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 L2（[0, 1], x） defined by a linear combination of Jo（μkX）, where J...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 L2（[0, 1], x） defined by a linear combination of Jo（μkX）, where Jo is the Bessel function of order 0 and {μk} is the strictly increasing sequence of all positive zeros of Jo. For f ∈ L^2（[0, 1], x）, let E（f, n） be the error of the best L2（[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^π f（√x^2 ＋t^2-2xtcosO）dθ The differences （I- T（t））^r/2f = ∑j=0^∞（-1）^j（j^r/2）T^j（t）f of order r ∈ （0, ∞） and the L^2（[0, 1],x）- modulus of continuity ωr（f,τ） = sup{｜｜（I- T（t））^r/2f｜｜：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） and ωr（f, τ） for some cases of r and τ. More precisely, we will find the smallest constant n（τ, r） which depends only on n, r, and % such that the inequality E（f, n）≤ n（τ, r）ωr（f, τ） is valid.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 investigate the relationship between best approximations by elements of closed convex cones and the estimation of functionals on an inner product space (X,<·,·>)in terms of the inner product on X.
In this paper, we prove that the best rational approximation of a given analytic function in Orlicz space L~*(G), where G = {|z|≤∈}, converges to the Pade approximants of the function as the measure of G approaches ...In this paper, we prove that the best rational approximation of a given analytic function in Orlicz space L~*(G), where G = {|z|≤∈}, converges to the Pade approximants of the function as the measure of G approaches zero.展开更多
Let f be a function, continuous and real valued on the segment △,△ (-∞,∞) and {Rn} be the sequence of the rational functions of best uniform approximation to fon △ of order (n,n). In the present work, the converg...Let f be a function, continuous and real valued on the segment △,△ (-∞,∞) and {Rn} be the sequence of the rational functions of best uniform approximation to fon △ of order (n,n). In the present work, the convergence of {Rn} in the complex plane is considered for the special caseswhen the poles (or the zeros, respectively) of {Rn} accumulate in the terms of weak convergence of measures to acompact set of zera capacity.As a consequence, sufficient conditions for the holomorphic and the meromorphic continuability of fare given.展开更多
Using reproducing kernels for Hilbert spaces, we give best approximation for Weierstrass transform associated with spherical mean operator. Also, estimates of extremal functions are checked.
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
We consider Jackson inequality in L^2 （B^d×T, Wκ,μ^B （x））, where the weight function Wκ,μ^B （X） is defined on the ball B^d and related to reflection group, and obtain the sharp Jackson inequalityEn-1,m-...We consider Jackson inequality in L^2 （B^d×T, Wκ,μ^B （x））, where the weight function Wκ,μ^B （X） is defined on the ball B^d and related to reflection group, and obtain the sharp Jackson inequalityEn-1,m-1（f）2≤κn,m（τ,r）ωr（f,t）2,τ≥2τn,λ,where Tn,λ is the first positive zero of the Gegenbauer cosine polynomial Cn^λ （cos θ）（n ∈ N）.展开更多
In this paper, we continue studying the so-called non-linear best m-term one-sided approximation problems and obtain the asymptotic estimations of non-linear best m-term one-sided trigonometric approximation under the...In this paper, we continue studying the so-called non-linear best m-term one-sided approximation problems and obtain the asymptotic estimations of non-linear best m-term one-sided trigonometric approximation under the norm Lp （1 ≤ p ≤ ∞） of multiplier function classes and the corresponding m-term Greedy-liked one-sided trigonometric approximation results.展开更多
Let be a hypercube in Rn. We prove theorems concerning mean-values of harmonic and polyharmonic functions on In(r), which can be considered as natural analogues of the famous Gauss surface and volume mean-value formul...Let be a hypercube in Rn. We prove theorems concerning mean-values of harmonic and polyharmonic functions on In(r), which can be considered as natural analogues of the famous Gauss surface and volume mean-value formulas for harmonic functions on the ball in and their extensions for polyharmonic functions. We also discuss an application of these formulas—the problem of best canonical one-sided L1-approximation by harmonic functions on In(r).展开更多
An inverse theorem for the best weighted polynomial approximation of a function in L<sub>w<sub>α</sub></sub><sup>p</sup>（S） is established. We also investigate Besov spaces gener...An inverse theorem for the best weighted polynomial approximation of a function in L<sub>w<sub>α</sub></sub><sup>p</sup>（S） is established. We also investigate Besov spaces generated by Freud weight and their connection with algebraic polynomial approximation in L<sub>p</sub>（R）w<sub>λ</sub>, where w<sub>α</sub> is a Jacobi-type weight on S, 0【p≤∞, S is a simplex and W<sub>λ</sub> is a Freud weight. For Ditzian-Totik K-functionals on L<sub>P</sub>（S）, 1≤P≤∞, we obtain a new equivalence expression.展开更多
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 L2（[0, 1], x） defined by a linear combination of Jo（μkX）, where Jo is the Bessel function of order 0 and {μk} is the strictly increasing sequence of all positive zeros of Jo. For f ∈ L^2（[0, 1], x）, let E（f, n） be the error of the best L2（[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^π f（√x^2 ＋t^2-2xtcosO）dθ The differences （I- T（t））^r/2f = ∑j=0^∞（-1）^j（j^r/2）T^j（t）f of order r ∈ （0, ∞） and the L^2（[0, 1],x）- modulus of continuity ωr（f,τ） = sup{｜｜（I- T（t））^r/2f｜｜：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） and ωr（f, τ） for some cases of r and τ. More precisely, we will find the smallest constant n（τ, r） which depends only on n, r, and % such that the inequality E（f, n）≤ n（τ, r）ωr（f, τ） is valid.
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 investigate the relationship between best approximations by elements of closed convex cones and the estimation of functionals on an inner product space (X,<·,·>)in terms of the inner product on X.
基金This research is suported by National Science foundation Grant.
文摘In this paper, we prove that the best rational approximation of a given analytic function in Orlicz space L~*(G), where G = {|z|≤∈}, converges to the Pade approximants of the function as the measure of G approaches zero.
基金The work is supported by Project 69 with Ministry of ScienceEducation, Bulgaria.
文摘Let f be a function, continuous and real valued on the segment △,△ (-∞,∞) and {Rn} be the sequence of the rational functions of best uniform approximation to fon △ of order (n,n). In the present work, the convergence of {Rn} in the complex plane is considered for the special caseswhen the poles (or the zeros, respectively) of {Rn} accumulate in the terms of weak convergence of measures to acompact set of zera capacity.As a consequence, sufficient conditions for the holomorphic and the meromorphic continuability of fare given.
文摘Using reproducing kernels for Hilbert spaces, we give best approximation for Weierstrass transform associated with spherical mean operator. Also, estimates of extremal functions are checked.
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.
基金supported by National Natural Science Foundation of China(11071019)Beijing Natural Science Foundation(1132001)
文摘We consider Jackson inequality in L^2 （B^d×T, Wκ,μ^B （x））, where the weight function Wκ,μ^B （X） is defined on the ball B^d and related to reflection group, and obtain the sharp Jackson inequalityEn-1,m-1（f）2≤κn,m（τ,r）ωr（f,t）2,τ≥2τn,λ,where Tn,λ is the first positive zero of the Gegenbauer cosine polynomial Cn^λ （cos θ）（n ∈ N）.
基金Supported by National Natural Science Foundation of China (Grant No. 10771016) supported by Shandong Agricultural University Youth Foundation
文摘In this paper, we continue studying the so-called non-linear best m-term one-sided approximation problems and obtain the asymptotic estimations of non-linear best m-term one-sided trigonometric approximation under the norm Lp （1 ≤ p ≤ ∞） of multiplier function classes and the corresponding m-term Greedy-liked one-sided trigonometric approximation results.
文摘Let be a hypercube in Rn. We prove theorems concerning mean-values of harmonic and polyharmonic functions on In(r), which can be considered as natural analogues of the famous Gauss surface and volume mean-value formulas for harmonic functions on the ball in and their extensions for polyharmonic functions. We also discuss an application of these formulas—the problem of best canonical one-sided L1-approximation by harmonic functions on In(r).
文摘An inverse theorem for the best weighted polynomial approximation of a function in L<sub>w<sub>α</sub></sub><sup>p</sup>（S） is established. We also investigate Besov spaces generated by Freud weight and their connection with algebraic polynomial approximation in L<sub>p</sub>（R）w<sub>λ</sub>, where w<sub>α</sub> is a Jacobi-type weight on S, 0【p≤∞, S is a simplex and W<sub>λ</sub> is a Freud weight. For Ditzian-Totik K-functionals on L<sub>P</sub>（S）, 1≤P≤∞, we obtain a new equivalence expression.
