L2中的Jackson不等式与函数类的Kolmogorov n-宽度

研究了L2空间中用三角多项式逼近周期可微函数时关于S-平均连续模的Jackson不等式,得到了Jackson不等式中最小常数的精确值.涉及的函数类(Fh,φ a,r)由Lr2中满足约束条件ωa,s(f(r);h)≤φ(h)的函数f组成,其中α≥1/2,r∈N,0＜h≤π/2n,φ(t)为0,+∞)上的连续增函数且φ(0)=0.并计算了(F h,φ a,r)在L2中的Kolmogorov n-宽度的精确值,同时得到(F h,φ a,r)中函数f的Fourier系数绝对值的上确界的精确值.其中L2=L20,2π]表示以2π为周期的勒贝格平方可积实函数空间,Lr2表示由L2中r-1阶导数f(r-1)绝对连续,r阶导数f(r)∈L2的函数f组成的集合.

The Jackson inequality in L2 and the Kolmogorov n-widths of classes of functions

Let L2 = L20,2π] be the space of 2π-periodic measurable functions square-integrable on 0,2π] , L_2^τ be the set of all functions f ∈ L2 whose （r-1）th derivative 7f^（τ-1） is absolutely continuous and f^（τ）∈ L2. In this paper, the Jackson inequality with the S-average modulus of continuity for the case of approximation of differentiable periodic functions by trigonometric polynomials is investigated in the space L2 ,the exact values of the lowest constants in Jackson inequality are obtained. The class F（α,r）（h,φ） concerned in this paper is consisted of functions f ∈ L2 satisfying the constraint ωa,s（f（τ）,h）≤φ（h）,where a≥ 1 /2,r∈N,0〈h≤π/2,φ（t） is an arbitrary increasing function such that φ（0） = 0. In the present paper, the exact value of the Kolmogorov n-width ofF（α,r）（h,φ）in L2 is calculated. At the same time ,the least upper bound of the absolute value of Fourier coefficient of fin F（α,r）（h,φ）is abtained.