Relative widths of smooth functions determined by fractional order derivatives

For two subsets W and V of a normed space X  . The relative Kolmogorov nn-width of W relative to V in X is defined byKn(W,V)X≔infLnsupf∈Winfg∈V∩Ln∥f-g∥X,where the infimum is taken over all nn-dimensional subspaces LnLn of X  . For α∈R+α∈R+, define Wpα(1⩽p⩽∞) to be the collection of 2π2π-periodic and continuous functions f representable as a convolutionf(t)=c+(Bα*g)(t),f(t)=c+(Bα*g)(t),where g∈Lp(T),T=[0,2π]g∈Lp(T),T=[0,2π], ∥g∥p⩽1∥g∥p⩽1, ∫Tg(x)dx=0∫Tg(x)dx=0, and Bα(t)∈L1(T)Bα(t)∈L1(T) with the Fourier expanded formBα(t)=12π∑k∈Z⧹{0}(ik)-αeikt.In this article, we discuss the relative Kolmogorov nn-width of Wpα relative to Wpα in the space Lq(T)Lq(T). For the case p=∞,1⩽q⩽∞p=∞,1⩽q⩽∞, and the case p=1,1⩽q⩽2,α>1-1q and the case p=1,23-1q, we obtain their weak asymptotic results. In addition, we also obtain the weak asymptotic result of Wpα relative to Wpα in the space Lp(T)Lp(T) for 0<α⩽20<α⩽2.