DSpace at VNU: Optimal adaptive sampling recovery

Adv Comput Math (2011) 34:1–41
DOI 10.1007/s10444-009-9140-9

Optimal adaptive sampling recovery
˜
Dinh Dung

Received: 29 April 2009 / Accepted: 25 August 2009 /
Published online: 16 September 2009
© Springer Science + Business Media, LLC 2009

Abstract 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 ⊂ Lq , 0 < q ≤ ∞, be a class of
functions on Id := [0, 1]d . For B a subset in Lq , 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 SnB by functions
in B. An efficient sampling recovery method should be adaptive to f . Given a
family B of subsets in Lq , we consider optimal methods of adaptive sampling
recovery of functions in W by B from B in terms of the quantity
Rn (W, B )q := inf sup inf
B∈B f ∈W SnB

f − SnB ( f ) q .

Denote Rn (W, B )q by en (W)q if B is the family of all subsets B of Lq such
that the cardinality of B does not exceed 2n , and by rn (W)q if B is the family
of all subsets B in Lq of pseudo-dimension at most n. Let 0 < p, q, θ ≤ ∞

and α satisfy one of the following conditions: (i) α > d/ p; (ii) α = d/ p, θ ≤
min(1, q), p, q < ∞. Then for the d-variable Besov class U αp,θ (defined as the
unit ball of the Besov space Bαp,θ ), there is the following asymptotic order
en U αp,θ

q

rn U αp,θ

q

n−α/d .

Communicated by Qiyu Sun.
˜ (B)
D. Dung
Information Technology Institute, Vietnam National University,
Hanoi 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
e-mail:


˜
D. Dung

2

To construct asymptotically optimal adaptive sampling recovery methods
for en (U αp,θ )q and rn (U αp,θ )q we use a quasi-interpolant wavelet representation of functions in Besov spaces associated with some equivalent discrete
quasi-norm.
Keywords Adaptive sampling recovery · Quasi-interpolant wavelet

representation · B-spline · Besov space
Mathematics Subject Classifications (2000) 41A46 · 41A05 · 41A25 · 42C40

1 Introduction
We are interested in problems of sampling recovery of functions defined on
the unit d-cube Id := [0, 1]d . Let Lq := Lq (Id ), 0 < q ≤ ∞, denote the quasinormed space of functions on Id with the usual qth integral quasi-norm · q
for 0 < q < ∞, and the normed space C(Id ) of continuous functions on Id with
the max-norm · ∞ for q = ∞. We consider sampling recoveries of functions
from a class of a certain smoothness W ⊂ Lq by functions from a subset B
in Lq . The recovery error will be measured in the norm · q . We will focus
our attention to optimal methods of adaptive sampling recovery of functions
in W by subsets B of a finite capacity which is measured by their cardinality
or pseudo-dimension. Let us first recall some well-known sampling recovery
methods.
Suppose that f is a function in W and ξ = {xk }nk=1 are n points in Id . We want
to approximately recover f from the sampled values f (x1 ), f (x2 ), ..., f (xn ). A
general sampling recovery method can be defined as
Rn ( f ) = Rn (H, ξ, f ) := H( f (x1 ), ..., f (xn )),

(1.1)

where H is a given mapping from Rn to Lq . Such a method is, in general, nonlinear. A classical formula of linear sampling recovery of the form (1.1) is
n

Ln ( f ) = Ln ( , ξ, f ) :=

f (xk )ϕk ,

(1.2)

k=1

where = {ϕk }nk=1 are given n functions on Id .
To study optimal sampling methods of recovery for f ∈ W from n their
values, we can use the quantity
gn (W)q := inf sup

H,ξ f ∈W

f − Rn (H, ξ, f ) q ,

where the infimum is taken over all sequences ξ = {xk }nk=1 and all mappings H
from Rn into Lq .
Similarly in non-linear approximations [7], in non-adaptive and adaptive
sampling recoveries of functions with a certain smoothness, it is convenient
to take functions to be recovered from the Besov class U αp,θ which is defined


Optimal adaptive sampling recovery

3

as the unit ball of the Besov space Bαp,θ , having a fractional smoothness α > 0
(the definition of this space is given in Section 2). Notice that in problems of
sampling recovery, other classes such as well-known Sobolev and LizorkinTriebel classes, etc can be considered (see [22]).
We use the notations: x+ := max(0, x) for x ∈ R; An ( f )
Bn ( f ) if
An ( f ) ≤ CBn ( f ) with C an absolute constant not depending on n and/or
f ∈ W, and An ( f ) Bn ( f ) if An ( f )
Bn ( f ) and Bn ( f )

An ( f ).
It is known the following result (see [10, 19, 21, 22, 26] and references there).
Let 0 < p, q ≤ ∞, 0 < θ ≤ ∞ and α > d/ p. Then there is a linear sampling
recovery method L∗n of the form (1.2) such that
gn U αp,θ

q

sup

f ∈U αp,θ

f − L∗n ( f )

q

n−α/d+(1/ p−1/q)+ .

(1.3)

This result says that the linear sampling recovery method L∗n is asymptotically
optimal in the sense that any sampling recovery method Rn of the form (1.1)
does not give the rate of convergence better than L∗n .
Sampling recovery methods of the form (1.1) which may be linear or
non-linear are non-adaptive, i.e., the points ξ = {xk }nk=1 at which the values
f (x1 ), ..., f (xn ) are sampled, and the recovery method Rn are the same for
all functions f ∈ W. Let us introduce a setting of adaptive sampling recovery
which will give the asymptotic order of the recovery error better than the nonadaptive sampling recovery in some cases.
Let B be a subset in Lq . We will define a sampling recovery method with
the free choice of sample points and recovering functions from B. Roughly

speaking, for each f ∈ W we choose a set of n sample points. This choice
defines a collection of n sampled values. Based on the information of 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 SnB by functions in B. Let us give a precise notion
of SnB .
Denote by I n the set of subsets ξ in Id of cardinality at most n. Let V n
be the set whose elements are collections of real numbers aξ = {a(x)}x∈ξ ,
ξ ∈ I n , a(x) ∈ R (for aξ , b η ∈ V n , we write by definition aξ = b η if and only
if ξ = η and a(x) = b (x) for any x ∈ ξ ). Let In be a mapping from W into I n
and P a mapping from V n into B. Then the pair (In , P) generates the mapping
SnB from W into B by the formula
SnB ( f ) := P { f (x)}x∈In ( f ) ,

(1.4)

which defines a general sampling recovery method with the free choice of n
sample points by functions in B.
A non-adaptive sampling recovery method of the form (1.1) is a particular
case of (1.4) for which In ( f ) ≡ ξ, f ∈ W, for some ξ = {xk }nk=1 , B = Lq and
P can be treated as a mapping H from Rn into Lq .
We want to choose a sampling recovery method SnB so that the error of
this recovery f − SnB ( f ) q is as smaller as possible. Clearly, such an efficient


˜
D. Dung

4

choice should be adaptive to f . The error of an optimal adaptive sampling
recovery method for each f ∈ W, is measured by
RnB ( f )q := inf
SnB

f − SnB ( f ) q ,

where the infimum is taken over all sampling recovery methods SnB of the form
(1.4). The worst case error for the function class W is expressed by the quantity
RnB (W)q := sup RnB ( f )q .
f ∈W

Given a family B of subsets in Lq , we consider optimal sampling recoveries by
B from B in terms of the quantity
Rn (W, B )q := inf RnB (W)q .
B∈B

(1.5)

We assume a restriction on the sets B ∈ B , requiring that they should have,
in some sense, a finite capacity. In the present paper, the capacity of B is
measured by its cardinality or pseudo-dimension. This reasonable restriction
would provide nontrivial lower bounds of asymptotic order of Rn (W, B )q for
well known function classes W. Denote Rn (W, B )q by en (W)q if B in (1.5) is
the family of all subsets B in Lq such that |B| ≤ 2n , where |B| denotes the
cardinality of B, and by rn (W)q if B in (1.5) is the family of all subsets B in Lq
of pseudo-dimension at most n.
The quantity en (W)q is related to the entropy n-width (entropy number)
εn (W)q which is the functional inverse of the classical ε-entropy introduced by
Kolmogorov and Tikhomirov [18]. The quantity rn (W)q is related to the nonlinear n-width ρn (W)q introduced recently by Ratsaby and Maiorov [24]. (See

the definition of εn (W)q and ρn (W)q in Appendix).
The pseudo-dimension of a set B of real-valued functions on a set , is
defined as follows. For a real number t, let sgn(t) be 1 for t > 0 and −1
otherwise. For x ∈ Rn , let sgn(x) = (sgn(x1 ), sgn(x2 ), ..., sgn(xn )). The pseudodimension of B is defined as the largest integer n such that there exist points
a1 , a2 , . . . , an in and b ∈ Rn such that the cardinality of the set
sgn(y) : y = f (a1 ) + b 1 , f (a2 ) + b 2 , . . . , f (an ) + b n , f ∈ B
is 2n . If n is arbitrarily large then the the pseudo-dimension of B is infinite.
Denote the pseudo-dimension of B by dimp (B). The notion of pseudodimension was introduced by Pollard [23] and later Haussler [16] as an
extension of the VC-dimension [28], suggested by Vapnik-Chervonekis for
sets of indicator functions. The pseudo-dimension and VC-dimension measure
the capacity of a set of functions and are related to its ε-entropy. They play
an important role in theory of pattern recognition and regression estimation,
empirical processes and computational learning theory. Thus, in the probably
approximately correct (PAC) learning model, if B is a set of real-valued
functions on having a finite VC or pseudo-dimension, and P is a probability
distribution on , then we can estimate any f ∈ B by some g to an arbitrary
accuracy ε and probability 1 − δ by just knowing its values at m randomly


Optimal adaptive sampling recovery

5

sample points from where m depends on ε and δ (see also [24, 25]). If B is a
n-dimensional linear manifold of real-valued functions on , then dimp (B) = n
(see [16]).
We say that p, q, θ, α satisfy Condition (1.6) if
0 < p, q, θ ≤ ∞, α < ∞, and there holds one of the following restrictions :
(i) α > d/ p;
(ii) α = d/ p, θ ≤ min(1, q), p, q < ∞.

(1.6)

The main results of the present paper are read as follows.
Theorem 1.1 Let p, q, θ, α satisfy Condition (1.6). Then for the d-variable
Besov class U αp,θ , there is the following asymptotic order
en U αp,θ

q

rn U αp,θ

q

n−α/d .

Let = {ϕk }k∈J be a family of elements in Lq . Denote by
linear set of linear combinations of n free terms from , that is
⎧
⎫
n
⎨
⎬
ϕ=
a jϕk j : k j ∈ J .
n ( ) :=
⎩
⎭

n(

) the non-

j=1

Consider adaptive sampling recovery methods of f ∈ W by functions from
n ( ) in terms of the quantity
sn (W, )q := Rn n ( ) (W)q .
The quantity sn (W, )q has been introduced in [14] in another equivalent
form (with the notation νn (W, )q ). Let us recall it. For each function f ∈ W,
we choose a sequence ξ = {xs }ns=1 of n points in Id , a sequence a = {as }ns=1 of
n functions on Rn and a sequence n = {ϕks }ns=1 of n functions from . This
choice defines a sampling recovery method given by
n

S(

n , a, ξ, f ) :=

as ( f (x1 ), ..., f (xn ))ϕks .
s=1

Then the quantity sn (W, )q can be defined by
sn (W, )q = sup inf
f ∈W

n ,a,ξ

f − S(

n , a, ξ,

f ) q,

where the infimum is taken over all sequences ξ = {xs }ns=1 of n points in Id ,
a = {as }ns=1 of n functions defined on Rn , and n = {ϕks }ns=1 of n functions
from .
The optimal adaptive sampling recovery in terms of the quantity sn (W, )q
is related to the quantity σn (W, )q of non-linear n-term approximation which
characterizes the approximation of W by functions from n ( ) (see the
definition in Appendix). The reader can find in [7, 27] surveys on various


˜
D. Dung

6

aspects of this approximation and its applications. Let us recall some results
in [15] on adaptive sampling recovery in regard to the quantity sn (W, )q .
For a given natural number r, let M be the centered B-spline of even order
2r with support [−r, r] and knots at the integer points −r, ..., 0, ..., r, and define
B-spline wavelets
Mk,s (x) := M(2k x − s),
for a non-negative integer k and s ∈ Z. Then M is the set of all Mk,s which do
not vanish identically on I. The following result was proven in [15].
Let 1 ≤ p, q ≤ ∞, 0 < θ ≤ ∞, and 1 < α < min(2r, 2r − 1 + 1/ p). Then for
the the univariate Besov class U αp,θ , there is the following asymptotic order
sn U αp,θ , M

q

n−α .

(1.7)

To construct an asymptotically optimal adaptive sampling recovery method
for sn (U αp,θ , M)q which gives the upper bound of (1.7) we used the following
quasi-interpolant wavelet representation of functions in the Besov space Bαp,θ
in terms of the B-spline wavelet system M associated with some equivalent
discrete quasi-norm. If 1 ≤ p ≤ ∞, 0 < θ ≤ ∞, and 1 < α < min(2r, 2r − 1 +
1/ p), then a function f in the Besov space Bαp,θ can be represented as a series
∞

f =

ck,s ( f )Mk,s

(1.8)

k=0 s∈J(k)

with the convergence in Bαp,θ , where J(k) is the set of s for which Mk,s do not
vanish identically on I, and ck,s ( f ) are functions of a finite number of values of
f which does not depend on neither k, s nor f . Moreover, the quasi-norm of
Bαp,θ is equivalent to the discrete quasi-norm
⎞1/ p ⎫θ ⎞1/θ
⎪
⎬
⎜

⎟
(α−1/ p)k ⎝
p⎠
|ck,s ( f )|
2
⎠ .
⎝
⎪
⎪
⎩
⎭
k=0
s∈J(k)
⎛

∞

⎧
⎪
⎨

⎛

(1.9)

Such a representation can be constructed by using a quasi-interpolant. See [15]
for details.
An asymptotically optimal non-linear sampling recovery method was constructed as the sum of a linear quasi-interpolant operator at a lower B-spline
dyadic scale of this representation, and non-linear adaptive operator which is
the sum of greedy algorithms at some higher dyadic scales of B-spline wavelets.

In the present paper, we also extend (1.7) to the case 0 < p, q ≤ ∞ and
α ≥ d/ p, and generalize it for multivariate functions on the d-cube Id . In
particular, important is the case 0 < p < 1 or 0 < q < 1 which are of great
interest in non-linear approximations (see [7, 9]). To get d-variable B-spline
wavelets we let
d

M(x) :=

M(xi ), x = (x1 , x2 , ..., xd ),
i=1


Optimal adaptive sampling recovery

7

and
Mk,s (x) := M(2k x − s),
for a non-negative integer k and s ∈ Zd . Denote again by M the set of all Mk,s
which do not vanish identically on Id . We prove the following theorem.
Theorem 1.2 Let p, q, θ, α satisfy Condition (1.6) and α < min(2r, 2r − 1 +
1/ p). Then for the d-variable Besov class U αp,θ , there is the following asymptotic
order
sn U αp,θ , M

q

n−α/d .

We will prove a multivariate generalization of the quasi-interpolant wavelet
representation (1.8–1.9) which plays an important role in the proofs of
Theorems 1.1 and 1.2 (see Theorem 2.2). The methods and techniques used
[15] for the proof of the representation (1.8–1.9) cannot be applied to this
generalization. Thus, to prove it we should overcome some difficulties by
employing a different technique. On the basic of this representation we
construct asymptotically optimal sampling recovery methods which give the
upper bound for en (U αp,θ )q , rn (U αp,θ )q and sn (U αp,θ , M)q . Their lower bounds are
established by the lower estimating of the smaller related quantities εn (U αp,θ )q ,
ρn (U αp,θ )q and σn (U αp,θ , M)q , respectively.
Notice that the quantities en (W)q and rn (W)q are absolute in the sense of
optimal sampling recovery methods, while the quantity sn (W, )q depends on
a system . However, Theorems 1.1 and 1.2 show that en (U αp,θ )q , rn (U αp,θ )q
and sn (U αp,θ , M)q (with the restriction α < min(2r, 2r − 1 + 1/ p)) have the same
asymptotic order.
For 0 < p < q ≤ ∞, the asymptotic order of optimal adaptive sampling recovery method in terms of the quantities en (U αp,θ )q , rn (U αp,θ )q and sn (U αp,θ , M)q
is better than the asymptotic order of any non-adaptive sampling recovery
method of the form (1.1).
It is known that the inequalities α ≥ d/ p and α > d/ p are a condition for
the embedding and the compact embedding of the Besov space Bαp,θ and other
function spaces of smoothness α into C(Id ), respectively. The previous papers
on sampling recovery considered only the case α > d/ p. In Theorems 1.1 and
1.2, we receive some results also for the case α = d/ p, θ ≤ min(1, q), 0 < p,
q < ∞ of the Besov class U αp,θ .
In the present paper, we consider optimal adaptive sampling recoveries for
the Besov class of multivariate functions. Results similar to Theorems 1.1 and
1.2 are also true for the Sobolev and Lizorkin-Triebel classes of multivariate
functions.
The paper is organized as follows.
In Section 2, we give a definition of quasi-interpolant form functions on

Id , construct a quasi-interpolant wavelet representation in terms of the Bspline dictionary M for Besov spaces and prove some quasi-norm equivalences
based on this representation, in particular, a discrete quasi-norm in terms of


˜
D. Dung

8

the coefficient functionals. In Sections 3 and 4, we prove Theorem 1.1. In
Section 3, we prove the asymptotic order of rn (U αp,θ )q in Theorem 1.1 and of
sn (U αp,θ , M)q in Theorem 1.2 and construct asymptotically optimal adaptive
sampling recovery methods which give the upper bound for rn (U αp,θ )q and
sn (U αp,θ , M)q . In Section 4, we prove the asymptotic order of en (U αp,θ )q in
Theorem 1.1 and construct asymptotically optimal adaptive sampling recovery
methods which give the upper bound for en (U αp,θ )q . In Appendix in Section 5,
we give some auxiliary notions and results on non-linear approximations which
are employed, in particular, in establishing the lower bounds in Theorems 1.1
and 1.2.

2 Quasi-interpolant wavelet representations in Besov spaces
Let
= {λ( j)} j∈Pd (μ) be a finite even sequence, i.e., λ(− j) = λ( j), where
Pd (μ) := { j ∈ Zd : | ji | ≤ μ, i = 1, 2, ..., d}. We define the linear operator Q
for functions f on Rd by
Q( f, x) :=

( f, s)M(x − s),

(2.1)

λ( j) f (s − j).

(2.2)

s∈Zd

where
( f, s) :=
j∈Pd (μ)

The operator Q is bounded in C(Rd ) and
Q( f )

C(Rd )

≤

f

C(Rd )

for each f ∈ C(Rd ), where
|λ( j)|.

=
j∈Pd (μ)

Moreover, Q is local in the following sense. There is a positive number
δ > 0 such that for any f ∈ C(Rd ) and x ∈ Rd , Q( f, x) depends only on the

value f (y) at a finite number of points y with |yi − xi | ≤ δ, i = 1, 2, ...d. We
d
will require Q to reproduce the space P2r−1
of polynomials of order at most
2r − 1 in each variable xi , that is,
d
Q( p) = p, p ∈ P2r−1
.
d
, is called a quasiAn operator Q of the form (2.1–2.2) reproducing P2r−1
d
interpolant in C(R ).
There are many ways to construct quasi-interpolants. A method of construction via Neumann series was suggested by Chui and Diamond [4] (see also [3,
p. 100–109]). De Bore and Fix [5] introduced another quasi-interpolant based
on the values of derivatives. The reader can see also the books [3, 6] for surveys
on quasi-interpolants.


Optimal adaptive sampling recovery

9

Let
= [a, b ]d be a d-cube in Rd . Denote by L p ( ) the quasi-normed
space of functions on
with the usual pth integral quasi-norm · p, for
0 < p < ∞, and the normed space C( ) of continuous functions on
with
the max-norm · ∞, for p = ∞.
If τ be a number such that 0 < τ ≤ min( p, 1), then for any sequence of

functions { fk } there is the inequality
τ

fk

≤

p,

fk

τ
p,

.

(2.3)

We will introduce Besov spaces of smooth functions and give necessary
knowledge of them. The reader can read this and more details about Besov
spaces in the books [1, 8, 20].
Let
ωl ( f, t) p := sup

|h|
l
h

f

p,Id (lh)

be the lth modulus of smoothness of f where Id (lh) := {x : x, x + lh ∈ Id }, and
the lth difference lh f is defined by
l
l
h

f (x) :=

l
f (x + jh).
j

(−1)l− j
j=0

For 0 < p, θ ≤ ∞ and 0 < α < l, the Besov space Bαp,θ is the set of functions
f ∈ L p for which the Besov quasi-semi-norm | f | Bαp,θ is finite. The Besov quasisemi-norm | f | Bαp,θ is given by
⎧
1/θ
∞
⎪
⎨
{t−α ωl ( f, t) p }θ dt/t
, θ < ∞,
| f | Bαp,θ :=
0
⎪

⎩sup
−α
θ = ∞.
t>0 t ωl ( f, t) p ,
The Besov quasi-norm is defined by
B( f ) =

f

Bαp,θ

:=

f

p

+ | f | Bαp,θ .

We will assume that continuous functions to be recovered are from the
Besov space Bαp,θ with the restriction on the smoothness α ≥ 1/ p which is a
condition for the embedding of this space into C(Id ).
If { fk }∞
k=0 is a sequence whose component functions fk are in L p , for
β
0 < p, θ ≤ ∞ and β ≥ 0 we use the b θ (L p ) “quasi-norms”
∞

{ fk }

β
b θ (L p )

:=

1/θ

2βk fk

p

θ

k=0

with the usual change to a supremum when θ = ∞. When { fk }∞
k=0 is a positive
sequence, we replace fk p by | fk | and denote the corresponding quasi-norm


˜
D. Dung

10

by { fk } b β . We will need the following discrete Hardy inequality. Let {ak }∞
k=0
θ
and {ck }∞
k=0 be two positive sequences and let for some M > 0, τ > 0

1/τ

∞

|ck | ≤ M

|ak |

τ

.

(2.4)

k=m

Then for β > 0, θ > 0,
{ck }

β

bθ

≤ CM {ak }

(2.5)

β

bθ

with C = C(β, θ) (see, e.g, [8]).
For the Besov space Bαp,θ , there is the following quasi-norm equivalence
B( f )

ωl f, 2−k

B1 ( f ) :=

p

b αθ

+

f

p.

If Q of is a quasi-interpolant of the form (2.1–2.2), for h > 0 and a function f
on Rd , we define the operator Qh by
Q( f ; h) = σh ◦ Q ◦ σ1/ h ( f ),
where σh ( f, x) = f (x/ h). By definition it is easy to see that
( f, k; h)M(h−1 x − k),

Q( f, x; h) =
k

where
λ( j) f (h(k − j)).

( f, k; h) :=
j∈Pd (μ)

The operator Q(·; h) has the same properties as Q: it is a local bounded
d
. Moreover, it
linear operator in Rd and reproduces the polynomials from P2r−1
gives a good approximation of smooth functions [6, p. 63–65]. We will also call
it a quasi-interpolant for C(Rd ).
The quasi-interpolant Q(·; h) is not defined for a function f on Id , and
therefore, not appropriate for an approximate sampling recovery of f from
its sampled values at points in Id . An approach to construct a quasi-interpolant
for a function on Id is to extend it by interpolation Lagrange polynomials. This
approach has been proposed in [15] for the univariate case. Let us recall it.
For a non-negative integer m, we put x j = j2−m , j ∈ Z. If f is a function
on I, let
2r−1

U m ( f, x) := f (x0 ) +

2sm

s
2−m

f (x0 )

s!

s=1
2r−1

Vm ( f, x) := f (x2m −2r+1 ) +
s=1

2sm

s−1

(x − x j ),
j=0

s
2−m

f (x2m −2r+1 )
s!

s−1

(x − x2m −2r+1+ j)
j=0

(2.6)
be the (2r − 1)th Lagrange polynomials interpolating f at the 2r left end
points x0 , x1 , ..., x2r−1 , and 2r right end points x2m −2r+1 , x2m −2r+3 , ..., x2m , of the


Optimal adaptive sampling recovery

11

interval I, respectively. The function f¯ is defined as an extension of f on R by
the formula
⎧
⎪
⎨U m ( f, x), x < 0
(2.7)
f¯(x) :=
f (x),
0≤x≤1
⎪
⎩
Vm ( f, x), x > 1.
Obviously, if f is continuous on I, then f¯ is a continuous function on R. Let Q
be a quasi-interpolant of the form (2.1–2.2) in C(R). We introduce the operator
Qm by putting
Qm ( f, x) = Q( f¯, x; 2−m ), x ∈ I,
for a function f on I. We have
Qm ( f, x) =

am,s ( f )Mm,s (x), ∀x ∈ I,

(2.8)

s∈J(m)

where J(m) := {s ∈ Z : −r < s < 2m + r} is the set of s for which Mm,s do not
vanish identically on I, and

am,s ( f ) :=

( f¯, s; 2−m ) =

λ( j) f¯(2−m (s − j)).

(2.9)

| j|≤μ

The operator Qm is called a quasi-interpolant for C(I).
We now give a multivariate generalization of the univariate quasiinterpolant Qm . For this purpose we rewrite the coefficient functionals am,s ( f )
for the definition of Qm , in a more suitable form. Let b be a function of
discrete variable k ∈ Z (m) where Z (m) := {s ∈ Z : 0 ≤ s ≤ 2m }. For nonnegative integer l, put Z (m, l) := {s ∈ Z : −l < s < 2m + l}. We extend b to
the function Ext(b ) on Z (m, r + μ) by the formula
⎧2r−1
s−1
s
⎪
b (0)
⎪
⎪
(k − j),
−r − μ < k < 0
⎪
⎪
⎪
s! j=0
⎪
s=0

⎨
Ext(b , k) := b (k),
0 ≤ k ≤ 2m
⎪
⎪
s−1
2r−1
⎪
s
⎪
b (2m − 2r + 1)
⎪
⎪
(2m − 2r + 1 − j), 2m < k < 2m + r + μ,
⎪
⎩
s!
s=0
j=0
(2.10)
where the sth difference

s

b (k) is defined by
s

s

b (k) :=

(−1)s− j
j=0

s
b (k + j).
j

A function f on I defines a function b f on Z (m) by b f (k) := f (2−m k). From
(2.6), (2.7) and (2.10) it is easily to see that
Ext(b f , k) := f¯(2−m k),


˜
D. Dung

12

and consequently, we can rewrite the coefficient functionals am,s ( f ) given in
(2.9), as follows
am,s ( f ) =

λ jExt(b f , s − j).

(2.11)

| j|≤μ

Let b be a function of d discrete variables k ∈ Z d (m) where Z d (m) :=
{k ∈ Zd , 0 ≤ ki ≤ 2m , i = 0, 1, ..., d}. For non-negative integer l, put

Z d (m, l) := {s ∈ Zd : −l < si < 2m + l, i = 0, 1, ..., d}. We extend b to the
function Ext(b ) on Z d (m, r + μ) by applying the tensor product of d such onedimensional extensions:
d

Ext(b ) :=

Exti (b ),
i=0

where the univariate extension Exti is applied to the univariate function b by
considering b as a function of variable ki with the other variables held fixed.
A function f on Id defines a function b f on Z d (m) by b f (k) := f (2−m k).
Let Q be a quasi-interpolant of the form (2.1–2.2) in C(Rd ). On the basic of
the formulas (2.8) and (2.11), we introduce the multivariate operator Qm by
Qm ( f, x) :=

am,s ( f )Mm,s (x),

∀x ∈ Id ,

(2.12)

s∈J(m)

where J(m) := Z d (m, r) = {s ∈ Zd : −r < si < 2m + r, i = 0, 1, ..., d} is the
set of s for which Mm,s do not vanish identically on Id , and
am,s ( f ) :=

λ( j)Ext(b f , s − j).

(2.13)

am,s ( f ) = am,s1 ((am,s2 (...am,sd ( f ))),

(2.14)

j∈Pd (μ)

Notice that

where the univariate functional am,si is applied to the univariate function f by
considering f as a function of variable xi with the other variables held fixed.
Moreover, the number of the terms in Qm ( f ) is of the size ≈ 2dm .
Similar to the quasi-interpolants Q and Q(·; h), the operator Qm is a local
d
bounded linear mapping in C(Id ) and reproducing P2r−1
. In particular,
Qm ( f )

C(Id )

≤C

f

C(Id )

(2.15)

for each f ∈ C(I ), with a constant C not depending on m, and,

d

d
,
Qm ( p∗ ) = p, p ∈ P2r−1
∗

(2.16)

where p is the restriction of p on I . The multivariate Qm is called a quasiinterpolant in C(Id ).
We will use the letter I to denote a dyadic cube of the form 2−m ( j + Id )
for some nonnegative integer m and j ∈ Zd . Its size is t I = 2−m and its volume
|I| = 2−dm . If m is indicated and I = 2−m ( j + Id ), sometimes we will also write
I = I j. For a d-cube ⊂ Rd , we use the following notations: D( , m) is the
d


Optimal adaptive sampling recovery

13

set dyadic cubes I of the size 2−m which are contained in ; J( , m) is the set
of s for which Mm,s do not vanish identically on ; ( , m) is the span of the
B-splines Mm,s , s ∈ J( , m). In these notations, will be dropped if = Id .
For each f ∈ L p , the error of the approximation of f by the B-splines from
(m) is given by
Em ( f ) p := inf

f −ϕ

ϕ∈ (m)

p.

On the basic of the quasi-interpolant of de Bore and Fix [5], DeVore and
Popov [9] constructed a linear bounded operator Pm in L p reproducing the
d
polynomials from P2r−1
, of the form
Pm ( f, x) :=

αm,s ( f )Mm,s (x),
s∈J(m)

where αm,s ( f ) are certain coefficient functionals. It was proven [9] that for
0 < p ≤ ∞ and each f ∈ L p
f − Pm ( f )

p

≤ Cω2r ( f, 2−m ) p ,

(2.17)

with a constant C depending on r, d, p only.
If 0 < p ≤ ∞, for all non-negative integers m and all functions
g=

as Mm,s

(2.18)

s∈J(m)

from

(m), there is the norm equivalence
g

p

2−dm/ p {as }

p,m ,

(2.19)

where
1/ p

{as }

p,m

:=

|as | p
s∈J(m)

with the corresponding change when p = ∞.

Let
pk ( f ) := Pk ( f ) − Pk−1 ( f ) with

P−1 ( f ) = 0.

The following representation and quasi-norm equivalences were proven
in [9].
If 0 < p, θ ≤ ∞ and 0 < α < min(2r, 2r − 1 + 1/ p), then for a function f on
Id belongs to the Besov space Bαp,θ if and only if f can be represented by the
wavelet series
∞

f =

∞

pk ( f ) =
k=0

fk,s Mk,s ,

(2.20)

k=0 s∈J(k)

for some coefficient functionals fk,s , satisfying the convergence condition
B2 ( f ) :=

{ pk ( f )}

b αθ (L p )

< ∞.

(2.21)


˜
D. Dung

14

Moreover, the Besov quasi-norm B( f ) is equivalent to one of the quasi-norms
Bi ( f ), i = 2, 3, 4, where
B3 ( f ) :=

{ f − Pk ( f )}

b αθ (L p )

B4 ( f ) :=

{Ek ( f ) p }

+

b αθ

f

+

f

p,

p.

Let us recall some well-known embeddings of spaces Bαp,θ . For 0 < p, q,
θ ≤ ∞, α > 0 and α > δ := d(1/ p − 1/q)+ , there is the inequality
f

≤ C f

α−δ
Bq,θ

Bαp,θ ,

(2.22)

α−δ
and consequently, Bαp,θ is continuously embedded into Bq,θ
. Further, if
α
0 < p, θ ≤ ∞ and α ≥ d/ p, then B p,θ is continuously embedded into C(Id ),
and there is the inequality

f

≤ C f

∞

Bαp,θ .

(2.23)

This means that a function in Bαp,θ will be continuous by correcting its values
in a set of zero measure. In this sense, we will consider Bαp,θ with α ≥ d/ p, as a
subset of C(Id ). Notice also that the inequality α > d/ p provides the compact
embedding of Bαp,θ into C(Id ).
For a I = Is ∈ D(m), let I˜ = I˜s be the d-cube which is the union of
the d-cubes I j ∈ D(m), j ∈ Z s (μ + r), where Z s (l) := { j ∈ J(m) : | ji − si | ≤ l,
i = 1, 2, ..., d}.
Lemma 2.1 Let 0 < p ≤ ∞ and I ∈ D(m). Then for any continuous function f
on Id , we have
Qm ( f )
and, if in addition, f ∈

∞,I

≤

f

∞, I˜ ,

≤

f

p, I˜ .

(m), we have
Qm ( f )

p,I

Proof Let I = Is ∈ D(m) and f be a continuous function on Id . From the
inequality s∈J(I,m) Mm,s ≤ 1 we have
Qm ( f )

∞,I

= sup
x∈I

≤
=
≤

am, j( f )Mm, j(x)
j∈J(I,m)

max |am, j( f )|

j∈J(I,m)

λ( j) f¯(2−m ( j − j ))

max

j∈J(I,m)

j

∈Pd (μ)

max

max

j∈J(I,m) j ∈Pd (μ)

f¯(2−m ( j − j )) .


Optimal adaptive sampling recovery

15

Therefore,
Qm ( f )

∞,I

If in addition, f ∈
Qm ( f )

p,Is

≤

f

max

j∈Z s (μ+r)

=

∞,I j

f

∞, I˜ .

(2.24)

(m), then from (2.24) it follows that

≤ |Is |1/ p Qm ( f )

≤

∞,Is

|Is |1/ p

max

j∈Z s (μ+r)

f

∞,I j .

Since f is a polynomial on each I j, j ∈ Z s (μ + r), there is the inequality
f

≤ |I j|−1/ p f

∞,I j

p,I j .

Hence, by the equality |I j| = |Is |, j ∈ Z s (μ + r), we obtain
p
p,Is

Qm ( f )

≤

p

≤

p

max

j∈Z s (μ+r)

f

p
p,I j

f

p
p,I j

j∈Z s (μ+r)

=

p

f

p
.
p, I˜

Lemma 2.2 Let 0 < p ≤ ∞. Then for any k ≥ m and any function f ∈
we have
Qm ( f )

p

≤ C2(k−m)d/ p f

with some constant C depending on r, μ, d and
Proof Let f ∈

(k),

p

only.

(k) and k ≥ m. We have

Qm ( f )

p
p

=

|Qm ( f, x)| p dx ≤
I∈D(m)

I

|I| Qm ( f )

p
∞,I .

I∈D(m)

Applying Lemma 2.1 to each I ∈ D(m) gives
Qm ( f )

p
p

≤

|I| f

p
I∈D(m)

p
∞, I˜

(2.25)
≤

p

dp −dm

(2μ + 2r) 2

f
I∈D(m)

Since f is a polynomial on each interval I ∈ D(I, k), we get
f

∞,I

≤ |I |−1/ p f

p,I

,

p
∞,I .


˜
D. Dung

16

and therefore,
f

p
∞,I

=

max

I ∈D(I,k)

f

p
∞,I

≤ 2dk

f

≤ 2dk max

I ∈D(I,k)

p
p,I

= 2dk f

f

p
p,I

p
p,I .

I ∈D(I,k)

Hence and from (2.25) it follows that
Qm ( f )

p
p

≤

p

(2μ + 2r)dp 2d(k−m)

f

p
p,I

I∈D(Id ,m)

= C p 2d(k−m) f

p
p

with the constant C depending on r, μ, d and

only.

Let ⊂ Id is a d-cube. We will need the following modified modulus of
smoothness
wl ( f, t, ) p :=

t−d

where
(lh) := {x : x, x + lh ∈
wl ( f, t, Id ) p . It is known that

1/ p
[−t,t]d

(lh)

|

l
h(

f, x)| p dx dh

,

}. As above we will write: wl ( f, t) p :=

C1 wl ( f, t) p ≤ ωl ( f, t) p ≤ C2 wl ( f, t) p
with constants C1 , C2 which depend on l, p, d only (see [9]).
Lemma 2.3 Let 0 < p ≤ ∞ and f ∈ L p . Then we have

Pm ( f ) − Qm (Pm ( f ))

p

≤ Cω2r ( f, 2−m ) p

with some constant C depending on r, μ, p, d and

only.

Proof Below we will denote by C j a constant depending at most on r, μ, p, d
and
. In order to prove this lemma we will use Lemma 2.1 and a technique
in the proof of Theorem 4.5 in [9]. For a d-cube , we define the quantity of
approximation of a function f on by polynomials as follows
e2r ( f ) p, := inf f − ϕ
ϕ

p,

,

d
where the inf is taken over all polynomials ϕ from P2r−1
. For f ∈ L p ( ), there
holds the inequality

e2r ( f ) p,
where t is the size of

(see [9]).

≤ C1 w2r ( f, t , ) p ,

(2.26)


Optimal adaptive sampling recovery

17

Let Is ∈ D(m), and g be the polynomial of best L p ( I˜s ) approximation by
to Pm ( f ), i.e., Pm ( f ) − g p, I˜s = e2r (Pm ( f )) p, I˜s . Then, we have

d
P2r−1

Pm (g) = Qm (g) = g.
Hence, we obtain
Pm ( f ) − Qm (Pm ( f ))
Since Pm ( f ) − g ∈

≤ C2 { Pm ( f ) − g

p,Is

p,Is

+ Qm (Pm ( f ) − g)

p,Is }.

(m), by Lemma 2.1 we get

Qm (Pm ( f ) − g)

≤ C3 Pm ( f ) − g

p,Is

p, I˜s .

Applying (2.17) gives
Pm ( f ) − Qm (Pm ( f ))

p,Is

≤ C4 Pm ( f ) − g

p,Is

≤ C5 e2r (Pm ( f )) p, I˜s .
d
. There holds the inequality
Let ϕ be any polynomial in P2r−1

≤ C6

Pm ( f ) − f

p
p, I˜s

+ f −ϕ

e2r (Pm ( f )) p, I˜ ≤ C7

Pm ( f ) − f

p
p, I˜s

+ e2r ( f ) p, I˜ .

Pm ( f ) − ϕ

p
p, I˜s

p
p, I˜s

.

Hence,
p

s

p

s

For I j ∈ D(m), let I¯ j be the cube which is the union of the cubes I j , j ∈ Z j.
Clearly, I¯ j ⊂ I˜ j. It was proven in [9] that
f − Pm ( f )

p,I j

≤ C8 e2r ( f ) p, I¯j

(2.27)

for any I j ∈ D(m). By (2.27) we have
Pm ( f ) − f

p
p, I˜s

=

Pm ( f ) − f

p
p,I j

j∈Z s (μ+r)
p

≤ C9

j∈Z s (μ+r)

e2r ( f ) p, I˜ .
j

Therefore,
Pm ( f ) − Qm (Pm ( f ))

p
p,Is

p

≤ C10
j∈Z s (μ+r)

e2r ( f ) p, I˜ .
j


˜
D. Dung

18

Using the last estimation and (2.26) we derive that
Pm ( f ) − Qm (Pm ( f ))

p
p

=

Pm ( f ) − Qm (Pm ( f ))

p
p,Is

s
p

≤ C11
j∈Z s (μ+r)

s

e2r ( f ) p, I˜

w2r ( f, t I˜ j , I˜ j ) pp

≤ C12
j∈Z s (μ+r)

s

≤ C13 t−d
= C14 t−d

I j (2rh)

≤ C16 t−d

I˜ j (2rh)

|

2r
h (

f, x)| p dx dh

d
j∈Z s (μ+r) [−t,t] j ∈Z j (μ+r)

s

≤ C15 t−d

[−t,t]d

j∈Z s (μ+r)

s

×

j

|

2r
h (

f, x)| p dx dh

[−t,t]d

Is (2rh)

s

[−t,t]d

Id (2rh)

|

|

2r
h (

2r
h (

f, x)| p dx dh

f, x)| p dx dh

= C17 w2r ( f, 2−m ) pp

≤ C p ω2r ( f, 2−m ) pp ,
where t := max j∈Z s (μ+r) t I˜ j ≤ C18 2−m .
Lemma 2.4 Let 0 < p ≤ ∞ and θ ≤ min( p, 1). Then for any f ∈ L p , there
holds the inequality
f − Qm ( f )

p

≤ C2−dm/ p

∞

1/θ
θ

2dk/ p ω2r ( f, 2−k ) p

(2.28)

k=m

with some constant C depending at most on r, μ, p and
in the right-hand side is finite.

, whenever the sum

Proof Let f ∈ L p be a function such that the sum in the right-hand side of
(2.28) is finite. We have by (2.3)
f − Qm ( f )

θ
p

≤

f − Pm ( f )

θ
p

+ Pm ( f ) − Qm (Pm ( f ))
+ Qm ( f − Pm ( f ))

θ
p.

θ
p

(2.29)


Optimal adaptive sampling recovery

19

We obtain by (2.17)
f − Pm ( f )

p

≤ Cω2r ( f, 2−m ) p ,

(2.30)

and by Lemma 2.3
Pm ( f ) − Qm (Pm ( f ))
Further, since pk ( f ) ∈

p

≤ Cω2r ( f, 2−m ) p .

(2.31)

(k) by (2.3) and Lemma 2.2 we derive that

Qm ( f − Pm ( f ))

θ
p

∞

≤

Qm ( pk ( f ))

θ
p

k=m+1
∞

2(k−m)d/ p ( pk ( f ))

≤C

θ

p

.

(2.32)

k=m+1

Again, by (2.3) and (2.17) we get
( pk ( f ))

θ
p

≤

f − Pk ( f )

θ
p

+ f − Pk−1 ( f )

θ
p

≤ Cω2r ( f, 2−k )θp .

(2.33)

Combining (2.29–2.33) proves the lemma.
The following corollary is immediately implied from the last lemma.
Corollary 2.1 Let 0 < p, q, θ ≤ ∞, 0 < α = d/ p < 2r, θ ≤ min(1, q). Then for
any f ∈ Bαp,θ , we have
f − Qm ( f )

q

≤ C2−(α−d(1/ p−1/q)+ )m f

Bαp,θ

.

with some constant C depending at most on d, r, p and
Let
qk ( f ) := Qk ( f ) − Qk−1 ( f )

Q−1 ( f ) := 0.

with

Theorem 2.1 Let 0 < p, θ ≤ ∞, d/ p < α < min(2r, 2r − 1 + 1/ p). Then for
the Besov space Bαp,θ , the following quasi-norms are equivalent to B( f ):
B5 ( f ) :=

{ f − Qk ( f )}

B6 ( f ) :=

{qk ( f )}

b αθ (L p )

+

f

p,

b αθ (L p ) .

Proof Fix a number 0 < τ ≤ min( p, 1). By Lemma 2.4 we have
1/τ

∞

2

dm/ p

f − Qm ( f )

p

≤ C

2

dk/ p

−k

ω2r ( f, 2 ) p

τ

.

k=m

Put ck := 2dk/ p f − Qk ( f ) p ; ak := 2dk/ p ω2r ( f, 2−k ) p ; β := α − d/ p. Then
applying the discrete Hardy inequality (2.4–2.5) gives
B6 ( f )

B1 ( f ).

(2.34)

˜
D. Dung

20

Further, from the inequality
it follows

qk ( f )

B6 ( f )

f − Qk ( f )

p

p+

f − Qk−1 ( f )

B5 ( f ).

p,

(2.35)

On the other hand, by (2.3) we obtain
1/τ

∞

f − Qm ( f )

p

≤

qk ( f )

τ
p

,

k=m+1

and consequently, by the discrete Hardy inequality (2.4–2.5)
B5 ( f ) ≤ B6 ( f ).

(2.36)

B4 ( f ) ≤ B5 ( f ).

(2.37)

Finally, by definition

Combining (2.34–2.37) completes the proof of Theorem 2.1.
According to Theorem 2.1, a function f ∈ Bαp,θ has the decomposition
∞

f =

qk ( f )

(2.38)

k=0

with the convergence in the quasi-norm B6 . We now deduce from this decomposition a quasi-interpolant wavelet representation of a function in Bαp,θ
in terms of the B-splines Mk,s ∈ M, and an associated discrete equivalent
quasi-norm for the functional coefficients. By using the B-spline refinement
equation, one can represent the component functions qk ( f ) as
qk ( f ) =

ck,s ( f )Mk,s ,

(2.39)

s∈J(k)

where ck,s are certain coefficient functionals of f, which are defined as follows.
For the univariate case, we put
ck,s ( f ) :=

ak,s ( f ),
ak,s ( f ) − ak,s ( f ),

if 2k−1 + r ≤ s < 2k + r,
if − r < s < 2k−1 + r,

(2.40)

and
2r

ak,s ( f ) := 2

−2r+1
j=0

2r
ak−1, j+s+r ( f )
j

(see [15]). For the multivariate case, we define ck,s similarly to the formula
(2.14) for ak,s , that is
cm,s ( f ) = cm,s1 ((cm,s2 (...cm,sd ( f ))),

(2.41)

where the univariate functional cm,si is applied to the univariate function f by
considering f as a function of variable xi with the other variables held fixed.


Optimal adaptive sampling recovery

21

From Theorem 2.1, (2.38), (2.40), (2.41) and (2.18–2.19) we obtain the following theorem on quasi-interpolant wavelet representation for Besov spaces.

Theorem 2.2 Under the assumptions of Theorem 2.1, a function f on Id belongs
to the Besov space Bαp,θ if and only if f has a quasi-interpolant wavelet
representation
∞

∞

qk ( f ) =

f =
k=0

ck,s ( f )Mk,s

(2.42)

k=0 s∈J(k)

Bαp,θ ,

with the convergence in the space
and in addition the quasi-norm of the
Besov space B( f ) is equivalent to the discrete quasi-norm
∞

B7 ( f ) :=

2(α−d/ p)k {ck,s ( f )}

p,k

θ

1/θ

.

k=0

with the usual change to a supremum when θ = ∞.
Remark From (2.12), (2.13), (2.40), (2.41) we can see that for each pair k, s the
coefficient ck,s ( f ) in the decomposition (2.42) is a linear combination of the
values f (2−k (s − j)), and f (2−k+1 (s − j)), j ∈ Pd (μ), s ∈ Z s (r). The number
of these values does not exceed the fixed number (2μ + 2r)d and, therefore, not
depend on neither functions f and nor k, s. This property is very important
in the constructions of asymptotically optimal adaptive recovery methods in
Sections 3 and 4.
From the proof of Theorem 2.1 it follows that for 0 < p, θ ≤ ∞, d/ p <
α < 2r, there holds the inequality
B7 ( f )

B( f )

(2.43)

for any f ∈ Bαp,θ .
Let us consider sampling recovery methods by using a quasi-interpolant
Qm . For a function f on Id , the formula (2.12–2.13) of Qm ( f ) defines a nonadaptive linear sampling recovery method with a nice approximation and local
properties. The function Qm ( f ) approximately recovers f from its sampled
values at the uniform dyadic points 2−m j, j ∈ Z d (m). The number of sampled

values is |Z d (m)| = (2m + 1)d . Further, the number of the B-splines Mm,s in
Qm ( f ) is |J(m)| = (2m + 2r − 1)d . As mentioned above, for each pair (m, s)
the coefficient am,s ( f ) is a linear combination of the values f (2−m (s − j)), j ∈
Pd (μ), and maybe, f (2−m j) for ji = 0, 1, ..., 2r − 1 or ji = 2m − 2r + 2, 2m −
2r + 3, ..., 2m , i = 1, 2, ..., d if the point 2−m s is near to the boundary of the dcube Id , respectively. Moreover, the number of these values does not exceed
the (2μ + 2r)d and not depend on neither functions f and nor m.
For each point x ∈ Id , we have
Qm ( f, x) =

am,s ( f )Mm,s (x).
|si −2m xi |<

r, i=1,2,...,d


˜
D. Dung

22

Hence, the points 2−m j, j ∈ Z d (m), at which the sampled values are taken in
the linear sampling method (2.12) for recovering f (x), are in the neighborhood
of x
U(x) := {y ∈ Id : |yi − xi | < 2−m (μ + r), i = 1, 2, ..., d}
whose size does not depend on x, and is 2−m multiplied by an absolute
constant. The number of these points does not exceed (2μ + 2r)d . From
(2.15–2.16) one can easily derive that if f is continuous on Id , then Qm ( f, x)
converges uniformly for x ∈ Id when m → ∞.
Consider a non-adaptive sampling recovery method of the form
n

Gn ( , ξ, a, f ) :=

ak ( f (x1 ), ..., f (xn ))ϕk ,

(2.44)

k=1

where a = {ak }nk=1 is a given sequence of n functions on Rn , and = {ϕk }nk=1
is a given sequence of n functions on Id . This is a particular case of (1.1). In
order to study optimal (non-linear) methods of the form (2.44) for recovering
f ∈ W, we introduce the quantity
γn (W)q := inf sup

,ξ,a f ∈W

f − Gn ( , ξ, a, f ) q ,

where the infimum is taken over all triples ( , ξ, a) with ξ = {xk }nk=1 , a =
{ak }nk=1 and = {ϕk }nk=1 .
As shown in the following two corollaries, the sampling recovery method
Qm∗ ( f ) with an appropriate choice of m∗ = m∗ (n) is asymptotically optimal for
gn (U αp,θ )q , γn (U αp,θ )q with 0 < p, q ≤ ∞, and sn (U αp,θ , M)q with 0 < q ≤ p ≤ ∞
(see Theorem 1.2).
Corollary 2.2 Let p, q, θ, α satisfy Condition (1.6) and α < 2r. Then there is the
inequality
sup

f ∈U αp,θ

f − Qm ( f )

q

2−(α−d(1/ p−1/q)+ )m .

(2.45)

If in addition, m∗ is the largest integer of m such that (2m + 2μ − 1)d ≤ n,
then Qm∗ is a linear sampling recovery method of the form (2.44), and we have
gn U αp,θ

q

γn (U αp,θ )q

sup

f ∈U αp,θ

f − Qm∗ ( f )

q

n−α/d+(1/ p−1/q)+ . (2.46)

Proof For 0 < p, q ≤ ∞ and α > d(1/ p − 1/q)+ , by (2.22) we have the inα−d(1/ p−1/q)+
clusion U αp,θ ⊂ CU q,θ
with a multiplier C. If in addition, m∗ is the

largest integer of m such that (2m + 2μ − 1)d ≤ n, then the number of sampling values and and the number of the B-spines Mm∗ ,s in Qm∗ ( f ) do not
exceed n and, consequently, Qm∗ is a linear sampling recovery method of


Optimal adaptive sampling recovery

23

the form (2.44). Hence, the inequality (2.45) and the upper bounds of (2.46)
follows Theorem 2.2 and (2.43) for the case (i) in Condition (1.6), and from
Corollary 2.1 for the case (ii) in Condition (1.6). The lower bound of (2.46)
already is in (1.3) for the case (i) in Condition (1.6). The lower bound of (2.46)
for the case (ii) in Condition (1.6) can be proven in a way similar to the proof
of the lower bound of Theorem 23 in [22].
Corollary 2.3 Under the assumptions of Corollary 2.2, let q ≤ p. If m∗ is the
∗
largest integer of m such that (2m + 2μ − 1)d ≤ n, then Qm∗ = Sn (m ) is a linear
sampling recovery method of the form (1.4) with (m∗ ) ⊂ n (M), and we have
sn (U αp,θ , M)q

sup

f ∈U αp,θ

f − Qm∗ ( f )

q

n−α/d .

3 Adaptive sampling recovery by sets of finite pseudo-dimension
We will need a concept of partitions of the unite d-cube Id into dyadic
coordinate d-cubes introduced by Birman and Solomyak [2] for adaptive
approximations by piecewise polynomials. Let be a partition of Id into half
open coordinate d-cubes. Denote by | | the number of d-cubes in . If a
partition
is obtained from by dividing some d-cubes in into 2d d-cubes
with the same size, then
is called an elementary extension of . The family
m consists of all partitions with | | ≤ m, which can be obtained from the
trivial partition ∗1 := {Id } by a finite number of elementary extensions. Note
that each partition ∈ m consists of m dyadic d-cubes with various sizes.
Denote by ∗2dk the uniform partition of Id into 2dk d-cubes with the size 2−k .
Let Q( ) be the set of all piecewise polynomial functions f on Id such that f
is a polynomial of order at most 2r − 1 on any d-cube of the partition , and
Qm the union of Q( ),
∈ m.
¯ k∗ be non-negative integers with k¯ ≤ k∗ , and {nk }k∗
Lemma 3.1 Let k,
a
¯
k=k+1
∗
¯
sequence of non-negative integers with nk ≤ |J(k)|, , k = k + 1, ..., k . Let B be
the set of all functions f of the form
k∗

nk

as Mk,s +

f =
¯
s∈J(k)

ck,s j Mk,s j

(3.1)

¯
j=1
k=k+1

with s j ∈ J(k). Then B ⊂ Qm , where
¯

k∗

m = 2dk + (2r)d
¯
k=k+1

¯
nk (k − k).

(3.2)


˜

D. Dung

24

Proof Let f ∈ B be any function of the form (3.1). Put
⎧
dν
¯
⎪
ν = 0, 1, ..., k,
⎪
⎨2 ,
mν =

ν

¯

2dk + (2r)d
⎪
⎪
⎩

¯
nk (k − k),
nk

fk¯ =

as Mk,s ;

¯
s∈J(k)

ν = k¯ + 1, ..., k∗ ,

¯
k=k+1

fk =

ck,s j Mk,s j ,

k = k¯ + 1, ..., k∗ ,

j=1

¯ Obviously, { m }k¯ is a sequence of partitions
and ∗2di = mi , i = 0, 1, ..., k.
i i=0
d
such that m0 = {I } and each mi is an elementary extension of mi−1 for
¯ Moreover, f ¯ ∈ Q( m ¯ ).
i = 1, ..., k.
k
k
Let I(k, s j ) be the support of the B-spline Mk,s j . Consider all the dcubes I(k, s j ) for j = 1, 2, ..., nk , k = k¯ + 1, ..., k∗ . For each k, there are nk
d-cubes I(k, s j ). Each I(k, s j ) can be divided into (2r)d half open dyadic
d-cubes Il (k, s j ), l ∈ (r), of the size 2−k , where Il (k, s j ) := 2−k (l + [0, 1)d ),
and (r) := {l ∈ Zd : −r ≤ li < r, i = 1, 2, ..., d}. Thus, the half open dyadic
¯

d-cubes Il (k, s j ) of the size 2−k < 2−k , j = 1, 2, ..., nk , l ∈ (r), k = k¯ +
∗
nk half open dyadic d-cubes
1, ..., k∗ , are contained in at most (2r)d kk=k+1
¯
¯

of the size 2−k . Hence, it is easy to derive that there is a partition mk+1
which
¯
is an elementary extension of mk¯ such that each Il (k, s j ) is contained in a
¯
dyadic d-cube of the size 2−(k+1) of mk+1
for l ∈ (r), j = 1, 2, ..., nk , k = k¯ +
¯
∗
¯ k¯ + 1,
1, ..., k , and in particular, Il (k, s j ), j = 1, 2, ..., nk+1
(r), k = k,
¯ ,l ∈
. Because the functions fk¯ and fk+1
are a linear combination of
are in mk+1
¯
¯
the B-splines Mk,s
,
j
=
1, 2, ..., nk+1

¯ j , j = 1, 2, ...nk¯ , and Mk+1,s
¯
¯ , respectively,
j
they and therefore, their sum fk¯ + fk+1
belong to Q( mk+1
).
¯
¯
ν
Suppose that for ν > k¯ + 1 there is a sequence of partitions { mi }i=0
such
¯
k+1
that it is an extension of the sequence { mi }i=0 and for for each i = 0, ..., ν:

(i)
mi is an elementary extension of
mi−1 ;
(ii) each Il (k, s j ) is contained in a dyadic d-cube of the size 2−i of mi for
l ∈ (r), j = 1, 2, ..., nk , k = i, ..., k∗ ;
¯ k¯ +
(iii) each Il (k, s j ) belongs to mi for l ∈ (r), j = 1, 2, ..., ni , k = k,
1, ..., i;
(iv) the sum fk¯ + fk+1
+ ... + fi belongs to Q( mi ) for i = k¯ + 1, ..., ν.
¯
The half open dyadic d-cubes Il (k, s j ), l ∈ (r), j = 1, 2, ..., nk , k = ν +
∗
1, ..., k∗ , are contained in at most (2r)d kk=ν+1 nk half open dyadic d-cubes of

the size 2−ν . Consequently, there is a partition mν+1 which is an elementary
ν+1
extension of mν such that the sequence { mi }i=0
has the properties (i)–(iv)
(with replacing ν by ν + 1).
k∗
of partitions
By induction we have proven that there is a sequence { mi }i=0
d
such that m0 = {I } and mi is an elementary extension of mi−1 , and the sum


Optimal adaptive sampling recovery

25

f = fk¯ + fk+1
+ ... + fk∗ belongs to Q(
¯
f ∈ Qm , and therefore, B ⊂ Qm .

mk∗ )

with mk∗ = m. This means that

Theorem 3.1 Let p, q, θ, α satisfy Condition (1.6). Then for the d-variable
Besov class U αp,θ , there is the following asymptotic order
rn (U αp,θ )q

n−α/d .

(3.3)

If in addition, α < 2r, we can explicitly construct a subset B in n (M) having
dimp (B) ≤ n, and a sampling recovery method SnB of the form (1.4), such that
sup

f ∈U αp,θ

f − SnB ( f )

q

n−α/d .

(3.4)

Proof The lower bound of (3.3) follows from the inequality rn (U αp,θ )q ≥
ρn (U αp,θ )q and Theorem 5.3 in the Appendix. Concerning the upper bound
of (3.3), we notice that since the pseudo-dimension of a function set B of
cardinality ≤ 2n is not greater than n, there holds the inequality
rn (W)q ≤ en (W)q .
Hence, the upper bound of rn (U αp,θ )q is implied from the upper bound of
en (U αp,θ )q which will be proved in Section 4, and an asymptotically optimal
sampling recovery method for en (U αp,θ )q is also asymptotically optimal for
rn (U αp,θ )q . However, we will prove the upper bound of rn (U αp,θ )q construct an
asymptotically optimal sampling recovery method for rn (U αp,θ )q in a different
way which is much simpler than those for en (U αp,θ )q . We will see, in particular,
that for p < q an asymptotically optimal adaptive sampling recovery method
is constructed as the sum of a linear quasi-interpolant operator at a lower

B-spline dyadic scale of the representation (2.42), and non-linear adaptive
operator which is the sum of greedy algorithms at some higher dyadic scales of
B-spline wavelets.
Let us prove (3.4) and therefore, the upper bound of (3.3). From embedding
theorems (see, e.g., [1]) we can see that the space Bαp,θ can be considered as a
subspace of the largest space Bαp,∞ . Hence, it is sufficient to prove (3.4) for
U := U αp,∞ . We will need the following inequality (see the proof of Theorem
2 in [25]). There is a constant C = C(r, d) such that for any non-negative
integer m,
dimp (Qm ) ≤ Cm.

(3.5)

We first consider the case p ≥ q. For any integer k¯ by Corollary 2.2 we have
for the non-adaptive sampling method Qk¯ ( f )
sup f − Qk¯ ( f )
f ∈U

q

¯

2−αk .

(3.6)