Continuous Algorithms in n-Term Approximation and Non-Linear Widths

Dũng, Đinh

doi:10.1006/jath.1999.3399

Cited by 31 publications

(37 citation statements)

References 11 publications

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“…The proof of this assertion is similar to the proof of Lemma 4 in [4]. We can also use the method in the proof of Theorem 2.2 from [15] to prove the lower estimate = n (SB r p, % , L 1 )> >n &r (log n) (d&1)(r+1Â2&1Â%) , which immediately yields the lower bound of = n (SB r p, % , L q ) and = n (SW r p , L q ).…”

Section: Lower Bound For Non-linear Widthsmentioning

confidence: 69%

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

Dũng¹

2001

Journal of Complexity

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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%) .

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Section: Lower Bound For Non-linear Widthsmentioning

confidence: 69%

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

Dũng¹

2001

Journal of Complexity

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“…If is a family of elements in X such that σ n (W, , X) > 0, we have [12] σ n+m (W, , Y) ≤ σ n (W, , X)σ m (SX, , Y).…”

mentioning

confidence: 97%

Optimal adaptive sampling recovery

Dũng

2009

Adv Comput Math

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We propose an approach to study optimal methods of adaptive sampling recovery of functions by sets of a finite capacity which is measured by their cardinality or pseudo-dimension. Let W ⊂ L q , 0 < q ≤ ∞, be a class of functions on I d := [0, 1] d . For B a subset in L q , we define a sampling recovery method with the free choice of sample points and recovering functions from B as follows. For each f ∈ W we choose n sample points. This choice defines n sampled values. Based on these sampled values, we choose a function from B for recovering f . The choice of n sample points and a recovering function from B for each f ∈ W defines a sampling recovery method S B n by functions in B. An efficient sampling recovery method should be adaptive to f . Given a family B of subsets in L q , we consider optimal methods of adaptive sampling recovery of functions in W by B from B in terms of the quantityDenote R n (W, B) q by e n (W) q if B is the family of all subsets B of L q such that the cardinality of B does not exceed 2 n , and by r n (W) q if B is the family of all subsets B in L q of pseudo-dimension at most n. Let 0 < p, q, θ ≤ ∞ and α satisfy one of the following conditions:Then for the d-variable Besov class U α p,θ (defined as the unit ball of the Besov space B α p,θ ), there is the following asymptotic order e n U α p,θ q r n U α p,θ q n −α/d .

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“…(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.…”

Section: Remark 24mentioning

confidence: 95%