Results regarding best approximation and best Simultaneous approximation on convex metric spaces are Obtained.Existence of fixed points for an ultimately nonexpansive semigroup of mappings is also shown.
Let L^2([0,1],x) be the space of the real valued,measurable,square summable functions on [0,1] with weight x,and let ■_n be the subspace of L^2([0,1],x) defined by a linear combination of J_0(μ_kx),where J_0 is the ...Let L^2([0,1],x) be the space of the real valued,measurable,square summable functions on [0,1] with weight x,and let ■_n be the subspace of L^2([0,1],x) defined by a linear combination of J_0(μ_kx),where J_0 is the Bessel function of order 0 and {μ_k} is the strictly increasing sequence of all positive zeros of J_0.For f∈L^2([0,1],x),let E(f,■_n) be the error of the best L^2([0,1],x),i.e.,approximation of f by elements of ■_n.The shift operator off at point x∈[0,1] with step t∈[0,1] is defined by T(t)f(x)=(1/π)∫_0~πf((x^2+t^2-2xtcosθ)^(1/2))dθ. The differences (1- T(t))^(r/2)f =∑_(j=0)~∞(-1)~j(_j^(r/2))T^j(t)f of order r∈(0,∞) and the L^2([0,1],x)-modulus of continuityω_r(f,τ)= sup{||(I-T(t))^(r/2)f||:0≤t≤τ}of order r are defined in the standard way,where T^0(t)=I is the identity operator.In this paper,we establish the sharp Jackson inequality between E(f,■_n) andω_r(f,τ) for some cases of r andτ.More precisely,we will find the smallest constant ■_n(τ,r) which depends only on n,r,andτ,such that the inequality E(f,■_n)≤■_n(τ,r)ω_r(f,τ) is valid.展开更多
In this paper, a new concept of weakly ,convex graph for set-valued mappings is introduced and studied. By using the concept , some new coincidence, the bestapproximation and fixed point-theorems are obta...In this paper, a new concept of weakly ,convex graph for set-valued mappings is introduced and studied. By using the concept , some new coincidence, the bestapproximation and fixed point-theorems are obtained.展开更多
Some new coincidence theorem s involving a new class of set_valued mappings containing composites of acyclic mappings defined in a contractible space are proved. For applications, some best approximation theorems an...Some new coincidence theorem s involving a new class of set_valued mappings containing composites of acyclic mappings defined in a contractible space are proved. For applications, some best approximation theorems and coincidence theorems for set-valued mappings are als o given. A number of known results in recent literature are improved and general ized by the theorems in this paper.展开更多
In this paper, we introduce a condition weaker than the LP differentiability, which we call Cp condition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation...In this paper, we introduce a condition weaker than the LP differentiability, which we call Cp condition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation at that point. We also give a necessary and sufficient condition for that a function be LP differentiable. In addition, we study the convexity of the set of cluster points of the net of best appoximations of f, {Pε(f)} asε→0.展开更多
In this paper, a new class of triangular summation operators based on the equidistant nodes was constructed. It is proved that this class of operators converges uniformly to arbitrary continuous fimctions with the per...In this paper, a new class of triangular summation operators based on the equidistant nodes was constructed. It is proved that this class of operators converges uniformly to arbitrary continuous fimctions with the period 2π on the whole axis, Fttrthermore, the best approximation order and the highest convergence order are obtained. In contrast to certain operators constructed by Bernstein and Kis in the previous works, the convergence properties of the new operator constructed in this paper are superior.展开更多
In numerical analysis, it is significant to approximate the linear functional Ef=sum from i=0 to m-1([integral from a to b(a<sub>1</sub>(x)f<sup>1</sup>(x)dx+ sum from f=0 to i<sub>1&...In numerical analysis, it is significant to approximate the linear functional Ef=sum from i=0 to m-1([integral from a to b(a<sub>1</sub>(x)f<sup>1</sup>(x)dx+ sum from f=0 to i<sub>1</sub>(b<sub>1</sub>f<sup>1</sup>(x<sub>1</sub>))]) by a simpler linear functional Lf=sum from i=1 to m(a<sub>1</sub>f(x<sub>1</sub>)) In this paper, making use of natural Tchebysheff spline function, we give existence theorem and uniqueness theorem of L that is exact for the degree m to F; we also give three sufficient and necessary conditions in which L is the Sard best approximation to F.展开更多
We propose a class of iteration methods searching the best approximately generalized polynomial, which has parallel computational function and converges to the exact solution quadratically. We first transform it into ...We propose a class of iteration methods searching the best approximately generalized polynomial, which has parallel computational function and converges to the exact solution quadratically. We first transform it into a special system of nonlinear equations with constraint, then by using to certain iteration method, we combine the two basic processes of the Remes method into a whole such that the iterative process of the system of nonlinear equations and the computation of the solution to the system of linear equations proceed alternately. A lot of numerical examples show that this method not only has good convergence property but also always converges to the exact solution of the problem accurately and rapidly for almost all initial approximations .展开更多
In an abstract set up, we get strong type inequalities in L^p+1 by assuming weak or extra-weak inequalities in Orlicz spaces. For some classes of functions, the number p is related to Simonenko indices. We apply the ...In an abstract set up, we get strong type inequalities in L^p+1 by assuming weak or extra-weak inequalities in Orlicz spaces. For some classes of functions, the number p is related to Simonenko indices. We apply the results to get strong inequal- ities for maximal functions associated to best Ф-approximation operators in an Orlicz space L^Ф.展开更多
In order to address the complex uncertainties caused by interfacing between the fuzziness and randomness of the safety problem for embankment engineering projects, and to evaluate the safety of embankment engineering ...In order to address the complex uncertainties caused by interfacing between the fuzziness and randomness of the safety problem for embankment engineering projects, and to evaluate the safety of embankment engineering projects more scientifically and reasonably, this study presents the fuzzy logic modeling of the stochastic finite element method (SFEM) based on the harmonious finite element (HFE) technique using a first-order approximation theorem. Fuzzy mathematical models of safety repertories were introduced into the SFEM to analyze the stability of embankments and foundations in order to describe the fuzzy failure procedure for the random safety performance function. The fuzzy models were developed with membership functions with half depressed gamma distribution, half depressed normal distribution, and half depressed echelon distribution. The fuzzy stochastic mathematical algorithm was used to comprehensively study the local failure mechanism of the main embankment section near Jingnan in the Yangtze River in terms of numerical analysis for the probability integration of reliability on the random field affected by three fuzzy factors. The result shows that the middle region of the embankment is the principal zone of concentrated failure due to local fractures. There is also some local shear failure on the embankment crust. This study provides a referential method for solving complex multi-uncertainty problems in engineering safety analysis.展开更多
This paper is part II of "On Copositive Approximation in Spaces of Contin- uous Functions". In this paper, the author shows that if Q is any compact subset of real numbers, and M is any finite dimensional strict Che...This paper is part II of "On Copositive Approximation in Spaces of Contin- uous Functions". In this paper, the author shows that if Q is any compact subset of real numbers, and M is any finite dimensional strict Chebyshev subspace of C (Q), then for any admissible function f ∈ C(Q)/M, the best copositive approximation to f from M is unique.展开更多
We pressent new Ky Fan type best approximation theorems for a discontinuous multivalued map on metrizable topological vector spaces and hyperconvex spaces. In addition, fixed point results are derived for the map stud...We pressent new Ky Fan type best approximation theorems for a discontinuous multivalued map on metrizable topological vector spaces and hyperconvex spaces. In addition, fixed point results are derived for the map studied. Our work generalizes severl results in approximation theory.展开更多
The relationship between the order of approximation by neural network based on scattered threshold value nodes and the neurons involved in a single hidden layer is investigated. The results obtained show that the degr...The relationship between the order of approximation by neural network based on scattered threshold value nodes and the neurons involved in a single hidden layer is investigated. The results obtained show that the degree of approximation by the periodic neural network with one hidden layer and scattered threshold value nodes is increased with the increase of the number of neurons hid in hidden layer and the smoothness of excitation function.展开更多
In this paper we investigate several solution algorithms for the convex fea- sibility problem(CFP)and the best approximation problem(BAP)respectively.The algorithms analyzed are already known before,but by adequately ...In this paper we investigate several solution algorithms for the convex fea- sibility problem(CFP)and the best approximation problem(BAP)respectively.The algorithms analyzed are already known before,but by adequately reformulating the CFP or the BAP we naturally deduce the general projection method for the CFP from well-known steepest decent method for unconstrained optimization and we also give a natural strategy of updating weight parameters.In the linear case we show the connec- tion of the two projection algorithms for the CFP and the BAP respectively.In addition, we establish the convergence of a method for the BAP under milder assumptions in the linear case.We also show by examples a Bauschke's conjecture is only partially correct.展开更多
Using a recent result regarding the fixed points of multivalued mappings, the existence of invariant best simultaneous approximation in chainable metric space is proved.
In this paper, Remes algorithm is applied to compute the numerical solution of the best chebyshev approximation from varisolvent family. Feasibility and convergence of the algorithm are discussed carefully.
In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the...In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.展开更多
A Bernstein type theorem and a converse theorem of best approximation by polynomials in Bergman spaces Hq^p(p>0,q>1) are proved.Some proofs and results in [1] are in proved.
An existence result on Ky Fan type best approximation is proved. For this pur- pose, a class of factorizable multifunctions and the other one being a demicontinuous, rela- tive almost quasi-convex, onto function on an...An existence result on Ky Fan type best approximation is proved. For this pur- pose, a class of factorizable multifunctions and the other one being a demicontinuous, rela- tive almost quasi-convex, onto function on an approximately weakly compact, convex sub- set of Hausdorff locally convex topological vector space are used. As consequence, this result extends the best approximation results of Basha and Veeramani[8] and many others.展开更多
文摘Results regarding best approximation and best Simultaneous approximation on convex metric spaces are Obtained.Existence of fixed points for an ultimately nonexpansive semigroup of mappings is also shown.
基金supported partly by National Natural Science Foundation of China (No.10471010)partly by the project"Representation Theory and Related Topics"of the"985 Program"of Beijing Normal University and Beijing Natural Science Foundation (1062004).
文摘Let L^2([0,1],x) be the space of the real valued,measurable,square summable functions on [0,1] with weight x,and let ■_n be the subspace of L^2([0,1],x) defined by a linear combination of J_0(μ_kx),where J_0 is the Bessel function of order 0 and {μ_k} is the strictly increasing sequence of all positive zeros of J_0.For f∈L^2([0,1],x),let E(f,■_n) be the error of the best L^2([0,1],x),i.e.,approximation of f by elements of ■_n.The shift operator off at point x∈[0,1] with step t∈[0,1] is defined by T(t)f(x)=(1/π)∫_0~πf((x^2+t^2-2xtcosθ)^(1/2))dθ. The differences (1- T(t))^(r/2)f =∑_(j=0)~∞(-1)~j(_j^(r/2))T^j(t)f of order r∈(0,∞) and the L^2([0,1],x)-modulus of continuityω_r(f,τ)= sup{||(I-T(t))^(r/2)f||:0≤t≤τ}of order r are defined in the standard way,where T^0(t)=I is the identity operator.In this paper,we establish the sharp Jackson inequality between E(f,■_n) andω_r(f,τ) for some cases of r andτ.More precisely,we will find the smallest constant ■_n(τ,r) which depends only on n,r,andτ,such that the inequality E(f,■_n)≤■_n(τ,r)ω_r(f,τ) is valid.
文摘In this paper, a new concept of weakly ,convex graph for set-valued mappings is introduced and studied. By using the concept , some new coincidence, the bestapproximation and fixed point-theorems are obtained.
文摘Some new coincidence theorem s involving a new class of set_valued mappings containing composites of acyclic mappings defined in a contractible space are proved. For applications, some best approximation theorems and coincidence theorems for set-valued mappings are als o given. A number of known results in recent literature are improved and general ized by the theorems in this paper.
基金supported by Universidad Nacional de Río Cuarto and Conicet
文摘In this paper, we introduce a condition weaker than the LP differentiability, which we call Cp condition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation at that point. We also give a necessary and sufficient condition for that a function be LP differentiable. In addition, we study the convexity of the set of cluster points of the net of best appoximations of f, {Pε(f)} asε→0.
文摘In this paper, a new class of triangular summation operators based on the equidistant nodes was constructed. It is proved that this class of operators converges uniformly to arbitrary continuous fimctions with the period 2π on the whole axis, Fttrthermore, the best approximation order and the highest convergence order are obtained. In contrast to certain operators constructed by Bernstein and Kis in the previous works, the convergence properties of the new operator constructed in this paper are superior.
文摘In numerical analysis, it is significant to approximate the linear functional Ef=sum from i=0 to m-1([integral from a to b(a<sub>1</sub>(x)f<sup>1</sup>(x)dx+ sum from f=0 to i<sub>1</sub>(b<sub>1</sub>f<sup>1</sup>(x<sub>1</sub>))]) by a simpler linear functional Lf=sum from i=1 to m(a<sub>1</sub>f(x<sub>1</sub>)) In this paper, making use of natural Tchebysheff spline function, we give existence theorem and uniqueness theorem of L that is exact for the degree m to F; we also give three sufficient and necessary conditions in which L is the Sard best approximation to F.
文摘We propose a class of iteration methods searching the best approximately generalized polynomial, which has parallel computational function and converges to the exact solution quadratically. We first transform it into a special system of nonlinear equations with constraint, then by using to certain iteration method, we combine the two basic processes of the Remes method into a whole such that the iterative process of the system of nonlinear equations and the computation of the solution to the system of linear equations proceed alternately. A lot of numerical examples show that this method not only has good convergence property but also always converges to the exact solution of the problem accurately and rapidly for almost all initial approximations .
基金supported by Consejo Nacional de Investigaciones Científicas y Técnicas(CONICET)and Universidad Nacional de San Luis(UNSL)with grants PIP 11220110100033CO and PROICO 317902
文摘In an abstract set up, we get strong type inequalities in L^p+1 by assuming weak or extra-weak inequalities in Orlicz spaces. For some classes of functions, the number p is related to Simonenko indices. We apply the results to get strong inequal- ities for maximal functions associated to best Ф-approximation operators in an Orlicz space L^Ф.
基金supported by the National Natural Science Foundation of China(Grant No.50379046)the Doctoral Fund of the Ministry of Education of China(Grant No.A50221)
文摘In order to address the complex uncertainties caused by interfacing between the fuzziness and randomness of the safety problem for embankment engineering projects, and to evaluate the safety of embankment engineering projects more scientifically and reasonably, this study presents the fuzzy logic modeling of the stochastic finite element method (SFEM) based on the harmonious finite element (HFE) technique using a first-order approximation theorem. Fuzzy mathematical models of safety repertories were introduced into the SFEM to analyze the stability of embankments and foundations in order to describe the fuzzy failure procedure for the random safety performance function. The fuzzy models were developed with membership functions with half depressed gamma distribution, half depressed normal distribution, and half depressed echelon distribution. The fuzzy stochastic mathematical algorithm was used to comprehensively study the local failure mechanism of the main embankment section near Jingnan in the Yangtze River in terms of numerical analysis for the probability integration of reliability on the random field affected by three fuzzy factors. The result shows that the middle region of the embankment is the principal zone of concentrated failure due to local fractures. There is also some local shear failure on the embankment crust. This study provides a referential method for solving complex multi-uncertainty problems in engineering safety analysis.
文摘This paper is part II of "On Copositive Approximation in Spaces of Contin- uous Functions". In this paper, the author shows that if Q is any compact subset of real numbers, and M is any finite dimensional strict Chebyshev subspace of C (Q), then for any admissible function f ∈ C(Q)/M, the best copositive approximation to f from M is unique.
文摘We pressent new Ky Fan type best approximation theorems for a discontinuous multivalued map on metrizable topological vector spaces and hyperconvex spaces. In addition, fixed point results are derived for the map studied. Our work generalizes severl results in approximation theory.
文摘The relationship between the order of approximation by neural network based on scattered threshold value nodes and the neurons involved in a single hidden layer is investigated. The results obtained show that the degree of approximation by the periodic neural network with one hidden layer and scattered threshold value nodes is increased with the increase of the number of neurons hid in hidden layer and the smoothness of excitation function.
基金supported by the National Natural Science Foundation of China,Grant 10571134
文摘In this paper we investigate several solution algorithms for the convex fea- sibility problem(CFP)and the best approximation problem(BAP)respectively.The algorithms analyzed are already known before,but by adequately reformulating the CFP or the BAP we naturally deduce the general projection method for the CFP from well-known steepest decent method for unconstrained optimization and we also give a natural strategy of updating weight parameters.In the linear case we show the connec- tion of the two projection algorithms for the CFP and the BAP respectively.In addition, we establish the convergence of a method for the BAP under milder assumptions in the linear case.We also show by examples a Bauschke's conjecture is only partially correct.
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.
文摘In this paper, Remes algorithm is applied to compute the numerical solution of the best chebyshev approximation from varisolvent family. Feasibility and convergence of the algorithm are discussed carefully.
文摘In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.
基金This paper is a part of the author's series of letures at the Mathematical Institute of the Hungarian Academy of Sciences while visiting Hungary sent by the state Education Committee,the People's Republic of China.
文摘A Bernstein type theorem and a converse theorem of best approximation by polynomials in Bergman spaces Hq^p(p>0,q>1) are proved.Some proofs and results in [1] are in proved.
文摘An existence result on Ky Fan type best approximation is proved. For this pur- pose, a class of factorizable multifunctions and the other one being a demicontinuous, rela- tive almost quasi-convex, onto function on an approximately weakly compact, convex sub- set of Hausdorff locally convex topological vector space are used. As consequence, this result extends the best approximation results of Basha and Veeramani[8] and many others.