Approximation Classes for Adaptive Methods

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Primer of Adaptive Finite Element Methods (2011, methodology): “We then have the following characterization of As in terms of the standard approximation classes [7, 8, 52]:”
Theory of adaptive finite element methods: An introduction (2009, methodology): “It is important to realize that the nonlinear interaction between data A and U must be accounted for, thereby leading to a new concept of approximation class As, which generalizes those in [13, 14, 69].”
Optimality of a Standard Adaptive Finite Element Method (2007, background): “A combination of the (near) characterization of As in terms of Besov spaces from [5], and Besov regularity theorems from [7,6], indicate that under very mild conditions the value of s is indeed only restricted by the polynomial order.”
Stochastic integration in quasi-Banach spaces (2006, unknown): “ce rate of wavelet or ﬁnite element approximation methods is determined by the regularity α>0 of the target function f: O → Rin the scale Bα τ,τ(O), 1 τ = α d + 1 p , α>0, (1.1) of Besov spaces [8, 37, 38, 28]; here, O ⊆ Rd is a Lipschitz domain and p>1 describes the Lp(O)-norm that is used”
Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method (2008, unknown): “We have the following characterization of As in terms of standard approximation classes [3, 4, 23]:”
Adaptive Finite Element Methods with convergence rates (2004, background): “The proofs of Theorems 9.1, 9.3 can be found in [ 3 ].”