All the results and proofs of [1] are unchanged and hold true for the new corrected definitions given below.. Recall that the definition of sampling recovery method S B n was given by 1.
Trang 1Adv Comput Math (2012) 36:605–606
DOI 10.1007/s10444-011-9198-z
E R R A T U M
Erratum to: Optimal adaptive sampling recovery
Dinh D ˜ung
Published online: 6 October 2011
© Springer Science+Business Media, LLC 2011
Erratum to: Adv Comput Math (2011) 34:1–41
DOI 10.1007/s10444-009-9140-9
We correct the definitions of the quantities of optimal sampling recovery
e n (W) q and r n (W) q which have been introduced in [1] All the results and proofs of [1] are unchanged and hold true for the new corrected definitions given below
Recall that the definition of sampling recovery method S B
n was given by (1.4)
in [1] The worst case error of the recovery by SB
n for the function class W , is
measured by
sup
f ∈W f − S B
n ( f) q
Given a familyB of subsets in L q, we consider optimal sampling recoveries by
B from Bin terms of the quantity
R n (W, B) q := inf
B ∈B infS B n
sup
f ∈W f − S B
n ( f) q (1)
We assume a restriction on the sets B∈B, requiring that they should have, in
some sense, a finite capacity The capacity of B is measured by its cardinality or
pseudo-dimension This reasonable restriction would provide nontrivial lower
bounds of asymptotic order of R n (W, B) q for well known function classes W.
Communicated by Charles Micchelli.
The online version of the original article can be found under doi: 10.1007/s10444-009-9140-9
D D ˜ung (B)
Information Technology Institute, Vietnam National University,
Hanoi 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
e-mail: dinhdung@vnu.edu.vn
Trang 2606 D D ˜ung
Denote R n (W, B) q by e n (W) qifBin (1) is the family of all subsets B in Lqsuch that|B| ≤ 2 n, where|B| denotes the cardinality of B, and by r n (W) qifBin (1)
is the family of all subsets B in L q of pseudo-dimension at most n.
Let = {ϕ k}k ∈J be a family of elements in L q Denote by n () the
non-linear set of non-linear combinations of n free terms from , that is n () := { ϕ =
n
j=1a j ϕ k j : k j ∈ J } The quantity s n (W, ) q which has been introduced in [1,2] (denoted byν n (W, ) qin [2]), can be equivalently redefined as
s n (W, ) q := inf
S B
n : B= n () supf ∈W f − S B
n ( f) q
A different definition of s n (W, ) q is as follows For each function f ∈ W,
we choose a sequence{x s}n
s=1 of n points inId This choice defines n sampled
values{ f(x s )} n
s=1 Then we choose a sequence a = {a s}n
s=1 of n numbers and a sequence{ϕ k s}n
s=1of n functions from , depending on the sampling
informa-tion of{x s}n
s=1and{ f(x s )} n
s=1 This choice defines a sampling recovery method given by
A n ( f) :=
n
s=1
Notice that the set of all sampling recovery methods A n coincides with the set
of all S B
n such that B = n () Therefore, there holds true the equality
s n (W, ) q= inf
A n sup
f ∈W f − A
where the infimum is taken over all sampling recovery methods A n of the form (2) Hence, we can take (3) as an alternative definition of sn (W, ) q
It is easy to verify that for the above corrected and new definitions, there
hold true the inequalities e n (W) q ≥ ε n (W) q , r n (W) q ≥ ρ n (W) q and s n (W, ) q≥
σ n (W, ) q which were used in [1, 2] for the proofs the lower bounds of
e n (U α
p,θ ) q , r n (U α
p,θ ) q and s n (U α
p,θ , M) q
References
1 D ˜ung, D.: Optimal adaptive sampling recovery Adv Comput Math 34, 1–41 (2011)
2 D ˜ung, D.: Non-linear sampling recovery based on quasi-interpolant wavelet representations.
Adv Comput Math 30, 375–401 (2009)