Relative average widths of Sobolev spaces in L 2(ℝ d )

Abstract

A b s t r a c t . In this paper, in order to consider the problems of relative width on R d , we proposed definitions of relative average width which combine the ideas of the relative width and the average width. We established the smallest number M which make the following equality2 ) is the Riesz potential or Bessel potential of the unit ball in L 2 (R k ) and the notations Kσ(·, ·, L2(R d )) and dσ(·, L2(R d )) denote respectively the relative average width in the sense of Kolmogorov and the average width in the sense of Kolmogorov in their given order. In 2001, Subbotin and Telyakovskii got similar results on the relative width of Kolmogorov type. We also proved that, U(W β 2 ) ∩ B(L2(R d ))L2(R d )) = dσ(U (W α 2 ), L2(R d )), where 0 < β < α.

Best restricted approximation of smooth function classes

“…Combining the ideas of Magaril-Il'yaev [16] and Konovalov [7], Liu and Xiao [14] introduced the problem to consider the quantityd…”

“…Lp is the average Kolmogorov σ-width of W in the space L p . In [14], Liu and Xiao gave some exact results on some classes of functions in L 2 (R d ) as follows.…”

Relative widths of function classes of L 2(T) defined by a linear differential operator in L q (T)

“…obtained many results in this field. And some Chinese mathematicians such as Yongping Liu, Lianhong Yang in [25], [26], [27], and Weiwei Xiao in [ 22], [23], [24] also did some work in connection with relative width.…”