Science in China Series A: Mathematics
Oct., 2007, Vol. 50, No. 10, 1427–1440
www.scichina.com
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Limit theory of restricted range approximations
of complex-valued continuous functions
Xian-fa LUO1 & Chong LI2†
1
2
School of Mathematics and Computer Science, Jishou University, Jishou 416000, China;
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
(email: luoxianfaseu@163.com, cli@zju.edu.cn)
Abstract
This paper is concerned with the problem of the best restricted range approximations
of complex-valued continuous functions. Several properties for the approximating set PΩ such that
the classical characterization results and/or the uniqueness results of the best approximations hold are
introduced. Under the very mild conditions, we prove that these properties are equivalent that P is a
Haar subspce.
Keywords:
restricted range approximation, Haar subspace, characterization, uniqueness
MSC(2000): 41A65, 41A50
1
Introduction
Recent attention is focused on the problem of the best restricted range approximations in the
space of complex-valued continuous functions. The setting is as follows. Let Q be a compact
metric space and let C(Q) denote the Banach space of all complex-valued continuous functions
on Q endowed with the uniform norm:
f = max |f (t)|,
t∈Q
∀f ∈ C(Q).
Let {Ωt : t ∈ Q} be a family of nonempty closed convex sets in the complex plane C and let
P be a finite dimensional subspace of C(Q). Set PΩ = {p ∈ P : p(t) ∈ Ωt , ∀ t ∈ Q}. Then,
for a given f ∈ C(Q), an element p∗ of PΩ is called a best (restricted range) approximation
to f from PΩ if and only if f − p∗ = d(f, PΩ ), where d(f, PΩ ) is defined by d(f, PΩ ) =
inf p∈PΩ f − p. This problem was first introduced and formulated by Smirnov and Smirnov in
[1, 2], where the characterization theorems and the uniqueness theorems of the best restricted
range approximation were obtained under the special case when {Ωt : t ∈ Q} is a system of
closed disks with centers and radii continuously depending on t, and P is a Haar subspace.
These results were extended to some more general cases in [3–5]. In particular, this problem
was considered in [6] for a very general case when the strong interior-point condition and the
lower semicontinuity of the set-valued function t → Ωt on Q are assumed.
Received December 2, 2005; accepted June 20, 2007
DOI: 10.1007/s11425-007-0121-5
† Corresponding author
This work was partially supported by the National Natural Science Foundation of China (Grant No.10671175),
the Program for New Century Excellent Talents in University and the Scientific Research Fund of Hunan
Provincial Education Department (Grant No. 06C651)
Xian-fa LUO & CHong LI
1428
The interest of the present paper is in the converse problem, that is, characterizing the
approximating set PΩ for which the corresponding characterization and/or the uniqueness theorems hold. A similar problem in the simple case for uniqueness was also considered in [4]. In
the present paper, making use of the introduced new notion of an extremal bi-support, we define
several properties such as C, C, Ci , C i , C ∗ and C ∗ as well as U and Ui . The characterization
results for the approximating set PΩ to have these properties are provided, respectively.
The paper is motivated by the ideas in [7, 8] where similar properties were introduced and
characterized respectively for the non-restricted and restricted range approximations of realvalued continuous functions by means of the notion of an alternation. However, this problem
for the case of complex-valued continuous function approximations is more difficult than that
for the case of real-valued continuous function approximations because the characterization of
the alternation type, which is a powerful tool and plays a key role in the case of real-valued
continuous function approximations, is invalid in the case of complex-valued continuous function
approximations. In fact, the technique used in the present paper is completely different from
that in [7, 8].
2
Notions and preliminary results
In this paper, we use C to denote the complex plane and B(z0 , δ) the open ball of C with a
center z0 and a radius δ. Let Z be a closed subset of C. The interior (resp. boundary, convex
hull) of Z is denoted by intZ (resp. bdZ, coZ). The normal cone of Z at z0 is denoted by
NZ (z0 ) and defined by NZ (z0 ) = {τ ∈ C : Reτ (z − z0 ) 0, ∀z ∈ Z}. The distance from z0 to
Z is denoted by d(z0 , Z) and defined by d(z0 , Z) = inf{|z0 − z| : z ∈ Z}.
One basic assumption in the study of the restricted range approximation problem of complexvalued continuous functions is that PΩ has an interior point or a strong interior point which
are defined as follows, see [1–6] for details.
Definition 2.1.
A point p ∈ PΩ is called
(i) An interior point of PΩ if
p(t) ∈ intΩt ,
∀ t ∈ Q;
(2.1)
(ii) A strong interior point of PΩ if there exists a positive number δ such that
B(p(t), δ) ⊆ Ωt ,
∀ t ∈ Q.
(2.2)
Moreover, PΩ is said to satisfy the interior-point (resp. strong interior-point) condition if there
exists p ∈ PΩ such that (2.1) (resp. (2.2)) holds.
Clearly, a strong interior point is an interior point. Proposition 2.1 below shows that the
converse is true provided that the set-valued function Ω : t → Ωt is lower semicontinuous on Q,
which is defined as follows (cf. [9]).
Definition 2.2.
Let Ω : t → Ωt be a set-valued function defined on Q. Then Ω is said to be
(i) Lower semicontinuous at t0 ∈ Q if, for any z0 ∈ Ωt0 and any > 0, there exists an open
neighborhood U (t0 ) of t0 such that Ωt ∩ B(z0 , ) = ∅ for all t ∈ U (t0 );
(ii) Lower semicontinuous on Q if it is lower semicontinuous at each point t0 ∈ Q.
Limit theory of restricted range approximations of complex-valued continuous functions
1429
Proposition 2.1. Suppose that the set-valued function Ω : t → Ωt is lower semicontinuous
on Q. Let p ∈ PΩ . Then p is an interior point of PΩ if and only if it is a strong interior point
of PΩ .
Proof. We only need to prove the necessity part. Suppose that p is an interior point of PΩ
but not a strong interior point of PΩ . Then, for every positive integer n, there exists tn ∈ Q
such that B(p(tn ), n1 ) Ωtn . Pick zn ∈ B(p(tn ), n1 )\Ωtn . Then |zn − p(tn )| < n1 . Without
loss of generality, we may assume tn → t0 . Then p(tn ) → p(t0 ) and zn → p(t0 ) ∈ Ωt0 . By the
assumption, B(p(t0 ), δ) ⊆ Ωt0 for some δ > 0. Let 0 < < 14 δ. Then there exists a natural
number N1 such that
|p(tn ) − p(t0 )| <
1
δ
4
and |zn − p(t0 )| < ,
∀ n > N1 .
(2.3)
For each n > N1 , define
xn = p(tn ) + λn (zn − p(tn ))
(2.4)
yn = p(tn ) + μn (zn − p(tn ))i,
√
where i = −1. Choose λn > 0 and μn > 0 such that
(2.5)
and
|xn − p(t0 )| = |yn − p(t0 )| =
3
δ.
4
(2.6)
Then, it follows from (2.3) and (2.6) that λn > 1 and μn > 1. Without loss of generality, we
may assume that xn → w1 and yn → w2 . Then by (2.6) one has that
|w1 − p(t0 )| = |w2 − p(t0 )| =
Since
yn − p(tn ) =
it follows from (2.8) that
μn
λn
3
δ.
4
μn
(xn − p(tn ))i,
λn
(2.7)
(2.8)
→ 1 and
w2 − p(t0 ) = (w1 − p(t0 ))i.
(2.9)
Set un = 2p(tn )−xn , vn = 2p(tn )−yn . Then un → w3 := 2p(t0 )−w1 and vn → w4 := 2p(t0 )−w2 .
Clearly,
w3 − p(t0 ) = −(w1 − p(t0 )),
w4 − p(t0 ) = −(w1 − p(t0 ))i,
(2.10)
where the second equality is by (2.9). Furthermore, by (2.7) and (2.10),
|w3 − p(t0 )| = |w4 − p(t0 )| =
3
δ.
4
(2.11)
Note that, by (2.7), (2.9), (2.10) and (2.11), co{wk }4k=1 is a square with the center p(t0 ) and
√
the length of each side 34 2δ. Hence, there exists > 0 such that, for any set {wk }4k=1 with
wk ∈ B(wk , ) for k = 1, . . . , 4,
B(p(t0 ), ) ⊆ co({wk }4k=1 ).
(2.12)
Xian-fa LUO & CHong LI
1430
Since B(p(t0 ), δ) ⊆ Ωt0 , it follows from (2.7) and (2.11) that
1 wk ∈ B wk , δ ⊆ Ωt0
4
∀ k = 1, . . . , 4.
As the set-valued function t → Ωt is lower semicontinuous at t0 , there exists natural a number
N2 > N1 such that for any n > N2 , B(wk , ) ∩ Ωtn = ∅ for each k = 1, . . . , 4. Take wk ∈
B(wk , ) ∩ Ωtn (∀ k = 1, . . . , 4). Then, (2.12) and the fact that Ωtn is convex,
B(p(t0 ), ) ⊆ co{wk }4k=1 ⊆ Ωtn .
(2.13)
Thanks to Combining (2.13) and (2.3) implies that zn ∈ Ωtn , which contradicts the choice of
zn . Hence p is a strong interior point of PΩ and the proof is complete.
Throughout the whole paper, we always assume that {Ωt } satisfies the following Hypotheses.
Hypothesis 1.
The set-valued function t → Ωt is lower semicontinuous on Q.
Hypothesis 2.
PΩ has an interior point p.
Let f ∈ C(Q) and p∗ ∈ PΩ . Following [3, 6], set
M (f − p∗ ) = {t ∈ Q : |f (t) − p(t)| = f − p∗ },
B(p∗ ) = {t ∈ Q : p∗ (t) ∈ bdΩt }.
Moreover, define
σ(p∗ , t) = sign (f (t) − p∗ (t)),
∀t∈Q
(2.14)
and
τ (p∗ , t) = {τ : τ ∈ −NΩt (p∗ (t)) \ 0},
∀ t ∈ Q,
(2.15)
where signz = z/|z| if z = 0 and 0 if z = 0. Note that τ (p∗ , t) = ∅ since NΩt (p∗ (t)) = {0} if
t ∈ B(p∗ ), and that, for each t ∈ B(p∗ ), τ ∈ τ (p∗ , t) if and only if τ ∈ −NΩt (p∗ (t)) \ {0}.
The following two classes of admissible functions were introduced by Smirnov and Smirnov
in [2, 4] respectively for the study of the uniqueness problem of the best restricted range
approximation of complex-valued continuous functions.
Definition 2.4. A function f ∈ C(Q) is called
(i) Admissible of type I if
f (t) ∈ Ωt , ∀ t ∈ Q
(2.16)
or there exists a best approximation p∗ to f from PΩ such that
M (f − p∗ ) ∩ B(p∗ ) = ∅;
(2.17)
(ii) Admissible of type II if f ∈ PΩ or
d(f, PΩ ) > sup d(f (t), Ωt ).
t∈Q
(2.18)
The set of all admissible functions of type I (resp. II) is denoted by Ca1 (Q) (resp. Ca2 (Q)).
Below we will prove that Ca1 (Q) is contained in Ca2 (Q). For this end, we first give two lemmas.
Lemma 2.1. Let F be the function on Q defined by F (t) = d(f (t), Ωt ) for each t ∈ Q. Then
F is upper semicontinuous on Q.
Limit theory of restricted range approximations of complex-valued continuous functions
1431
Proof. Let t0 ∈ Q. Then there exists z0 ∈ Ωt0 such that F (t0 ) = |f (t0 ) − z0 |. Let > 0.
Since f is continuous at t0 , it follows from Hypothesis 1 that there exists an open neighborhood
U (t0 ) of t0 such that Ωt ∩ B(z0 , 2 ) = ∅ and |f (t) − f (t0 )| < 2 for all t ∈ U (t0 ). Thus, for each
t ∈ U (t0 ) and zt ∈ Ωt ∩ B(z0 , 2 ), we have that
F (t) |f (t) − zt | |f (t) − f (t0 )| + |f (t0 ) − z0 | + |z0 − zt | < F (t0 ) + .
This shows that F (t) is upper semicontinuous at t0 since > 0 is arbitrary.
Lemma 2.2.
PΩ such that
Proof.
f ∈ Ca2 (Q) \ PΩ if and only if there exists a best approximation p∗ to f from
f − p∗ > d(f (t), Ωt ),
∀ t ∈ M (f − p∗ ).
(2.19)
It suffices to prove the “if” part since the proof of the “only if” part is trivial. Suppose
that there exists a best approximation p∗ to f from PΩ such that (2.19) holds. Noting that the
function t → d(f (t), Ωt ) is upper semicontinuous on Q by Lemma 2.1, one has that
d(f, PΩ ) = f − p∗ > max{d(f (t), Ωt ) : t ∈ M (f − p∗ )}
(2.20)
since M (f −p∗ ) is compact. Consequently, there exists an open set Q1 ⊆ Q with Q1 ⊇ M (f −p∗)
such that
d(f, PΩ ) > sup d(f (t), Ωt ).
(2.21)
t∈Q1
/ M (f − p∗ ). It follows that
Let Q2 = Q \ Q1 . Then, for each t ∈ Q2 , one has t ∈
f − p∗ > |f (t) − p∗ (t)| d(f (t), Ωt ),
∀ t ∈ Q2 .
Since Q2 is compact and the function t → d(f (t), Ωt ) is upper semicontinuous by Lemma 2.1,
we have that d(f, PΩ ) > maxt∈Q2 d(f (t), Ωt ). This together with (2.21) gives (2.18) and so
f ∈ Ca2 (Q) \ PΩ .
Proposition 2.2.
Ca1 (Q) ⊆ Ca2 (Q).
Proof. Let f ∈ Ca1 (Q). Then (2.16) or (2.17) holds. Without loss of generality, we may assume
that f ∈
/ PΩ . Hence d(f, PΩ ) > 0. Thus, to complete the proof, it suffices to show (2.19). Since
(2.19) is clear in the case when (2.16) holds, it remains to show (2.19) in the case when (2.17)
holds. Let t ∈ M (f − p∗ ). Then t ∈
/ B(p∗ ) by (2.17); hence p∗ (t) ∈ intΩt , which implies that
∗
|f (t)−p (t)| > d(f (t), Ωt ). Consequently, f −p∗ = |f (t)−p∗ (t)| > d(f (t), Ωt ), ∀ t ∈ M (f −p∗ ),
that is, (2.19) holds.
Remark 2.1.
In general, Ca1 (Q) = Ca2 (Q) as shown by the following
Example 2.1. Let Q = {−1, 0, 1}, Ω−1 = Ω1 = {z : Rez 1} and Ω0 = C. Let P =
span{1, t}. Then PΩ = {p : p(t) = a + bt, Re(a + b) 1, Re(a − b) 1}. Now, define f ∈ C(Q)
by f (−1) = 1/2, f (0) = 0 and f (1) = 2. Then d(f, PΩ ) = 1. In fact, for each p(t) = a+bt ∈ PΩ ,
we have
Re(a + b) 1, Re(a − b) 1;
(2.22)
hence Re a 1 and
f − p |f (0) − p(0)| = |a| Re a 1.
(2.23)
Xian-fa LUO & CHong LI
1432
Set p∗ = 1. Then p∗ ∈ PΩ and f −p∗ = 1. This together with (2.23) implies that d(f, PΩ ) = 1
and p∗ is a best approximation to f from PΩ . Moreover, we have that f ∈ Ca2 (Q) because
d(f, PΩ ) = 1 >
1
= sup d(f (t), Ωt ).
2 t∈Q
However f ∈
/ Ca1 (Q). In fact, note that (2.16) does not hold and M (f − p∗ ) ∩ B(p∗ ) = {1}. It
suffices to prove that p∗ is a unique best approximation to f from PΩ . For this purpose, let
p ∈ PΩ satisfy f − p = 1. Assume that p(t) = a + bt. Then a = 1 by (2.23) and so Reb = 0
by (2.22). Thus
1 = f − p |f (1) − p(1)| = |b − 1| = 1 + (Im b)2 1.
This yields Im b = 0, and hence p = p∗ = 1.
Let
M̃ (p∗ ) = {t ∈ M (f − p∗ ) ∩ B(p∗ ) : σ(p∗ , t) ∈ NΩt (p∗ (t))}.
The following proposition shows that the elements of Ca2 (Q) can be characterized by the emptyness property of the set M̃ (p∗ ).
Proposition 2.3. Let f ∈ C(Q) \ PΩ . Then the following statements are equivalent:
(i) f ∈ Ca2 (Q);
(ii) M̃ (p∗ ) = ∅ for any best approximation p∗ to f from PΩ ;
(iii) M̃ (p∗ ) = ∅ for some best approximation p∗ to f from PΩ .
Proof. (i) =⇒ (ii). Let f ∈ Ca2 (Q) \ PΩ . Then (2.18) holds. Suppose on the contrary that
M̃ (p∗ ) = ∅ for some best approximation p∗ to f from PΩ . Let t ∈ M̃ (p∗ ). Then f − p∗ =
|f (t) − p∗ (t)| and σ(p∗ , t) ∈ NΩt (p∗ (t)). Thus, for any z ∈ Ωt , we have that
d(f, PΩ ) = f − p∗ = Reσ(p∗ , t)(z − p∗ (t)) + Reσ(p∗ , t)(f (t) − z)
Reσ(p∗ , t)(f (t) − z) |f (t) − z|.
This implies that d(f, PΩ ) d(f (t), Ωt ) supt∈Q d(f (t), Ωt ), which contradicts (2.18). Thus
the implication (i)=⇒(ii) is proved.
(ii)=⇒(iii). It is trivial.
(iii)=⇒(i). Suppose that (iii) holds. By Lemma 2.2, we only need to verify that d(f, PΩ ) >
d(f (t), Ωt ) for each t ∈ M (f − p∗ ). To do this, suppose on the contrary that there exists
some t0 ∈ M (f − p∗ ) such that d(f, PΩ ) d(f (t0 ), Ωt0 ). Then |f (t0 ) − p∗ (t0 )| = f − p∗ d(f (t0 ), Ωt0 ) so that p∗ (t0 ) is a best approximation to f (t0 ) from Ωt0 since p∗ (t0 ) ∈ Ωt0 . Thus
f (t0 ) − p∗ (t0 ) ∈ NΩt0 (p∗ (t0 )) by [10, Theorem 2.1, p. 6]. In particular, p∗ (t0 ) ∈ bdΩt0 .
Furthermore, σ(p∗ , t0 ) ∈ NΩt0 (p∗ (t0 )). Hence t0 ∈ M̃ (p∗ ). The proof is complete.
Recall that the notion of the Chebeshev alternation plays an important role in characterizing
the best approximations of real-valued continuous functions on a bounded interval [a, b], see
for example [7, 8]. Unfortunately, this notion does not make sense for the approximations of
complex-valued continuous functions. Motivated by the equivalent relationship between the
Chebeshev alternation and the extremal signature in a real case, we extend the notion of the
extremal support to study the problem of the best approximations of complex-valued continuous
functions. We first recall the notion of the extremal signature, see for exampl [11, 12].
Limit theory of restricted range approximations of complex-valued continuous functions
1433
Let A ⊆ Q be a finite subset and σ a function defined on Q. σ is said to have the finite
support A if σ(t) = 0 for each t ∈ A and 0 for each t ∈ Q \ A. Furthermore, σ is said to be a
signature with the support A if |σ(t)| = 1 for each t ∈ A. Then, σ is said to be an extremal
signature, if there exists a function μ with the support A such that sign μ(t) = σ(t) for each
t ∈ A and t∈A p(t)μ(t) = 0 for each p ∈ P. The following definition is an extension of this
notion.
Definition 2.5. Let (A, B) be a pair of finite subsets of Q. Let σ and τ be the functions
defined on Q. Then
(i) (σ, τ ) is called a bi-signature with the support (A, B) if σ and τ are the signatures with
the supports A and B, respectively;
(ii) A bi-signature (σ, τ ) with the support (A, B) is called an extremal bi-signature if there
exist two functions μ and ν on Q such that
(sign μ(t1 ), sign ν(t2 )) = (σ(t1 ), τ (t2 )),
and
t∈A
p(t)μ(t) +
p(t)ν(t) = 0,
∀ (t1 , t2 ) ∈ A × B
(2.24)
∀ p ∈ P.
(2.25)
t∈B
We still require the notion of the extremal bi-support with respect to (f, p∗ ).
Definition 2.6. Let A and B be the finite subsets of Q and A ∪ B = ∅. (A, B) is said to be
an extremal bi-support with respect to (f, p∗ ) if A ⊆ M (f − p∗ ), B ⊆ B(p∗ ) and there exists an
extremal bi-signature (σ, τ ) with the support (A, B) such that
(σ(t1 ), τ (t2 )) ∈ (σ(p∗ , t1 ), τ (p∗ , t2 )),
∀ (t1 , t2 ) ∈ A × B.
(2.26)
It is clear that if (A, B) is an extremal bi-support with respect to (f, p∗ ), then A = ∅ by
the interior-point condition. Thus, using Proposition 2.1 and [6, Theorem 5.1] (noting that the
strong interior-point condition mentioned there is satisfied, thanks to Hypothesis 2), we obtain
the following characterization theorem of the best restricted range approximation in view of an
extremal bi-signature.
Theorem 2.1. Let f ∈ C(Q) \ PΩ and p∗ ∈ PΩ . Then p∗ is a best restricted range approximation to f from PΩ if and only if there exists an extremal bi-support with respect to
(f, p∗ ).
In particular, when P is a Haar subspace, Theorem 2.1 can be improved to Theorem 2.2
below. Recall that an n-dimensional subspace P ⊆ C(Q) is called a Haar subspace if every
element p ∈ P \ {0} has at most n − 1 zeros in Q.
Theorem 2.2. Let f ∈ C(Q) \ PΩ and p∗ ∈ PΩ . If P is a Haar subspace, then the following
statements are equivalent:
(i) p∗ is a best restricted range approximation to f from PΩ ;
(ii) Either M̃ (p∗ ) = ∅ or any extremal bi-support (A, B) with respect to (f, p∗ ) satisfies
|A ∪ B| n + 1;
(iii) Either M̃ (p∗ ) = ∅ or there exists an extremal bi-support (A, B) with respect to (f, p∗ )
satisfying |A ∪ B| n + 1.
Xian-fa LUO & CHong LI
1434
Proof. (i)=⇒(ii) Suppose that p∗ is a best approximation to f from PΩ and M̃ (p∗ ) = ∅.
Suppose on the contrary that there exists an extremal bi-support (A, B) with respect to (f, p∗ )
such that |A ∪ B| n. In view of Definition 2.6, we have that A ⊆ M (f − p∗ ), B ⊆ B(p∗ )
and there exists an extremal bi-signature (σ, τ ) with the support (A, B) such that (2.26) holds.
Thus, by Definition 2.5, there are two functions μ and ν on Q such that (2.24) and (2.25) hold.
It follows from (2.24) and (2.26) that
(sign μ(t1 ), sign ν(t2 )) ∈ (σ(p∗ , t1 ), τ (p∗ , t2 )),
∀ (t1 , t2 ) ∈ (A, B).
(2.27)
Also from (2.25), we have that
t∈A∩B
p(t)μ(t) + ν(t) +
p(t)μ(t) +
t∈A\B
p(t)ν(t) = 0,
∀ p ∈ P.
(2.28)
t∈B\A
Noting that P is a Haar subspace and that |A ∩ B| + |A \ B| + |B \ A| n, we may pick p1 ∈ P
such that
p1 (t) = μ(t) + ν(t),
∀ t ∈ A ∩ B,
(2.29)
p1 (t) = μ(t),
∀ t ∈ A \ B,
(2.30)
p1 (t) = ν(t),
∀ t ∈ B \ A.
(2.31)
Since M̃ (p∗ ) = ∅, it follows that p1 (t) = 0 for each t ∈ A ∩ B. In fact, suppose on the contrary
that p1 (t1 ) = 0 for some t1 ∈ A ∩ B. Then μ(t1 ) + ν(t1 ) = 0. Note that sign μ(t1 ) = σ(p∗ , t1 )
and sign ν(t1 ) ∈ −NΩt1 (p∗ (t1 )) by (2.27). We have that
σ(p∗ , t1 ) = signμ(t1 ) = −sign ν(t1 ) ∈ NΩt1 (p∗ (t1 )),
so that t1 ∈ M̃ (p∗ ), which is a contradiction. Thus, (2.28) is not true for p = p1 . The implication
(i)=⇒(ii) is proved.
(ii)=⇒(iii) It is trivial.
(iii)=⇒(i) Suppose that (iii) holds. Then (i) holds by Theorem 2.1 in the case when M̃ (p∗ ) =
∅. It remains to prove (i) in the case when M̃ (p∗ ) = ∅. Take t2 ∈ M̃ (p∗ ). Then t2 ∈
M (f − p∗ ) ∩ B(p∗ ) and σ(p∗ , t2 ) ∈ NΩt2 (p∗ (t2 )). Let A = B = {t2 }. Define μ(t2 ) = σ(p∗ , t2 )
and ν(t2 ) = −σ(p∗ , t2 ). Then sign μ(t2 ) = σ(p∗ , t2 ), sign ν(t2 ) = −σ(p∗ , t2 ) ∈ τ (p∗ , t2 ) and
p(t2 )μ(t2 ) + p(t2 )ν(t2 ) = 0, ∀ p ∈ P. This means that (A, B) is an extremal bi-support with
respect to (f, p∗ ), and hence p∗ is a best approximation to f from PΩ by Theorem 2.1. The
proof is complete.
3
The limit theory for characterizations
We begin with some additional notions which will be used in this section.
Definition 3.1. PΩ is said to have Property C (resp. C) if, for any f ∈ C(Q) \ PΩ , p∗ is
a best approximation to f from PΩ if and only if any (resp. at least one) extremal bi-support
(A, B) with respect to (f, p∗ ) satisfies |A ∪ B| n + 1.
Remark 3.1. Clearly Property C implies Property C, but the converse is not true, in general,
as shown by the following
Limit theory of restricted range approximations of complex-valued continuous functions
1435
Example 3.1. Let Q = {−1, 1}, Ω−1 = {z ∈ C : |z + 1| 1} and Ω1 = {z ∈ C : |z − 1| 1}.
Let p̄ ∈ C(Q) be defined by p̄(t) = t for each t ∈ Q and let P = {p : p = ap̄, a ∈ C}. Then P is a
1-dimensional Haar subspace, and PΩ = {p : p(t) = at, |a − 1| 1}. Hence Hypotheses 1 and 2
in the previous section are satisfied (noting that p̄ is an interior point of PΩ ). Let f ∈ C(Q)\PΩ ,
and let p∗ be a best approximation to f from PΩ . We claim that there exists an extremal bisupport (A, B) with respect to (f, p∗ ) such that |A ∪ B| n + 1. Note that the claim holds by
Theorem 2.2 in the case when M̃ (p∗ ) = ∅. It remains to consider the case when M̃ (p∗ ) = ∅.
Without loss of generality, we may assume that 1 ∈ M̃ (p∗ ). Then 1 ∈ M (f − p∗ ) ∩ B(p∗ )
and σ(p∗ , 1) ∈ NΩ1 (p∗ (1)). Therefore, −1 ∈ B(p∗ ). Let A = {1}, B = {1, −1}. Then
A ⊆ M (f − p∗ ) and B ⊆ B(p∗ ). Thus to complete the proof of the claim, it suffices to verify
that (A, B) is an extremal bi-support with respect to (f, p∗ ). To do this, define
μA (1) = 2σ(p∗ , 1),
νB (1) = −σ(p∗ , 1),
νB (−1) = σ(p∗ , 1).
Then
p(1)μA (1) + p(1)νB (1) + p(−1)νB (−1) = 0,
⎧
⎨ σ(p∗ , 1),
σA (t) =
⎩ 0,
Define
∀ p ∈ P.
(3.1)
t = 1,
t ∈ Q \ {1}
and τB (t) = −tσ(p∗ , 1), t ∈ Q = {−1, 1}. Then
(sign μA (t1 ), signνB (t2 )) = (σA (t1 ), τB (t2 )),
∀ (t1 , t2 ) ∈ A × B.
This and (3.1) imply that (σA , τB ) is an extremal bi-signature with the support (A, B). Moreover, it is easy to see that
NΩ1 (p∗ (1)) = {λσ(p∗ , 1) : λ 0},
NΩ−1 (p∗ (−1)) = {−λσ(p∗ , 1) : λ 0}.
Hence
τ (p∗ , 1) = {−λσ(p∗ , 1) : λ > 0},
τ (p∗ , −1) = {λσ(p∗ , 1) : λ > 0},
which implies that
(σA (t1 ), τB (t2 )) ∈ (σ(p∗ , t1 ), τ (p∗ , t2 )), ∀ (t1 , t2 ) ∈ A × B.
(3.2)
Thus the claim stands. Consequently, PΩ has Property C.
Below we will show that PΩ does not have Property C. Define f by f (t) = −t for each t ∈ Q.
Then f ∈ C(Q) \ PΩ . Furthermore, for any p = ap̄ ∈ PΩ , one gets that Rea 0 because
1 |a − 1| 1 − Rea. Hence f − p = |1 + a| Re(1 + a) 1 = f − q ∗ and q ∗ = 0 is a best
approximation to f from PΩ . Let A1 = B1 = {1}. Then A1 ⊆ M (f − q ∗ ) and B1 ⊆ B(q ∗ ).
Define μA1 (1) = σ(q ∗ , 1) and νB1 (1) = −σ(q ∗ , 1). Then p(1)μA1 (1) + p(1)νB1 (1) = 0, ∀ p ∈ P.
This shows that (A1 , B1 ) is an extremal bi-support with respect to (f, q ∗ ) since sign μA1 (1) =
σ(q ∗ , 1) and signνB1 (1) ∈ τ (q ∗ , 1). However, |A ∪ B| = 1 < n + 1. In view of Definition 3.1, PΩ
does not have Property C.
Xian-fa LUO & CHong LI
1436
Definition 3.2. PΩ is said to have Property Ci (resp. C i ) if, for each f ∈ Cai (Q) \ PΩ , p∗ is
a best approximation to f from PΩ if and only if any (resp. at least one) extremal bi-support
(A, B) with respect to (f, p∗ ) satisfies |A ∪ B| n + 1 for i = 1, 2.
Definition 3.3. PΩ is said to have Property C ∗ (resp. C ∗ ) if, for each f ∈ C(Q) \ PΩ , p∗ is
a best approximation to f from PΩ if and only if either M̃ (p∗ ) = ∅ or any (resp. at least one)
extremal bi-support (A, B) with respect to (f, p∗ ) satisfies |A ∪ B| n + 1.
Remark 3.2. (i) PΩ has Property C (resp. C) =⇒ PΩ has Property C2 (resp. C 2 ) =⇒ PΩ
has Property C1 (resp. C 1 );
(ii) PΩ has Property C ∗ (resp. C ∗ ) =⇒ PΩ has Property C2 (resp. C 2 ) =⇒ PΩ has Property
C1 (resp. C 1 ).
To establish the characterizations for PΩ to have Property C (resp. Ci , Ci , C ∗ and C ∗ ), we
need to verify a lemma, which is also used in Sec. 4.
Lemma 3.1. Suppose that P is not a Haar subspace. Then there exists f ∈ Ca1 (Q) \ PΩ
which has the following properties:
(i) There are at least two best approximations to f from PΩ ;
(ii) There exists a best approximation p∗ to f from PΩ such that |M (f − p∗ ) ∪ B(p∗ )| n.
Proof. By the assumption, there exists p1 ∈ P \ {0} such that p1 (t) has n distinct zeros
t1 , . . . , tn in Q. Let {φ1 , . . . , φn } be a basis of P. Consider the following system of the equations
with the unknown complex variable (c1 , . . . , cn )
n
ck φi (tk ) = 0,
i = 1, . . . , n.
(3.3)
k=1
Then (3.3) has a nonzero solution (c1 , . . . , cn ) because det(φi (tk ))ni,k=1 = 0. Let N = {k : ck =
0}. Then N = ∅. Since Q is a compact metric space, by the Tietze Extension Theorem, there
exists f0 ∈ C(Q) such that
ck
, ∀k∈N
(3.4)
f0 (tk ) =
|ck |
and
|f0 (t)| < 1,
∀ t ∈ Q \ {tk : k ∈ N }.
(3.5)
Recall that p is an interior point of PΩ . Thus, by Proposition 2.1, there is a positive number δ
such that
B(p(t), δ) ⊆ Ωt , ∀ t ∈ Q.
(3.6)
Set M = maxt∈Q |p1 (t)|. Define the function f1 on Q by
1
|p1 (t)| f0 (t),
f1 (t) = 1 −
M
∀t∈Q
and set f = δf1 + p. Note that |f (t) − p(t)| δ for all t ∈ Q by (3.4) and (3.5). Hence f (t) ∈ Ωt
for each t ∈ Q by the closeness of Ωt . Furthermore, we claim that d(f, PΩ ) = δ and p is a best
approximation to f from PΩ . In fact, for each p ∈ PΩ , write p − p = ni=1 bi φi . Then by (3.3)
we have that
Re
n
k=1
ck [p(tk ) − p(tk )] = Re
n
k=1
ck
n
i=1
bi φi (tk ) = Re
n
i=1
bi
n
k=1
ck φi (tk ) = 0.
Limit theory of restricted range approximations of complex-valued continuous functions
1437
Thus there exists k0 ∈ N such that Reck0 [p(tk0 ) − p(tk0 )] 0. It follows that Ref1 (tk0 )[p(tk0 ) −
p(tk0 )] 0 due to (3.4). This implies that
f − p2 |f (tk0 ) − p(tk0 )|2 = |δf1 (tk0 ) + p(tk0 ) − p(tk0 )|2
= δ 2 |f1 (tk0 )|2 + 2δRef1 (tk0 )[p(tk0 ) − p(tk0 )] + |p(tk0 ) − p(tk0 )|2
δ2.
Hence d(f, PΩ ) δ and the claim stands because f − p δ d(f, PΩ ). Consequently,
f ∈ Ca1 (Q) \ PΩ since B(p) = ∅ by (3.6). Below we will show that f has at least two best
λ
approximations from PΩ . To do this, define pλ = M
p1 + p, ∀ λ ∈ [0, δ). Then |pλ (t) − p(t)| < δ
for each t ∈ Q and so pλ ∈ PΩ by (3.6). Moreover,
1
λ
δ−λ
λ
|p1 (t)| +
|p1 (t)| = δ −
|p1 (t)| δ.
|f (t) − p (t)| δ 1 −
M
M
M
Hence f − pλ δ = d(f, PΩ ) and pλ is a best approximation to f from PΩ for each λ ∈ [0, δ).
Thus (i) holds. It remains to show that f satisfies (ii). To this end, let p∗ = p. Then (3.4),
(3.5) and (3.6) imply that M (f − p∗ ) = {tk : k ∈ N } and B(p∗ ) = ∅. Therefore,
|M (f − p∗ ) ∪ B(pλ )| = |M (f − p∗ )| = |N | n,
and (ii) is proved. The proof is complete.
The first result of this section is as follows.
Theorem 3.1. The following statements are equivalent:
(i) P is a Haar subspace;
(ii) P has Property C ∗ ;
(iii) P has Property C ∗ ;
(iv) PΩ has Property C2 ;
(v) PΩ has Property C1 ;
(vi) PΩ has Property C 2 ;
(vii) PΩ has Property C 1 .
Proof.
By Theorem 2.2, Proposition 2.3 and Remark 3.2, the following implications hold:
(i)
(vii)
=⇒
⇐=
(ii)
=⇒ (iv)
⇓
⇓
(iii)
(v)
=⇒
(vi)
⇓
=⇒ (vii).
Thus, to complete the proof of Theorem 3.1, it suffices to prove the implication (vii)=⇒(i).
Suppose on the contrary that (i) is not true. Then, by Lemma 3.1, there exist f ∈ Ca1 (Q) \ PΩ
and a best approximation p∗ to f from PΩ such that |M (f − p∗ ) ∪ B(p∗ )| n. Thus if (A, B)
is an extremal bi-support with respect to (f, p∗ ) then |A ∪ B| n. Hence PΩ does not have
Property C 1 . The proof is complete.
Remark 3.3. By Remark 3.2 (i) and Theorem 3.1, PΩ has Property C such that P is a Haar
subspace. However, the converse is not true, in general, as illustrated in the following.
1438
Xian-fa LUO & CHong LI
Example 3.2. Let Q = {−1, 0, 1} and let Ω−1 = Ω0 = Ω1 = {z : |z| 1}. Let P = {p :
p(t) = a + bt, a, b ∈ C}. Then P is a Haar subspace and
PΩ = {p : p(t) = a + bt, |a| 1, |a − b| 1, |a + b| 1}.
Clearly Hypotheses 1 and 2 in sec. 2 are satisfied. To verify that PΩ does not have Property
C, define f by f (0) = 1, f (−1) = 0 and f (1) = 2 + 2i. Then, for each p ∈ PΩ , assuming
p(t) = a + bt, one has that
f − p = max{|1 − a|, |a − b|, |2 + 2i − (a + b)|}
√
|2 + 2i| − |a + b| 2 2 − 1.
√
√
√
Set p∗ (t) = 22 + 22 it, t ∈ Q. Then p∗ ∈ PΩ and f − p∗ = 2 2 − 1. Hence p∗ is a best
approximation to f from PΩ . Furthermore, M (f − p∗ ) = {1} and B(p∗ ) = {−1, 1}. Hence
|M (f − p∗ ) ∪ B(p∗ )| = 2, which implies that PΩ does not have Property C.
For the characterization of Property C, we need to establish the following.
Proposition 3.1. Suppose that P is a Haar subspace. Then the following statements are
equivalent:
(i) PΩ = P;
(ii) There exist p∗ ∈ PΩ and t0 ∈ Q such that p∗ (t0 ) ∈ bdΩt0 ;
(iii) There exists t1 ∈ Q such that Ωt1 = C.
Proof. (i)=⇒(ii) Suppose that (i) holds and set PintΩ = {p ∈ P : p(t) ∈ intΩt , ∀ t ∈ Q}. Then
PintΩ is a nonempty subset of PΩ since p ∈ PintΩ . We claim that PintΩ is an open subset of P
in the uniform norm. Indeed, let p0 ∈ PintΩ . Then p0 is an interior point of PΩ in the sense of
Definition 2.1; hence p0 is a strong interior point of PΩ by Proposition 2.1. This means that
there is δ > 0 such that B(p0 (t), δ) ⊆ Ωt for each t ∈ Q. Let p ∈ P with p − p0 < δ2 . Then
|p(t) − p0 (t)| < 2δ ; hence B p(t), δ2 ⊆ B(p0 (t), δ) ⊆ Ωt , ∀ t ∈ Q.
It follows that p(t) ∈ intΩt for each t ∈ Q and so p ∈ PintΩ . Consequently, B(p0 , δ2 ) ⊆ PintΩ
and the claim stands. Since PΩ is a proper closed subset of P, we get that PintΩ = PΩ . Pick
p∗ ∈ PΩ \ PintΩ . Then there exists t0 ∈ Q such that p∗ (t0 ) ∈ bdΩt0 by the definition of PintΩ .
Therefore, (ii) holds.
(ii)=⇒(iii). It is trivial.
(iii)=⇒(i). By the assumption, we may take z1 ∈ C \ Ωt1 . Since P is a Haar subspace, there
/ PΩ as p1 (t1 ) ∈
/ Ωt1 , that is, PΩ = P. The
exists p1 ∈ P such that p1 (t1 ) = z1 . Hence p1 ∈
proof is complete.
The second result of this section is stated as follows:
Theorem 3.2.
PΩ has Property C if and only if P is a Haar subspace and PΩ = P.
Proof. Suppose that P is a Haar subspace and PΩ = P. Then by Proposition 3.1, Ωt = C for
each t ∈ Q; hence B(p∗ ) = ∅ for each p∗ ∈ P. Therefore, PΩ has Property C by Theorem 2.2.
Conversely, suppose that PΩ has Property C. Then P is a Haar subspace by Remark 3.2 (i)
and Theorem 3.1. Thus, it remains to prove that PΩ = P. To do this, suppose on the
contrary that PΩ = P. Then, by Proposition 3.1, there exist p∗ ∈ PΩ and t0 ∈ Q such that
p∗ (t0 ) ∈ bdΩt0 , which implies that NΩt0 (p∗ (t0 )) = {0}. Let τ ∈ NΩt0 (p∗ (t0 )) \ {0}. By the
Limit theory of restricted range approximations of complex-valued continuous functions
1439
Tietze Extension Theorem there exists a function g ∈ C(Q) such that g(t0 ) = τ and |g(t)| |τ |
for each t ∈ Q. Define f ∈ C(Q) by f = p∗ + g. Then t0 ∈ M (f − p∗ ). Noting that
p∗ (t0 ) ∈ bdΩt0 and σ(p∗ , t0 ) = |ττ | ∈ NΩt0 (p∗ (t0 )), we have that t0 ∈ M̃ (p∗ ). Hence p∗ is a best
approximation to f from PΩ by Theorem 2.2. Furthermore, let A = B = {t0 }. It is easy to see
that (A, B) is an extremal bi-support with respect to (f, p∗ ). This implies that PΩ does not
have Property C since |A ∪ B| = 1, which is a contradiction. Thus PΩ = P.
4
The limit theory for uniqueness
We begin with the following definitions.
Definition 4.1. PΩ is said to have Property U (resp. U1 , U2 ) if, for each f ∈ C(Q) (resp.
Ca1 (Q), Ca2 (Q)), f has a unique best approximation from PΩ .
Definition 4.2. PΩ is said to have Property K with respect to Q if, for any p1 , p2 ∈ PΩ and
t0 ∈ Q, the condition p1 (t0 ) = p2 (t0 ) ∈ bdΩt0 implies that p1 = p2 .
Remark 4.1. Property K was first introduced by Shi[9] in the case of real-valued continuous function approximations and by Smirnov and Smirnov[4,5] in the case of complex-valued
continuous function approximations.
The first theorem addresses Property U, the proof of which is almost the same as that of
Theorem 4.2 in [4] and so omitted here.
Theorem 4.1.
PΩ has Property U if and only if the following conditions hold:
(i) P is a Haar subspace;
(ii) PΩ has Property K with respect to Q.
The second theorem addresses Properties U1 and U2 . Let G be a subset of C. Recall that
G is said to be strictly convex if, for any distinct elements g1 , g2 ∈ G, (g1 + g2 )/2 ∈ intG.
Note that the notion of the strict convexity plays a basic role in the study of the uniqueness of
approximations of complex-valued continuous functions, see, for example [3–5].
Hypothesis 3.
Ωt is strictly convex for each t ∈ Q.
Theorem 4.2.
Consider the following statements:
(i) P is a Haar subspace;
(ii) PΩ has Property U1 ;
(iii) PΩ has Property U2 .
Then (iii) =⇒ (ii) =⇒ (i). If, in addition, Hypothesis 3 is satisfied, then (i) ⇐⇒ (ii) ⇐⇒ (iii).
Proof.
(iii)=⇒(ii) This results from Proposition 2.2.
(ii)=⇒(i). Suppose that (ii) holds but (i) does not. Then P is not a Haar subspace. By
Lemma 3.1, there exists f ∈ Ca1 (Q) \ PΩ such that f has at least two best approximations from
PΩ , which contradicts (ii). Hence (ii)=⇒(i) holds.
Finally, suppose that, in addition, Hypothesis 3 is satisfied. To complete the proof of the
theorem, it suffices to prove (i)=⇒(iii). To this end, suppose that (i) holds. Let f ∈ Ca2 (Q) \ PΩ
and let p∗ and q ∗ be the best approximations to f from PΩ . Set r∗ = 12 (p∗ + q ∗ ). Then r∗
is a best approximation to f from PΩ . Using the standard technique, we get the following
Xian-fa LUO & CHong LI
1440
inclusions:
M (f − r∗ ) ⊆ M (f − p∗ ) ∩ M (f − q ∗ ) ⊆ Z(p∗ − q ∗ ),
∗
∗
∗
B(r ) ⊆ B(p ) ∩ B(q ),
(4.1)
(4.2)
where Z(p) stands for the set of all zeros of p in Q. By Proposition 2.3 we have that M̃ (r∗ ) = ∅.
Thus, Theorem 2.2 implies that
|M (f − r∗ ) ∪ B(r∗ )| n + 1.
(4.3)
Since each Ωt is strictly convex, (4.2) yields that B(r∗ ) ⊆ Z(p∗ − q ∗ ). Combining this and (4.1)
gives that M (f − r∗ ) ∪ B(r∗ ) ⊆ Z(p∗ − q ∗ ). Hence, |Z(p∗ − q ∗ )| n + 1 due to (4.3) and so
p∗ = q ∗ because P is a Haar subspace. Thus PΩ has Property U2 and the proof of Theorem 4.2
is complete.
The following corollary, which was obtained in [2–5] respectively under some stronger conditions, is now an immediate consequence of Theorem 4.1.
Corollary 4.1. Suppose that P is a Haar subspace and that Hypothesis 3 is satisfied. Let
f ∈ Ca2 (Q) and let p∗ be a best approximation to f from PΩ . Then p∗ is the unique best
approximation to f from PΩ provided that M̃ (p∗ ) = ∅.
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