The Jackson Inequality for the Best L^2-Approximation of Functions on [0,1] with the Weight x  

作　　者：Jian Li Yongping Liu 

机构地区：[1]School of Mathematical Sciences, Beijing Normal University, Beijing 100875,China

出　　处：《Numerical Mathematics(Theory,Methods and Applications)》2008年第3期340-356,共17页高等学校计算数学学报（英文版）

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).

摘　　要：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.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.

关 键 词：Jackson inequality modulus of continuity best approximation Bessel function. 

分 类 号：O17[理学—数学]