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 by K n ( W , V , X ) : = inf L n sup f ∈ W inf g ∈ V ∩ L n ‖ f − g ‖ X , where the infimum is taken over all n-dimensional subspaces L n of X. Let P r ( t ) = t σ ∏ j = 1 l ( t 2 − t j 2 ) , t j ⩾ 0 , j = 1 , 2 , … , l , l ⩾ 1 , σ = 0 or 1, r = 2 l + σ . Denote by P r ( D ) ( D = d d t ) the self-conjugate differential operator induced by P r ( t ) , and by K q ( P r ) the generalized Sobolev class of 2π-periodic smooth functions defined by K q ( P r ) = { f ∈ L ˜ q , 2 π ( r ) : ‖ P r ( D ) f ‖ q ⩽ 1 } , where L ˜ q , 2 π ( r ) = { f ∈ L q ( T ) : f ( r − 1 ) is absolutely continuous on T = [ 0 , 2 π ] and f ( r ) ∈ L q ( T ) } . In this paper, we consider the relative Kolmogorov n-width of K p ( P r ) relative to K p ( P r ) in the space L q ( T ) , and obtain the asymptotic orders of relative widths K n ( K ∞ ( P r ) , K ∞ ( P r ) , L q ( T ) ) and K n ( K 1 ( P r ) , K 1 ( P r ) , L q ( T ) ) for 1 ⩽ q ⩽ ∞ .