authors":"Traian Ionut Luca, D. Duca","doi":"10.33993/jnaat472-1153","DOIUrl":"https://doi.org/10.33993/jnaat472-1153","url":null,"abstract":"In this paper we study approximation methods for solving bi-criteria optimization problems. \u0000Initial problem is approximated by a new one which has the components of the objective and the constraints are replaced by their approximation functions. Components of the objective function are first and second order approximated and constraints are first order approximated. Conditions such that efficient solution of the approximate problem will remain efficient for initial problem and reciprocally are studied. \u0000Numerical examples are developed to emphasize the importance of these conditions.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"95 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127076886","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 present a theoretical and numerical study of linear complementary problems solvable as linear programs. We give several examples of linear complementarity problems which can be solved as linear programs using linear programming appraoches. Also, we propose two examples solved by the simplex and Karmarkar's method, while the most widely know method for solving linear complementarity problems "the complementarity pivoting algorithm due to Lemke" has failed to find a solution.
{"title":"Linear complementarity problems solvable as linear programs","authors":"Z. Kebbiche","doi":"10.33993/jnaat472-1156","DOIUrl":"https://doi.org/10.33993/jnaat472-1156","url":null,"abstract":"In this paper, we present a theoretical and numerical study of linear complementary problems solvable as linear programs. \u0000We give several examples of linear complementarity problems which can be solved as linear programs using linear programming appraoches. Also, we propose two examples solved by the simplex and Karmarkar's method, while the most widely know method for solving linear complementarity problems \"the complementarity pivoting algorithm due to Lemke\" has failed to find a solution.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115794779","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}
Some approximation properties of hexagonal Fourier series are investigated. The order of approximation by Nörlund means of hexagonal Fourier series is estimated in terms of modulus of continuity.
研究了六方傅里叶级数的一些近似性质。通过Nörlund方法对六边形傅立叶级数的逼近阶用连续模估计。
{"title":"Approximation of continuous functions on hexagonal domains","authors":"A. Guven","doi":"10.33993/jnaat471-1128","DOIUrl":"https://doi.org/10.33993/jnaat471-1128","url":null,"abstract":"\u0000 \u0000 \u0000Some approximation properties of hexagonal Fourier series are investigated. The order of approximation by Nörlund means of hexagonal Fourier series is estimated in terms of modulus of continuity. \u0000 \u0000 \u0000","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123986399","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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