由分数次导数所确定的L2(T)中的一元周期函数类在Lq(T)中的相对宽度

作者研究了相对宽度K_n(W₂^α(T), MW₂^β(T), L₂(T)), T=0,2π], 确定了使等式K_n(W₂^α(T), MW₂^β(T), L₂(T))=d_n(W₂^α(T), L₂(T))成立的最小M值, 得到了相对宽度K_n(W₂^α(T), W₂^α(T), L_q(T))的渐近阶, 其中α≥β>0, 1≤q≤∞, K_n(., ., L_q(T)) 和 d_n(., L_q(T))分别表示Kolmogorov意义下L_q(T)尺度下的相对宽度和宽度, MW_p^α(T), 1≤ p≤∞, 表示有如下卷积表达式的2π 周期函数类, f(t)=c+(B_α* g)(t),c∈ R, B_α*g 表示 B_α 和g 的卷积, g∈L_p(T) 满足∫₀^2πg(τ)dτ=0 和||g||_p≤M, B_α∈ L₁(T) 有如下Fourier展开: B_α(t)=1/2π∑' _{k∈ Z}(ik)^-αe^ikt,∑^'表示去掉 k=0的项.

Relative Widths of Function Classes of L2(T) Determined by Fractional Order Derivatives in Lq(T)

The relative widths K_n(W₂^α(T), MW₂^β(T), L₂(T)),  T=0, 2π], is studied and the smallest number M which makes the equality K_n(W₂^α(T), MW₂^β(T), L₂(T))=d_n(W₂^α(T), L₂(T)) valid is obtained, and the asymptotic order of relative widths K_n(W₂^α(T), W₂^α(T), L_q(T) ) is obtained, where α≥β>0, 1≤q ≤ ∞ , K_n (., ., L_q(T)) and d_n(., L_q(T)) denote respectively the relative widths and the widths in the sense of Kolmogorov in L_q(T), and MW_p^α(T), 1≤ p ≤ ∞ , denotes the collection of 2π-periodic and continuous functions f representable as a convolution f(t)=c+(B_α* g)(t), where B_α* g denotes the convolution of B_α and g, for g ∈ L_p(T) satisfying ∫₀^2πg(τ)dτ=0 and ||g||_p ≤ M. Here B_α is in L₁(T) with the Fourier expansion B_α(t) )=1/2π ∑' _{k∈ Z}(ik)^-αe^ikt, where ∑^' means that  the term is omitted  when k=0.