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

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.

The Jackson Inequality for the Best L2-Approximation of Functions on [0, 1] with the Weight x

Let L2(0, 1], x) be the space of the real valued, measurable, square summable functions on 0,1] with weight χ, and let N, be the subspace of L2(0, 1], X) defined by a linear combination of Jo(μkχ), where J0 is the Bessel function of order 0 and {μk}is the strictly increasing sequence of all positive zeros of J0. For f∈L2(0,1],x), let E(f, N) be the error of the best L2(0, 1],x), I.e., approximation off by elements of N. The shift operator off at point x∈0,1] with step t∈0, 1] is defined by () The differences () of order r∈(0, ∞) and the L2(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 T0(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 τ, such that the inequality E(f, N) ≤ N(τ, r)ωr(f, τ) is valid.