Non-linear Approximations Using Sets of Finite Cardinality or Finite Pseudo-dimension

Abstract

We investigate optimal non-linear approximations of multivariate periodic functions with mixed smoothness. In particular, we study optimal approximation using sets of finite cardinality (as measured by the classical entropy number), as well as sets of finite pseudo-dimension (as measured by the non-linear widths introduced by Ratsaby and Maiorov). Approximation error is measured in the and r>0 satisfying some restrictions, we establish asymptotic orders of these quantities, as well as construct asymptotically optimal approximation algorithms. We particularly prove that for either r>1Âp and % p or r>(1Âp&1Âq) + and % min [q, 2], the asymptotic orders of these quantities for the Besov class SB r p, % are both n &r (log n) (d&1)(r+1Â2&1Â%) .

Optimal adaptive sampling recovery

“…For the proof of the following lemma see [17] and [14]. Then, it is easy to see that m −1/q · l m q = · L q ( ,μ) for some probability distribution μ.…”

“…By Lemma 2 in [14], there is a subset ⊂ B m ∞ of cardinality at most 2 m/16 such that for any x, y ∈ , x = y, we have…”

“…The quantity s n (W, ) q has been introduced in [14] in another equivalent form (with the notation ν n (W, ) q ). Let us recall it.…”

Best m-Term Approximation and Sobolev–Besov Spaces of Dominating Mixed Smoothness—the Case of Compact Embeddings

“…(ii) Under certain additional restrictions, the periodic counterpart of (2.2) has been proved by Dinh Dũng [20][21][22]. He applied a similar discretization argument to reformulate the approximation problem as one for sequence spaces.…”

“…In this connection, we also refer to Belinsky [8], Dinh Dũng [21], and Temlyakov [54] for corresponding estimates in the periodic situation, see [62,Sect. 4.6] for a detailed comparison.…”