Applied Mathematics Letters 21 (2008) 383–387
www.elsevier.com/locate/aml
Uniqueness of simultaneous approximations in continuous
function spacesI
Lihui Peng a , Chong Li b,∗
a Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China
b Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Received 26 October 2006; received in revised form 25 March 2007; accepted 21 May 2007
Abstract
The present work is concerned with the uniqueness problem of best simultaneous approximation. An n-dimensional l1 - or
l∞ -simultaneous unicity space is characterized in terms of Property A.
c 2007 Elsevier Ltd. All rights reserved.
Keywords: Uniqueness; Simultaneous approximation; Unicity space; Property A; Continuous function spaces
1. Introduction
Let D be a compact subset of Rd satisfying D = intD and C(D) the space of continuous real-valued functions
defined on D. Let A denote the set of all non-atomic, positive finite measures on D and µ ∈ A . Let C1 (D, µ) denote
the linear space C(D) equipped with the norm k · k1 , which is defined by
Z
| f (x)|dµ for each f ∈ C1 (D, µ).
k f k1 =
D
m
Let 1 ≤ q ≤ +∞ and m ∈ N
{+∞}, where N denotes the set of all positive integers. Let {λi }i=1
be a
P∪
m
m
positive-valued sequence satisfying i=1 λi = 1. For any real-valued sequence {ai }i=1 , we define

!1/q
m

X

q

λi |ai |
1 ≤ q < +∞;
k(ai )kq =
i=1


 max |ai |
q = +∞.
1≤i≤m
Note that the norm defined above k · kq depends upon the weights (λi ). Throughout the whole of this work, we use Yq
to denote the normed linear space consisting of all function sequences f := ( f 1 , f 2 , . . . , f m ) with each f i ∈ C1 (D, µ)
I Supported in part by the National Natural Science Foundation of China (Grant No. 10671175) and Program for New Century Excellent Talents
in University.
∗ Corresponding author.
E-mail addresses: penglihui790124@sohu.com (L. Peng), cli@zju.edu.cn (C. Li).
c 2007 Elsevier Ltd. All rights reserved.
0893-9659/$ - see front matter doi:10.1016/j.aml.2007.05.009
384
L. Peng, C. Li / Applied Mathematics Letters 21 (2008) 383–387
R
such that D k( f i (x))kq dµ(x) < +∞, and { f 1 , f 2 , . . . , f m } is additionally equi-continuous when q = +∞. Let Yq
be endowed with the norm k · kYq defined by
Z
kfkYq =
k( f i (x))kq dµ(x) for each f ∈ Yq .
D
Clearly, C1 (D, µ) can be isometrically viewed as a linear subspace of Yq in a natural way where, for any f ∈ C1
(D, µ), f ∈ Yq is defined by f := ( f, f, . . . , f ).
Let U be an n-dimensional subspace of C(D) and let f = ( f 1 , f 2 , . . . , f m ) ∈ Yq . Then the simultaneous
approximation problem that we consider here is that of finding an element u 0 ∈ U such that
kf − u 0 kYq ≤ kf − ukYq
for each u ∈ U.
(1.1)
Any element u 0 ∈ U satisfying (1.1) is called a best lq -simultaneous approximation to f from U . Moreover, U is
called an lq -simultaneous unicity space if each f ∈ Yq possesses a unique best lq -simultaneous approximation to f
from U .
The study of the simultaneous approximation problem has a long history; see for example [1–9] and references
therein. Such problems can be viewed as special cases of vector-valued approximation, and in the case when m < +∞
and λi = 1 for each i = 1, . . . , m, interest in this more general area was stimulated by Pinkus in [10], where he was
mainly concerned with the question of when a finite dimensional subspace is a unicity space, and he pointed out that
many questions remain unresolved. In particular, he showed in [10] that U is an l1 -simultaneous unicity space for
all µ ∈ A if and only if m is odd and U satisfies Property A; and pointed out that little seems to be known about
characterizing l∞ -simultaneous unicity space.
In the present work we will continue to carry out investigation in this direction, which covers the more general
case when m = ∞. The main result is TheoremP
2.1 which shows that U is an l1 -simultaneous unicity space for all
m
m with each ε ∈ {1, −1}, and that U is
µ ∈ A if and only if U satisfies Property A and i=1
λi εi 6= 0 for any {εi }i=1
i
an l∞ -simultaneous unicity space for all µ ∈ A if and only if U satisfies Property A.
2. Main results
Let U be an n-dimensional subspace of C(D). We begin with the notion of Property A for U . For f ∈ C(D), recall
that Z ( f ) := {x ∈ D : f (x) = 0}.
Definition 2.1. U is said to satisfy Property A if for each u ∈ U \ {0} with D \ Z (u) = ∪ri=1 Ai , where 1 ≤ r ≤ +∞
and Ai are open, disjoint and connected, and for any {i }ri=1 with each i ∈ {1, −1}, there exists v ∈ U \ {0} such that
v = 0 a.e. on Z (u) and i v ≥ 0 on Ai for each i = 1, . . . , r , where a.e. is with respect to Lebesgue measure.
The notion of Property A has been extensively studied in [10–18] etc. In particular, the following proposition due to
Kroó (cf. [11]) characterizes the unicity space in C1 (D, µ) in terms of Property A.
Proposition 2.1. U is a unicity space for C1 (D, µ) for all µ ∈ A if and only if U satisfies Property A.
The following characterization result for the best l1 -simultaneous approximation is an extension of [10, Theorem
6.1] to the general case.
Proposition 2.2. Let f = ( f 1 , . . . , f m ) ∈ Y1 and u 0 ∈ U . Then u 0 is a best l1 -simultaneous approximation to f from
m ⊆ L (D, µ) with each kh ∗ k ≤ 1 such that
U if and only if there exist {h i∗ }i=1
∞
i ∞
Z
Z
m
m
X
X
λi h i∗ (x)u(x)dµ = 0 for each u ∈ U.
(2.1)
λi sgn( f i − u 0 )(x)u(x)dµ +
i=1
D\Z ( f i −u 0 )
i=1
Z ( f i −u 0 )
Proof. The sufficient part is clear and hence we only prove the necessity part. Let B∞ denote the closed unit ball of
L ∞ (D, µ) endowed with the weak* topology. Let
m
z
}|
{
Ω = B∞ × · · · × B∞ .
(2.2)
L. Peng, C. Li / Applied Mathematics Letters 21 (2008) 383–387
385
and let Ω be endowed with the product topology. Then Ω is a compact Hausdorff space. Let g = (g1 , . . . , gm ) ∈ Y1
and define the function Φ(g) : Ω → R by
m Z
X
Φ(g)(h 1 , . . . , h m ) =
λi h i (x)gi (x)dµ for each (h 1 , . . . , h m ) ∈ Ω .
(2.3)
i=1
D
Then Φ(g) ∈ C(Ω ). In fact, without loss of generality, assume that m = ∞. Let ε > 0 and w0 = (h 01 , h 02 , . . .) ∈ Ω .
Then there exists m 0 such that
∞
X
λi kgi k1 <
i=m 0 +1
.
4
(2.4)
Set
O(ε, w0 ) = w := (h 1 , h 2 , . . .) ∈ Ω
Z
: λi (h i − h i0 )(x)gi (x)dµ <
D
ε
for each i ≤ m 0 .
2m 0
Then O(ε, w0 ) is an open subset of Ω containing w0 and, for each w := (h 1 , h 2 , . . .) ∈ O(ε, w0 ), one has that
∞ Z
X
0
|Φ(g)(w) − Φ(g)(w0 )| = λi (h i − h i )(x)gi (x)dµ
i=1 D
Z
m
X
ε
0
0
≤
λi (h i − h i )(x)gi (x)dµ +
i=1 D
2
< ε.
This shows Φ(g) ∈ C(Ω ). In particular, Φ(f) ∈ C(Ω ) and Φ(U ) ⊆ C(Ω ), where
Φ(U ) = {Φ(u) ∈ C(Ω ) : u ∈ U }.
Moreover, for each u ∈ U , one has that
kΦ(f) − Φ(u)kC =
=
=
∞ Z
X
max
(h 1 ,...,h m )∈Ω
∞
X
i=1
∞
X
i=1
Z
λi h i (x)( f i − u)(x)dµ
max
h i ∈B∞
λi h i (x)( f i − u)(x)dµ
D
D
λi k f i − uk1 dµ
i=1
= kf − ukY1 .
Therefore, u 0 ∈ U is a best l1 -simultaneous approximation to f from U if and only if Φ(u 0 ) is a best Chebyshev
approximation to Φ(f) from Φ(U ). Then [12, Theorem 1.3] is applicable
P and hence there exist k points q1 , . . . , qk
∈ Ω , where 1 ≤ k ≤ n + 1, and k non-zero numbers c1 , . . . , ck with kj=1 |c j | = 1 such that
k
X
c j (Φ(f) − Φ(u 0 ))(q j ) = kf − u 0 kY1
j=1
and
k
X
c j Φ(u)(q j ) = 0
for each u ∈ U.
(2.5)
j=1
Assume that
j
j
q j = (h 1 , . . . , h m )
for each j = 1, . . . , k
and set
h i∗ =
k
X
j=1
j
c j hi
for each i = 1, 2, . . . , m.
(2.6)
386
L. Peng, C. Li / Applied Mathematics Letters 21 (2008) 383–387
Then each kh i∗ k ≤ 1 and (2.5) can be equivalently rewritten as
m Z
m Z
X
X
∗
0
0
λi h i (x)( f i − u )(x)dµ = kf − u kY1 and
λi h i∗ (x)u(x)dµ = 0
i=1
D
i=1
for each u ∈ U.
(2.7)
D
Furthermore, it follows from (2.7) that
m Z
X
0
kf − u kY1 =
λi h i∗ (x)( f i − u 0 )(x)dµ
≤
∗
i=1 D\Z ( f i −u )
Z
m
X
i=1
=
D\Z ( f i −u ∗ )
m Z
X
i=1
D\Z ( f i −u ∗ )
λi |( f i − u 0 )(x)|dµ
λi sgn( f i − u 0 )(x)( f i − u 0 )(x)dµ
= kf − u 0 kY1 .
This implies that
h i∗ (x) = sgn( f i − u 0 )(x)
for each x ∈ D \ Z ( f i − u 0 ) and i = 1, 2, . . . , m.
Consequently, for each u ∈ U ,
m Z
m Z
X
X
0
λi sgn( f i − u )(x)u(x)dµ +
i=1
D\Z ( f i −u 0 )
and the proof is complete.
i=1
Z ( f i −u 0 )
λi h i∗ (x)u(x)dµ
=
(2.8)
m Z
X
i=1
D
λi h i∗ (x)u(x)dµ = 0
Using almost the same argument as in the proof of [10, Theorem 6.3], one has the following characterization result
for l1 -simultaneous unicity subspaces.
m
Proposition 2.3. U is an l1 -simultaneous unicity space if and only if there do not exist {h i∗ }i=1
⊆ L ∞ (D, µ) and
u 0 ∈ U \ {0} such that
(a) |h i∗ (x)| = 1 for each x ∈ D and each i = 1, 2, . . . , m;
(b) P
h i∗ |u 0 |R∈ C1 (D, µ) for each i = 1, 2, . . . , m;
m
∗
(c) i=1
D λi h i udµ = 0 for each u ∈ U.
Now we are ready to give the main result of the present work.
Theorem 2.1. Let U be an n-dimensional subspace of C(D). Then the following statements hold. P
m
λi εi 6= 0
(i) U is an l1 -simultaneous unicity space for all µ ∈ A if and only if U satisfies Property A and i=1
m
for any {εi }i=1 with each εi ∈ {1, −1}.
(ii) U is an l∞ -simultaneous unicity space for all µ ∈ A if and only if U satisfies Property A.
Proof. (i) For each µ ∈ A , assume that U is an l1 -simultaneous unicity space. Note that C1 (D, µ) is an isometric
subspace of Y1 . It follows from Proposition 2.1 that U satisfies PropertyPA.
m
m be a choice with each ε ∈ {1, −1} and suppose that
Now let {εi }i=1
i
i=1 λi εi = 0. For each i = 1, . . . , m, define
m
∗
∗
0
∞
h i ≡ εi on D and let u ∈ U . Then {h i }i=1 ⊆ L (D, µ) and the conditions (a), (b) and (c) are satisfied; hence U is
not an l1 -simultaneous unicity subspace, which leads to P
a contradiction.
m
m with each ε ∈ {1, −1} but U
λi εi 6= 0 for any {εi }i=1
Conversely, suppose that U satisfies Property A and i=1
i
m ⊆ L ∞ (D, µ)
is not an l1 -simultaneous unicity space for some µ ∈ A . Then by Proposition 2.3, there exist {h i∗ }i=1
r
0
0
and u ∈ U \ {0} such that the conditions (a)–(c) are satisfied. Assume D \ Z (u ) = ∪i=1 Ai , where Ai are open,
h ∗ |u 0 |
i
disjoint and connected and let j = 1, 2, . . . , r . Then each h i∗ (= |u
0 | ) is continuous on A j by the condition (b) and so
a constant
function
on
A
thanks
to
the
condition
(a).
Write
ε
:=
h i∗ for each i = 1, . . . , m. Then each εi ∈ {−1, 1}
j
i
Pm
Pm
and so i=1 λi εi 6= 0 by the assumption. Consequently, i=1 λi h i∗ is a non-zero constant on each A j . Let j denote
L. Peng, C. Li / Applied Mathematics Letters 21 (2008) 383–387
387
its sign. Since U satisfies Property A, there exists v ∈ U \ {0} such that v = 0 a.e. on Z (u 0 ) and j v ≥ 0 on A j for
each 1 ≤ j ≤ r . Thus
!
Z
m
m Z
X
X
∗
∗
λi h i (x)v(x)dµ =
λi h i (x) v(x)dµ > 0,
i=1
D
D
i=1
which contradicts the condition (c). The proof of (i) is complete.
(ii) Note that C1 (D, µ) is also an isometric subspace of Y∞ and so U satisfies Property A by Proposition 2.1
provided that U is an l∞ -simultaneous unicity space for all µ ∈ A .
Conversely, assume that U satisfies Property A. Suppose on the contrary that U is not an l∞ -simultaneous unicity
space for some µ ∈ A . Then there exist f = ( f 1 , f 2 , . . . , f m ) ∈ Y∞ and ±u 0 ∈ U such that ±u 0 are best
l∞ -simultaneous approximations to f from U . Define
f − (x) = inf f i (x)
i
and
f + (x) = sup f i (x)
for each x ∈ D.
(2.9)
i
It follows from the equi-continuity of { f 1 , f 2 , . . . , f m } that f − , f + ∈ C1 (D, µ). Set f = ( f + + f − )/2 and g = ( f +
− f − )/2. Then f, g ∈ C1 (D, µ). On the other hand, since
sup | f i (x) − u(x)| = max{| f − (x) − u(x)|, | f + (x) − u(x)|} = | f (x) − u(x)| + |g(x)|
i
for each x ∈ D and u ∈ U , we have that
kf − ukY∞ = k f − uk1 + kgk1
for each u ∈ U.
This yields that
k f − (±u 0 )k1 ≤ k f − uk1
for each u ∈ U.
This means that U is not a unicity space in C1 (D, µ), which is a contradiction by Proposition 2.1. Pm
m with each
Note that in the case when m < ∞ and λi = 1 for i = 1, . . . , m, i=1
εi 6= 0 for any {εi }i=1
εi ∈ {1, −1}, if and only if m is odd. Therefore, we have the following corollary which is known from [10].
Corollary 2.1. Let m < ∞ and λi = 1 for each i = 1, . . . , m. Then U is an l1 -simultaneous unicity space for all
µ ∈ A if and only if U satisfies Property A and m is odd.
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