In this paper, we investigate the degree of approximation by Baskakov_Durrmeyer operator for functions which derivatives have only discontinuity points of the first kind on [0,∞) with exponential growth.
In this paper we investigate simultaneous approximation for arbitrary system of nodes on smooth domain in complex plane. Some results which are better than those of known theorems are obtained.
We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W ＋ M a...We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W ＋ M and the quotient space W/M.展开更多
This paper considers to replace △_m(x)=(1-x^2)~2(1/2)/n +1/n^2 in the following result for simultaneous Lagrange interpolating approximation with (1-x^2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then |f^(k)(x)-P_^...This paper considers to replace △_m(x)=(1-x^2)~2(1/2)/n +1/n^2 in the following result for simultaneous Lagrange interpolating approximation with (1-x^2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then |f^(k)(x)-P_^(k)(f,x)|=O(1)△_(n)^(a-k)(x)ω(f^(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q, where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_n U Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.展开更多
In this paper we give equivalent theorems on simultaneous approximation for the combinations of Bernstein operators by r-th Ditzian- Totik modulus of smoothness w^rφλ （f, t）（0 ≤ λ≤ 1）. We also investigate th...In this paper we give equivalent theorems on simultaneous approximation for the combinations of Bernstein operators by r-th Ditzian- Totik modulus of smoothness w^rφλ （f, t）（0 ≤ λ≤ 1）. We also investigate the relation between the derivatives of the combinations of Bernstein operators and the smoothness of derivatives of functions.展开更多
The aim of this work is to generalize Szasz-Mirakian operator in the sense of Stancu-Durrmeyer operators. We obtain approximation properties of these operators. Here we study asymptotic as well as rate of convergence ...The aim of this work is to generalize Szasz-Mirakian operator in the sense of Stancu-Durrmeyer operators. We obtain approximation properties of these operators. Here we study asymptotic as well as rate of convergence results in simultaneous approximation for these modified operators.展开更多
In the present paper, the shape-preserving properties and the monotonicity for convex functions of Stancu operator are given. Moreover, the simultaneous approximation problems of this operator are also considered.
Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P&...Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +展开更多
In this paper, we investigate the simultaneous approximation of Bernstein- Sikkema operators, and establish the direct and equivalent theorems by using the Ditzian-Totik modulus of smoothness.
In this paper, some equivalent theorems on simultaneous approximation for combinations of Gamma operators by weighted moduli of smoothness ωφλ^r（f,t）wφ^s（0≤λ≤1）are given. The relation between derivatives ...In this paper, some equivalent theorems on simultaneous approximation for combinations of Gamma operators by weighted moduli of smoothness ωφλ^r（f,t）wφ^s（0≤λ≤1）are given. The relation between derivatives of combinations of Gamma operators and smoothness of derivatives of functions is also investigated.展开更多
Recently some classical operator quasi-interpolants were introduced to obtain much faster convergence. We consider left Gamma quasi-interpolants and give a pointwise simultaneous approximation equivalence theorem with...Recently some classical operator quasi-interpolants were introduced to obtain much faster convergence. We consider left Gamma quasi-interpolants and give a pointwise simultaneous approximation equivalence theorem with ωφλ^2r（f,t）∞ by means of unified the classical modulus and Ditzian-Totick modulus.展开更多
We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F<sub>1</sub> and F<sub>2</sub> are cont...We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F<sub>1</sub> and F<sub>2</sub> are continuous functions on a closed interval [a,b], S is an n-dimensional Chebyshev subspace of C<span style="white-space:normal;"> [<em style="white-space:normal;">a<span style="white-space:normal;">,<em style="white-space:normal;">b<span style="white-space:normal;">] and s<sub>1</sub>* & <span style="white-space:normal;">s<sub>2</sub>* are the best uniform approximations to F<sub>1</sub> and F<sub>2</sub> from S respectively. The characterization of the best approximation solution is used to show that, under some restrictions on the point set of alternations of F<sub>1</sub><span style="white-space:nowrap;">−s<sub>1</sub>* and <em style="white-space:normal;">F<sub style="white-space:normal;">2</sub>−<em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">*, <em style="white-space:normal;">s<sub style="white-space:normal;">1</sub><span style="white-space:normal;">* or <em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">* is also a best A(1) simultaneous approximation to F<sub>1</sub> and F<sub>2</sub> from S with F<sub>1</sub><span style="white-space:nowrap;">≥F<sub>2</sub> and n=2.展开更多
In this paper, we study the characterization of f-Chebyshev radius and f-Chebyshev centers and the existence of f-Chebyshev centers in locally convex spaces.
Using a recent result regarding the fixed points of multivalued mappings, the existence of invariant best simultaneous approximation in chainable metric space is proved.
The aim of the present paper is to prove direct and converse results for simultaneous approximation by modified Bernstein-Durrmeyer operators. A point-wise equivalence characterization of simultaneous approximation is...The aim of the present paper is to prove direct and converse results for simultaneous approximation by modified Bernstein-Durrmeyer operators. A point-wise equivalence characterization of simultaneous approximation is obtained.展开更多
The well posedness of best simultaneous approximation problems is considered. We establish the generic results on the well posedness of the best simultaneous approximation problems for any closed weakly compact nonemp...The well posedness of best simultaneous approximation problems is considered. We establish the generic results on the well posedness of the best simultaneous approximation problems for any closed weakly compact nonempty subset in a strictly convex Kadec Banach space. Further, we prove that the set of all points inE(G) such that the best simultaneous approximation problems are not well posed is a u- porous set inE(G) whenX is a uniformly convex Banach space. In addition, we also investigate the generic property of the ambiguous loci of the best simultaneous approximation.展开更多
Let(S, Σ, μ) be a complete positive σ-finite measure space and let X be a Banach space. We consider the simultaneous proximinality problem in Lp(S, Σ, X) for 1 p < +∞. We establish some N-simultaneous proximin...Let(S, Σ, μ) be a complete positive σ-finite measure space and let X be a Banach space. We consider the simultaneous proximinality problem in Lp(S, Σ, X) for 1 p < +∞. We establish some N-simultaneous proximinality results of Lp(S, Σ0, Y) in Lp(S, Σ, X) without the Radon-Nikody′m property(RNP) assumptions on the space span Y and its dual span Y*, where Σ0is a sub-σ-algebra of Σ and Y a nonempty locally weakly compact closed convex subset of X. In particular, we completely solve one open problem and partially solve another one in Luo et al.(2011).展开更多
This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the...This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth.展开更多
In this paper, we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated. We give so...In this paper, we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated. We give some direct and inverse results on pointwise simultaneous approximation by the combinations of Baskakov operators. We also give a new equivalent result on pointwise approximation by these operators.展开更多
Motivated by the problem to approximate all feasible schedules by one schedule in a given scheduling environment,we introduce in this paper the concepts of strong simultaneous approximation ratio and weak simultaneous...Motivated by the problem to approximate all feasible schedules by one schedule in a given scheduling environment,we introduce in this paper the concepts of strong simultaneous approximation ratio and weak simultaneous approximation ratio.Then we study the two variants under various scheduling environments,such as non-preemptive,preemptive or fractional scheduling on identical,related or unrelated machines.展开更多
文摘In this paper, we investigate the degree of approximation by Baskakov_Durrmeyer operator for functions which derivatives have only discontinuity points of the first kind on [0,∞) with exponential growth.
文摘In this paper we investigate simultaneous approximation for arbitrary system of nodes on smooth domain in complex plane. Some results which are better than those of known theorems are obtained.
文摘We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W ＋ M and the quotient space W/M.
文摘This paper considers to replace △_m(x)=(1-x^2)~2(1/2)/n +1/n^2 in the following result for simultaneous Lagrange interpolating approximation with (1-x^2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then |f^(k)(x)-P_^(k)(f,x)|=O(1)△_(n)^(a-k)(x)ω(f^(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q, where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_n U Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.
基金Supported by the Key Academic Discipline of Zhejiang Provincial of China under Grant No.2005.
文摘In this paper we give equivalent theorems on simultaneous approximation for the combinations of Bernstein operators by r-th Ditzian- Totik modulus of smoothness w^rφλ （f, t）（0 ≤ λ≤ 1）. We also investigate the relation between the derivatives of the combinations of Bernstein operators and the smoothness of derivatives of functions.
文摘The aim of this work is to generalize Szasz-Mirakian operator in the sense of Stancu-Durrmeyer operators. We obtain approximation properties of these operators. Here we study asymptotic as well as rate of convergence results in simultaneous approximation for these modified operators.
文摘In the present paper, the shape-preserving properties and the monotonicity for convex functions of Stancu operator are given. Moreover, the simultaneous approximation problems of this operator are also considered.
文摘Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +
基金the National Natural Science Foundation of China (10631080)the Zhejiang Provincial Key Basic Subject Foundation of China(10571014)
文摘In this paper, we investigate the simultaneous approximation of Bernstein- Sikkema operators, and establish the direct and equivalent theorems by using the Ditzian-Totik modulus of smoothness.
文摘In this paper, some equivalent theorems on simultaneous approximation for combinations of Gamma operators by weighted moduli of smoothness ωφλ^r（f,t）wφ^s（0≤λ≤1）are given. The relation between derivatives of combinations of Gamma operators and smoothness of derivatives of functions is also investigated.
基金the NSF of Zhejiang Province(102005)the Foundation of Key Discipline of ZhejiangProvince(2005)
文摘Recently some classical operator quasi-interpolants were introduced to obtain much faster convergence. We consider left Gamma quasi-interpolants and give a pointwise simultaneous approximation equivalence theorem with ωφλ^2r（f,t）∞ by means of unified the classical modulus and Ditzian-Totick modulus.
文摘We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F<sub>1</sub> and F<sub>2</sub> are continuous functions on a closed interval [a,b], S is an n-dimensional Chebyshev subspace of C<span style="white-space:normal;"> [<em style="white-space:normal;">a<span style="white-space:normal;">,<em style="white-space:normal;">b<span style="white-space:normal;">] and s<sub>1</sub>* & <span style="white-space:normal;">s<sub>2</sub>* are the best uniform approximations to F<sub>1</sub> and F<sub>2</sub> from S respectively. The characterization of the best approximation solution is used to show that, under some restrictions on the point set of alternations of F<sub>1</sub><span style="white-space:nowrap;">−s<sub>1</sub>* and <em style="white-space:normal;">F<sub style="white-space:normal;">2</sub>−<em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">*, <em style="white-space:normal;">s<sub style="white-space:normal;">1</sub><span style="white-space:normal;">* or <em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">* is also a best A(1) simultaneous approximation to F<sub>1</sub> and F<sub>2</sub> from S with F<sub>1</sub><span style="white-space:nowrap;">≥F<sub>2</sub> and n=2.
基金Research supported by the National Science Foundation of P.R.China
文摘In this paper, we study the characterization of f-Chebyshev radius and f-Chebyshev centers and the existence of f-Chebyshev centers in locally convex spaces.
文摘Using a recent result regarding the fixed points of multivalued mappings, the existence of invariant best simultaneous approximation in chainable metric space is proved.
基金Supported by Natural Science Foundation of Zhejiang Province(102002)
文摘The aim of the present paper is to prove direct and converse results for simultaneous approximation by modified Bernstein-Durrmeyer operators. A point-wise equivalence characterization of simultaneous approximation is obtained.
基金the National Natural Science Foundation of China (Grant No. 19971013) and Natural Science Foundation of Jiangsu Province (Grant No. BK99001) .
文摘The well posedness of best simultaneous approximation problems is considered. We establish the generic results on the well posedness of the best simultaneous approximation problems for any closed weakly compact nonempty subset in a strictly convex Kadec Banach space. Further, we prove that the set of all points inE(G) such that the best simultaneous approximation problems are not well posed is a u- porous set inE(G) whenX is a uniformly convex Banach space. In addition, we also investigate the generic property of the ambiguous loci of the best simultaneous approximation.
基金supported by National Natural Science Foundation of China(Grant Nos.11101363,11171300 and 11371325)Natural Science Foundation of Zhejiang Province(Grant No.LY12A01029)
文摘Let(S, Σ, μ) be a complete positive σ-finite measure space and let X be a Banach space. We consider the simultaneous proximinality problem in Lp(S, Σ, X) for 1 p < +∞. We establish some N-simultaneous proximinality results of Lp(S, Σ0, Y) in Lp(S, Σ, X) without the Radon-Nikody′m property(RNP) assumptions on the space span Y and its dual span Y*, where Σ0is a sub-σ-algebra of Σ and Y a nonempty locally weakly compact closed convex subset of X. In particular, we completely solve one open problem and partially solve another one in Luo et al.(2011).
基金Scientific Research Fund of Hunan Provincial Education Department (Grant No.06C651)the National Natural Science Foundation of China (Grant Nos.10671175,10731060)+1 种基金Program for New Century Excellent Talents in UniversityProjects MTM2006-13997-C02-01 and FQM-127 of Spain
文摘This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth.
基金This research is supported by the National Natural Science Foundation of Chinathe Zhejiang Provincial Natural ScienCe Foundation of China
文摘In this paper, we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated. We give some direct and inverse results on pointwise simultaneous approximation by the combinations of Baskakov operators. We also give a new equivalent result on pointwise approximation by these operators.
基金The first author was supported by the National Natural Science Foundation of China(Nos.11601198 and 71761015)The second author was supported by the National Natural Science Foundation of China(No.11671368).
文摘Motivated by the problem to approximate all feasible schedules by one schedule in a given scheduling environment,we introduce in this paper the concepts of strong simultaneous approximation ratio and weak simultaneous approximation ratio.Then we study the two variants under various scheduling environments,such as non-preemptive,preemptive or fractional scheduling on identical,related or unrelated machines.
