Entropy Numbers In Probabilistic Setting And Average Setting

In this paper, we introduce the conception of the entropy numbers in probabilistic setting and average setting. Letδ∈(0,1] be an arbitrary number. The corresponding entropy number in probabilistic setting and average setting, are defined bywhere G runs through all possible subset in B with measureμ(G )≤δ, and M (?) X, log|M|:=log₂|M|≤n.,|M| denote cardinality of M .Next, we determine the exact order of the entropy numbers in probabilistic setting and average setting of the finite-dimensional space R^m equipped with the standard Gaussian measure in l_q^m-metric, 1≤q≤2. Moreover, we also calculate the entropy numbers in probabilistic setting and average setting of multivariate Sobolev space with mixed derivative MW₂^r (T^d),r = ( r,...,r_d), 1 /2< r1 =L =rv