Nonlinear Wavelet Approximation in the Space C(R d )

A Bernstein-type inequality associated with wavelet decomposition

“…Our results are an extension of the recent work of DeVore, Jawerth and Popov [8] and that of DeVore, Petrushev and Yu [9].…”

“…For the L ∞ approximation, a result similar to Theorem 1.1 was established in [9]. The inequality (1.5) is a direct theorem of approximation (the Jackson inequality), and the inequality (1.6) is an inverse theorem (the Bernstein inequality).…”

“…Under these assumptions, it was shown in [8] and [9] that any function f ∈ L p (IR d ) (0 < p ≤ ∞) has a representation by means of the functions φ I :…”

“…Thus Σ n is a nonlinear manifold. The main results of the papers [8] and [9] are a characterization of the order of L p approximation (0 < p ≤ ∞) by the elements of Σ n in terms of Besov spaces.…”

“…This follows by noting that ω r (f, t) is an increasing function of t and then discretizing the integral in (1.3) at the points 2 −k (see [9]). …”