Relative widths of smooth functions determined by linear differential operator

Let W and V be centrally symmetric sets in a normed space X. The relative Kolmogorov n-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 n  -dimensional subspaces LnLn of X  . Let Pr(t)=tσ∏j=1l(t2−tj2), tj⩾0tj⩾0, j=1,2,…,lj=1,2,…,l, l⩾1l⩾1, σ=0σ=0 or 1, r=2l+σr=2l+σ. Denote by Pr(D)Pr(D) (D=ddt) the self-conjugate differential operator induced by Pr(t)Pr(t), and by Kq(Pr)Kq(Pr) the generalized Sobolev class of 2π-periodic smooth functions defined byKq(Pr)={f∈L˜q,2π(r):‖Pr(D)f‖q⩽1}, whereL˜q,2π(r)={f∈Lq(T):f(r−1) is absolutely continuous on T=[0,2π] and f(r)∈Lq(T)}. In this paper, we consider the relative Kolmogorov n  -width of Kp(Pr)Kp(Pr) relative to Kp(Pr)Kp(Pr) in the space Lq(T)Lq(T), and obtain the asymptotic orders of relative widths Kn(K∞(Pr),K∞(Pr),Lq(T))Kn(K∞(Pr),K∞(Pr),Lq(T)) and Kn(K1(Pr),K1(Pr),Lq(T))Kn(K1(Pr),K1(Pr),Lq(T)) for 1⩽q⩽∞1⩽q⩽∞.