In this paper, a modification of Szász-Mirakyan operators is studied [1] which generalizes the Szász-Mirakyan operators with the property that the linear combination (e_2 + alpha e_1) of the Korovkin's test functions (e_1) and (e_2) are reproduced for (alphageq 0). After providing some computational results, shape preserving properties of mentioned operators are obtained. Moreover, some estimations for the rate of convergence of these operators by using different type modulus of continuity are shown. Furthermore, a Voronovskaya-type formula and an approximation result for derivative of operators are calculated. Finally, some graphics which are based on our main results are shown.
{"title":"On Szász-Mirakyan type operators preserving polynomials","authors":"O. Yilmaz, A. Aral, Fatma Taşdelen Yeşildal","doi":"10.33993/jnaat461-1087","DOIUrl":"https://doi.org/10.33993/jnaat461-1087","url":null,"abstract":"In this paper, a modification of Szász-Mirakyan operators is studied [1] which generalizes the Szász-Mirakyan operators with the property that the linear combination (e_2 + alpha e_1) of the Korovkin's test functions (e_1) and (e_2) are reproduced for (alphageq 0). After providing some computational results, shape preserving properties of mentioned operators are obtained. Moreover, some estimations for the rate of convergence of these operators by using different type modulus of continuity are shown. Furthermore, a Voronovskaya-type formula and an approximation result for derivative of operators are calculated. Finally, some graphics which are based on our main results are shown.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131639607","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 a Kantorovich generalization of q-Stancu-Lupa¸s operators and investigate their approximation properties. The rate of convergence of these operators are obtained by means of modulus of continuity, functions of Lipschitz class and Peetre's K-functional. We also investigate the convergence of the operators in the statistical sense and give a numerical example in order to estimate the error in the approximation.
{"title":"Approximation theorems for Kantorovich type Lupaș-Stancu operators based on (q)-integers","authors":"S. K. Serenbay, Özge Dalmanoglu","doi":"10.33993/jnaat461-1108","DOIUrl":"https://doi.org/10.33993/jnaat461-1108","url":null,"abstract":"In this paper, we introduce a Kantorovich generalization of q-Stancu-Lupa¸s operators and investigate their approximation properties. The rate of convergence of these operators are obtained by means of modulus of continuity, functions of Lipschitz class and Peetre's K-functional. We also investigate the convergence of the operators in the statistical sense and give a numerical example in order to estimate the error in the approximation.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130113356","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}
Schur's [20] Markov-type extremal problem is to determine (i) (M_n= sup_{-1leq xileq 1}sup_{P_ninmathbf{B}_{n,xi,2}}(|P_n^{(1)}(xi)| / n^2)), where (mathbf{B}_{n,xi,2}={P_ninmathbf{B}_n:P_n^{(2)}(xi)=0}subset mathbf{B}_n={P_n:|P_n(x)|leq 1 ;textrm{for}; |x| leq 1}) and (P_n) is an algebraic polynomial of degree (leq n). Erdos and Szego [4] found that for (ngeq 4) this maximum is attained if (xi=pm 1) and (P_ninmathbf{B}_{n,pm 1,2}) is a (unspecified) member of the one-parameter family of hard-core Zolotarev polynomials. An extremal such polynomial as well as the constant (M_n) we have explicitly specified for (n=4) in [17], and in this paper we strive to obtain an analogous amendment to the Erdos-Szego solution for (n = 5). The cases (n>5) still remain arcane.Our approach is based on the quite recently discovered explicit algebraic power form representation [6], [7] of the quintic hard-core Zolotarev polynomial, (Z_{5,t}), to which we add here explicit descriptions of its critical points, the explicit form of Pell's (aka: Abel's) equation, as well as an alternative proof for the range of the parameter, (t). The optimal (t=t^*) which yields (M_5 = |Z_{5,t^*}^{(1)}(1)|/25) we identify as the negative zero with smallest modulus of a minimal (P_{10}). We then turn to an extension of (i), to higher derivatives as proposed by Shadrin [22], and we provide an analogous solution for (n=5). Finally, we describe, again for (n = 5), two new algebraic approaches towards a solution to Zolotarev's so-called first problem [2], [24] which was originally solved by means of elliptic functions.
{"title":"The second Zolotarev case in the Erdös-Szegö solution to a Markov-type extremal problem of Schur","authors":"H. Rack","doi":"10.33993/jnaat461-1100","DOIUrl":"https://doi.org/10.33993/jnaat461-1100","url":null,"abstract":"Schur's [20] Markov-type extremal problem is to determine (i) (M_n= sup_{-1leq xileq 1}sup_{P_ninmathbf{B}_{n,xi,2}}(|P_n^{(1)}(xi)| / n^2)), where (mathbf{B}_{n,xi,2}={P_ninmathbf{B}_n:P_n^{(2)}(xi)=0}subset mathbf{B}_n={P_n:|P_n(x)|leq 1 ;textrm{for}; |x| leq 1}) and (P_n) is an algebraic polynomial of degree (leq n). Erdos and Szego [4] found that for (ngeq 4) this maximum is attained if (xi=pm 1) and (P_ninmathbf{B}_{n,pm 1,2}) is a (unspecified) member of the one-parameter family of hard-core Zolotarev polynomials. An extremal such polynomial as well as the constant (M_n) we have explicitly specified for (n=4) in [17], and in this paper we strive to obtain an analogous amendment to the Erdos-Szego solution for (n = 5). The cases (n>5) still remain arcane.Our approach is based on the quite recently discovered explicit algebraic power form representation [6], [7] of the quintic hard-core Zolotarev polynomial, (Z_{5,t}), to which we add here explicit descriptions of its critical points, the explicit form of Pell's (aka: Abel's) equation, as well as an alternative proof for the range of the parameter, (t). The optimal (t=t^*) which yields (M_5 = |Z_{5,t^*}^{(1)}(1)|/25) we identify as the negative zero with smallest modulus of a minimal (P_{10}). We then turn to an extension of (i), to higher derivatives as proposed by Shadrin [22], and we provide an analogous solution for (n=5). Finally, we describe, again for (n = 5), two new algebraic approaches towards a solution to Zolotarev's so-called first problem [2], [24] which was originally solved by means of elliptic functions.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131763575","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 are concerned with accurate Chebyshev collocation (ChC) solutions to fourth order eigenvalue problems. We consider the 1D case as well as the 2D case. In order to improve the accuracy of computation we use the precondtitioning strategy for second order differential operator introduced by Labrosse in 2009. The fourth order differential operator is factorized as a product of second order operators. In order to asses the accuracy of our method we calculate the so called drift of the first five eigenvalues. In both cases ChC method with the considered preconditioners provides accurate eigenpairs of interest.
{"title":"Accurate Chebyshev collocation solutions for the biharmonic eigenproblem on a rectangle","authors":"I. Boros","doi":"10.33993/jnaat461-1124","DOIUrl":"https://doi.org/10.33993/jnaat461-1124","url":null,"abstract":"We are concerned with accurate Chebyshev collocation (ChC) solutions to fourth order eigenvalue problems. We consider the 1D case as well as the 2D case. In order to improve the accuracy of computation we use the precondtitioning strategy for second order differential operator introduced by Labrosse in 2009. The fourth order differential operator is factorized as a product of second order operators. In order to asses the accuracy of our method we calculate the so called drift of the first five eigenvalues. In both cases ChC method with the considered preconditioners provides accurate eigenpairs of interest.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121303625","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}
{"title":"Professor Costică Mustăța at his 75th anniversary","authors":"E. Catinas, I. Pavaloiu","doi":"10.33993/jnaat461-1131","DOIUrl":"https://doi.org/10.33993/jnaat461-1131","url":null,"abstract":"","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127630415","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 local and semilocal convergence results for Newton’s method in order to approximate solutions of subanalytic equations. The local convergence results are given under weaker conditions than in earlier studies such as [9], [10], [14], [15], [24], [25], [26], resulting to a larger convergence ball and a smaller ratio of convergence. In the semilocal convergence case contravariant conditions not used before are employed to show the convergence of Newton’s method. Numerical examples illustrating the advantages of our approach are also presented in this study.
{"title":"On Newton's method for subanalytic equations","authors":"I. Argyros, S. George","doi":"10.33993/jnaat461-1132","DOIUrl":"https://doi.org/10.33993/jnaat461-1132","url":null,"abstract":"We present local and semilocal convergence results for Newton’s method in order to approximate solutions of subanalytic equations. The local convergence results are given under weaker conditions than in earlier studies such as [9], [10], [14], [15], [24], [25], [26], resulting to a larger convergence ball and a smaller ratio of convergence. In the semilocal convergence case contravariant conditions not used before are employed to show the convergence of Newton’s method. Numerical examples illustrating the advantages of our approach are also presented in this study.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125516228","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}
{"title":"In memoriam dr. Călin Vamoș, member of Tiberiu Popoviciu Institute of Numerical Analysis","authors":"E. Catinas, N. Suciu","doi":"10.33993/jnaat461-1129","DOIUrl":"https://doi.org/10.33993/jnaat461-1129","url":null,"abstract":"","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121929322","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 consider the fundamental polynomials associated with the Bernstein operators of second kind. They form a blending system for which we study some shape preserving properties.Modified operators are introduced; they have better interpolation properties. The corresponding blending system is also studied.
{"title":"Bernstein operators of second kind and blending systems","authors":"D. Inoan, Fadel Nasaireh, I. Raşa","doi":"10.33993/jnaat461-1103","DOIUrl":"https://doi.org/10.33993/jnaat461-1103","url":null,"abstract":"We consider the fundamental polynomials associated with the Bernstein operators of second kind. They form a blending system for which we study some shape preserving properties.Modified operators are introduced; they have better interpolation properties. The corresponding blending system is also studied.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114281306","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, using majorization theorems and Lidstone's interpolating polynomials we obtain results concerning Jensen's and Jensen-Steffensen's inequalities and their converses in both the integral and the discrete case. We give bounds for identities related to these inequalities by using Chebyshev functionals. We also give Grüss type inequalities and Ostrowsky type inequalities for these functionals. Also we use these generalizations to construct a linear functionals and we present mean value theorems and n-exponential convexity which leads to exponential convexity and then log-convexity for these functionals. We give some families of functions which enable us to construct a large families of functions that are exponentially convex and also give Stolarsky type means with their monotonicity.
{"title":"Generalization of Jensen's and Jensen-Steffensen's inequalities and their converses by Lidstone's polynomial and majorization theorem","authors":"G. Aras-Gazić, J. Pečarić, A. Vukelic","doi":"10.33993/jnaat461-1111","DOIUrl":"https://doi.org/10.33993/jnaat461-1111","url":null,"abstract":"In this paper, using majorization theorems and Lidstone's interpolating polynomials we obtain results concerning Jensen's and Jensen-Steffensen's inequalities and their converses in both the integral and the discrete case. We give bounds for identities related to these inequalities by using Chebyshev functionals. We also give Grüss type inequalities and Ostrowsky type inequalities for these functionals. Also we use these generalizations to construct a linear functionals and we present mean value theorems and n-exponential convexity which leads to exponential convexity and then log-convexity for these functionals. We give some families of functions which enable us to construct a large families of functions that are exponentially convex and also give Stolarsky type means with their monotonicity.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131930288","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}
Many important applications of the class of convex sequences came across in several branches of mathematics as well as their generalizations. In this paper, we have introduced a new class of convex sequences, the class of (alpha)-convex sequences of higher order. In addition, the characterizations of sequences belonging to this class have been shown.
{"title":"On (alpha)-convex sequences of higher order","authors":"X. Krasniqi","doi":"10.33993/jnaat452-1093","DOIUrl":"https://doi.org/10.33993/jnaat452-1093","url":null,"abstract":"Many important applications of the class of convex sequences came across in several branches of mathematics as well as their generalizations. In this paper, we have introduced a new class of convex sequences, the class of (alpha)-convex sequences of higher order. In addition, the characterizations of sequences belonging to this class have been shown.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"122 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125917058","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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