On the Problem of Optimal Reconstruction

Kushpel, Alexander; Tozoni, Sérgio Antonio

doi:10.1007/s00041-006-6902-3

Cited by 11 publications

(11 citation statements)

References 24 publications

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“…example, in estimating the Levy means; see [5]), where (Ω , dµ) is a measurable space. Theorem 1 can be used to interchange the order of integration as follows:…”

Section: Resultsmentioning

confidence: 99%

Best approximation of functions on the ball on the weighted Sobolev space equipped with a Gaussian measure

Wang

Zhai

2010

Journal of Approximation Theory

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This paper contains two parts. In the first part, we obtain the relations between the classical and paverage Kolmogorov widths for all p, 0 < p < ∞, which is a generalization of the corresponding results of J. Creutzig given in [J. Creutzig, Relations between classical, average and probabilistic Kolmogorov widths, J. Complexity 18 (2002) 287-303]. In the second part, we investigate the best approximation of functions on the weighted Sobolev space BW r 2,µ (B d ) equipped with a centered Gaussian measure by polynomial subspaces in the L q,µ metric for 1 ≤ q < ∞, where L q,µ , 1 ≤ q < ∞, denotes the weighted L q space of functions on the unit ball B d with respect to the weight (1 − |x| 2 ) µ− 1 2 , µ ≥ 0. The asymptotic orders of the average error estimations are obtained. We find a striking fact that, in the average case setting, the polynomial subspaces are the asymptotically optimal linear subspaces in the L q,µ metric only for 1 ≤ q < 2 + 1/µ, which means that 2 + 1/µ is the critical value and is independent of dimension d.

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“…example, in estimating the Levy means; see [5]), where (Ω , dµ) is a measurable space. Theorem 1 can be used to interchange the order of integration as follows:…”

Section: Resultsmentioning

confidence: 99%

Best approximation of functions on the ball on the weighted Sobolev space equipped with a Gaussian measure

Wang

Zhai

2010

Journal of Approximation Theory

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show abstract

“…where dµ denotes the normalised invariant measure on S n−1 , the unit sphere in R n . We are interested in the [19]. For an arbitrary index set the respective result was established in [20].…”

Section: Proof Letmentioning

confidence: 99%

“…is the sequence of Rademacher functions [21], [19]. To extend our estimates to the case p = ∞ we apply Lemma 3.1 which gives a useful inequality between norms of polynomials on M d with an arbitrary spectrum.…”

Section: Introductionmentioning

confidence: 99%

Widths and entropy of sets of smooth functions on compact homogeneous manifolds

Kushpel¹,

Taş²,

Levesley³

2021

Turk J Math

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“…We employ estimates of Levy means in combination with the Bieberbach inequality and the Brunn-Minkowski theorem. Note that estimates of Levy means connected with different orthonormal systems have been obtained in [10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%