Average Widths of Sobolev Classes on Rn Original Research Article

In this paper, we introduce the concept of φ-average dimension far some subspaces of L[formula](Rn) and define the corresponding Kolmogorov φ-average ν-width of a set in L[formula](Rn). For the Sobolev class Wr[formula](Rn) in L[formula](Rn) we find necessary and sufficient conditions for this quantity to be finite and determine its asymptotic behaviour as ν → ∞. We also obtain the exact value of the average ν-widths of some classes of functions in L2(Rn).