Journal of Approximation Theory AT3102
Journal of Approximation Theory 91, 332348 (1997)
Article No. AT963102
On Best Simultaneous Approximation
Chong Li*
Department of Mathematics, Hangzhou Institute of Commerce, Hangzou,
People's Republic of China
and
G. A. Watson
Department of Mathematics and Computer Science, University of Dundee,
Dundee DD1 4HN, Scotland
Communicated by Andras Kroo
Received May 17, 1995; accepted in revised form December 3, 1996
The problem is considered of best approximation of finite number of functions
simultaneously. For a very general class of norms, characterization results are
derived. The main part of the paper is concerned with proving uniqueness and
strong uniqueness theorems. For a particular subclass, which includes the important special case of the Chebyshev norm, a characterization is given of a uniqueness
element. 1997 Academic Press
1. INTRODUCTION
Let X be a compact Hausdorff space and Y a normed linear space with
norm & } & Y . Let C(X, Y ) denote the set of all continuous functions from X
to Y, and let & } & A be a norm on C(X, Y ). Let U be defined by
U=[a # R l, &a& B 1],
where & } & B is a given norm on R l. Define a norm on l-tuples of elements
of C(X, Y ) as follows: for any F=(, 1 , ..., , l ) # C(X, Y ) l define
&F&=max
a#U
"
l
: ai ,i
i=1
",
(1.1)
A
where a=(a 1 , ..., a l ).
* Present address: Center for Mathematical Sciences, Zhejiang University, Hangzhou
310027, P. R. China.
332
0021-904597 25.00
Copyright 1997 by Academic Press
All rights of reproduction in any form reserved.
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ON BEST SIMULTANEOUS APPROXIMATION
333
Now suppose that functions , 1 , ..., , l in C(X, Y ) are given. Then the
problem is considered here of approximating these functions simultaneously
by functions in S, a subspace of C(X, Y ), in the sense of the minimization
of the norm (1.1). In other words, we want to find an l-tuple f =(,, ..., ,),
where , # S, to minimize
&F& f &.
(1.2)
If such a function f * exists, it is called a best simultaneous approximation to
F=(, 1 , ..., , l ). Problems of simultaneous approximation can be viewed as
special cases of vector-valued approximation, and some recent work in this
area is due to Pinkus [6], who points out that many questions remain
unresolved. He is concerned with the question of when a finite dimensional
subspace is a unicity space, for some different norms from those considered
here. We are also primarily interested in uniqueness questions. Characterization results for linear problems were recently given in [9] based on the
derivation of an expression for the directional derivative, and these
generalized earlier work of [8]. We being by showing how these results can
be obtained in a simpler and more direct manner, which permits their extension to some nonlinear problems. The rest of the paper is concerned with
uniqueness and strong uniqueness of best approximations.
In what follows, finite sequences of identical elements identified by
(,, ..., ,) will be assumed to be l-tuples.
2. CHARACTERIZATION OF BEST APPROXIMATIONS
Let C*(X, Y ) denote the dual space of C(X, Y ), and let W denote the
dual unit ball. For F=(, 1 , ..., , l ) # C(X, Y ) l define
l
g F (a, w)= : a i(w, , i ),
for all
(a, w) # U_W,
i=1
where the inner product notation links elements of C(X, Y ) and its dual.
Note that U_W is endowed with the product topology, while W is
endowed with the weak * topology. Since for any (a 0, w 0 ) # U_W,
}
l
l
| g F (a, w)& g F (a 0, w 0 )| = : a i( w, , i ) & : a 0i ( w 0, , i )
i=1
l
i=1
}
l
: |a i &a 0i | |( w, , i ) | + : |a 0i | |( w&w 0, , i ) |,
i=1
i=1
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334
LI AND WATSON
it follows that g F ( } , } ) # C(U_W ) (the space of continuous functions
defined on U_W ). Note that for any , # S, f =(,, ..., ,),
\
l
+
g f (a, w)= : a i ( w, ,) # C(U_W ).
i=1
Further for any such f,
&F& f &=max
a#U
"
l
: a i (, i &,)
i=1
}
"
A
l
=max max : a i( w, , i &,)
a#U w#W
i=1
}
=&g F ( } , } )& g f ( } , } )& C ,
where & } & C denotes the uniform norm on C(U_W ). Now define
Sg =[g f : f =(,, ..., ,), , # S].
It follows that f *=(,*, ..., ,*), ,* # S is a best simultaneous approximation
to F=(, 1 , ..., , l ) if and only if g f * # S g is a best approximation to g F in the
uniform norm of C(U_W ). Let P S (F ) denote the set of all best
simultaneous approximations f =(,, ..., ,), where , # S, to F. In addition,
let
d(F, S)=inf [&F& f & : f =(,, ..., ,), , # S],
d(F, C)=d(F, C(X, Y )).
Definition 1. A set S is a sunset for simultaneous approximation if for
any F=(, 1 , ..., , l ), and f *=(,*, ..., ,*), ,* # S, f * # P S (F ) implies that
f * # P S (F : ) for F : = f *+:(F& f *), and :0.
Descriptions of sunsets (or strict suns), and solar properties, are given in
[3]. Linear sets are examples of suns, as are convex sets, but also some nonconvex sets, for example rational functions.
Theorem 1. Let S/C(X, Y ) be a sunset of simultaneous approximation.
Then f * # P S (F ) if and only if for any f =(,, ..., ,), , # S, there exists
a # ext(U ), w # ext(W ) such that
g F& f *(a, w)=&F& f *&,
g f *& f (a, w)0,
where ``ext'' denotes the set of extreme points.
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ON BEST SIMULTANEOUS APPROXIMATION
Proof. If S is a sunset for simultaneous approximation, then S g is a strict
sun for uniform approximation in C(U_W ). The result then follows from
the generalized Kolmogorov criterion characterizing a best approximation
in C(U_W ) with the uniform norm (see, for example [3, Theorem I.2.4]),
using the KreinMilman Theorem. K
A special case of (1.1) is given by
&,& A =max &,(t)& Y ,
(2.1)
t#X
for any , # C(X, Y ). This includes the important case of the Chebyshev
norm on l-tuples. Let F=(, 1 , ..., , l ) # C(X, Y ) l. Then a Chebyshev norm
may be defined by
&F&= max max &, i (t)& Y ,
(2.2)
1il t # X
which is the special case of (1.1) when & } & A is given by (2.1) and & } & B is the
l 1 norm [see 8, 9]. Define
{
H(F, f )= t # X: max
&a&B =1
"
l
"
i=1
=
=&F& f & .
: a i (, i (t)&,(t))
Y
(2.3)
Let Z denote the unit ball in Y*, the dual space of Y, and let ( } , } ) Y denote
the inner product linking Y and Y*. Then using the form of points in
ext(W ) in this case, we have the following corollary of Theorem 1.
Corollary 1. Let & } & A be given by (2.1), and let S/C(X, Y ) be a sunset for simultaneous approximation. Then f * # P S (F ) if and only if for any
f =(,, ..., ,), , # S, there exists a # ext(U ), t # H(F, f ), v(t) # ext(Z) such that
l
: a i( v(t), , i (t)&,*(t)) Y =&F& f *&,
i=1
\
l
+
: a i ( v(t), ,*(t)&,(t)) Y 0.
i=1
Returning to the general problem, standard linear theory (for example
[7]) gives the following result.
Theorem 2. Let S be an n-dimensional subspace of C(X, Y ). Then
f * # P S (F) if and only if there exists a j # ext(U ), w j # ext(W ), : j >0,
j=1, ..., r with rj=1 : j =1 and 1rn+1 such that
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LI AND WATSON
g F& f *(a j, w j )=&F& f *&,
j=1, ..., r,
r
: : j g f (a j, w j )=0
for all f # S.
j=1
This is just the result given as Theorem 1 in [9].
3. UNIQUENESS OF BEST APPROXIMATIONS
For the general case uniqueness is a consequence of strict convexity of the
norm & } & A . This is established next. It is convenient to extract the following
result as a preliminary lemma.
Lemma 1. Let S/C(X, Y ), and let F=(, 1 , ..., , l ) # C(X, Y ) l. Let
f *=(,*, ..., ,*) # P S (F), with a* # U such that
"
l
&F& f *&= : a *(,
i
i &,*)
".
(3.1)
A
i=1
Then if d(F, C)<d(F, S), li=1 a *{0.
i
Proof. Assume that d(F, C)<d(F, S) and also that a* satisfying (3.1) is
such that li=1 a*i =0. Then there exists , # C(X, Y ) such that
max
a#U
"
l
l
" "
"
= : a *(, &,) .
"
"
: a i (, i &,)
i=1
< : a *(,
i
i &,*)
A
A
i=1
l
i
i
A
i=1
This is a contradiction which proves the result.
K
Theorem 3. Let & } & A be strictly convex, and let S/C(X, Y ) be a sunset
for simultaneous approximation. Then for any F=(, 1 , ..., , l ), d(F, C)=
d(F, S) or P S (F ) contains at most one element.
Proof. Let F=(, 1 , ..., , l ) be such that d(F, C)<d(F, S). Suppose that
f *=(,*, ..., ,*) # P S (F), f =(,, ..., , ) # P S (F ) with ,*{,. Let , 0i =2, i &,,
i=1, ..., l, F 0 =(, 01 , ..., , 0l ). Then it follows from the definition of a sunset
that f # P S (F 0 ). Also
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ON BEST SIMULTANEOUS APPROXIMATION
&F 0 & f *&=&2F& f & f *&
&F& f &+&F& f *&
=2 &F& f &
=&F 0 & f &.
(3.2)
Therefore f * # P S (F 0 ).
Further
l
"
"
max
_" : a (, &,)" +" : a (, &,*)" &
&2F& f & f *&=max
a#U
: a i (2, i &, &,*)
A
i=1
l
l
i
a#U
i
i
A
i=1
0
&F & f &,
i
A
i=1
using (3.2).
(3.3)
Thus equality holds to (3.3). Let a # U be chosen so that
"
l
: (2, i &, &,*)
" =&2F& f & f *&=2d(F, C).
A
i=1
Thus
"
l
l
"
: a i (, i &, )+ : a i (, i &,*)
i=1
i=1
"
l
A
l
= : a i (, i &, )
" +" : a (, &,*)" ,
i
A
i=1
i
i=1
A
which, using the assumption of strict convexity, implies that
l
l
: a i (, i &, )= : a i (, i &,*)
i=1
i=1
or
\
l
+
: a i (,*&, )=0.
i=1
(3.4)
Since by assumption d(F, C)<d(F, S), it follows from Lemma 1 that
( li=1 a i ){0, and so ,*=, and the result is established. K
For the special case covered by (2.1), it is possible to give a precise
characterization of a uniqueness element, which is defined as follows.
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LI AND WATSON
Definition 2. Let S be a sunset of C(X, Y ), and f *=(,*, ..., ,*), with
,* # S. Then ,* is called a uniqueness element of S if for any
F=(, 1 , ..., , l ) # C(X, Y ) l with d(F, C)<d(F, S), f * # P S (F ), then f * is a
unique best approximation to F from S.
Theorem 4. Let Y be strictly convex, let & } & A be given by (2.1), and let
S be a sunset for simultaneous approximation. Let f *=(,*, ..., ,*) # S l. Then
,* is a uniqueness element of S if and only if for any F=
(, 1 , ..., , l ) # C(X, Y ) l, with f * # P S (F ) and d(F, C)<d(F, S), ,* is uniquely
determined by the set of values in H(F, f *) (that is, if , # S and ,(t)=,*(t)
for all t # H(F, f *), then ,=,*).
Proof. Let ,* be a uniqueness element of S. Suppose that for some
F=(, 1 , ..., , l ) # C(X, Y ) l, with f * # P S (F ) and d(F, C)<d(F, S), there
exists f =(,, ..., , ), , # S, , {,*, such that
,(t)=,*(t),
for all t # H(F, f *).
Define for all t # X
_
, 0i (t)=,*(t)+ &,*&,& A & max
_
&a&B =1
\
l
+
: a j &,*(t)&,(t)& Y
j=1
, i (t)&,*(t)
,
&F& f *&
&
(3.5)
and let
F 0 =(, 01 , ..., , 0l ).
It is easy to verify directly that
&F 0 & f *&&,*&,& A .
(3.6)
Since for any t # H(F, f *),
max
&a&B =1
"
l
: a i (, 0i (t)&,*(t))
i=1
"
=&,*&,& A ,
Y
then
&F 0 & f *&&,*&,& A .
(3.7)
It follows from (3.6) and (3.7) that
&F 0 & f *&=&,*&,& A .
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(3.8)
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ON BEST SIMULTANEOUS APPROXIMATION
From Theorem 1, because f * # P S (F ), for any , # S, there exists a # ext(U ),
w # ext(W ), such that
l
\ : a + ( w, ,*&,) 0,
w, : a i (, i &,*) =&F& f *&,
i=1
(3.9)
l
i
i=1
and so using Corollary 1, for some t # H(F, f *), v(t) # ext(Z),
l
: a i( v(t), , i (t)&,*(t)) Y =&F& f *&.
i=1
From (3.5),
l
: a i( v(t), , 0i (t)&,*(t)) Y =&,*&,& A
i=1
=&F 0 & f *&,
using (3.8).
Thus t # H(F 0, f *), and we must have
l
w, : a i (, 0i &,*) =&F 0 & f *&.
i=1
(3.10)
Equations (3.9) and (3.10) show, using Theorem 1, that f * # P S (F 0 ).
Now for any t # X,
max
&a&B =1
"
l
"
: a i (, 0i (t)&,(t))
i=1
Y
l
"\ : a + (,*(t)&,(t))"
+ max
"_&,*&,& & max "\ : a + (,*(t)&,(t))" &
1
: a (, (t)&,*(t))
_
"
&F& f *&
max
&a&B =1
i
Y
i=1
l
A
&a&B =1
&a&B =1
i
i=1
l
i
i=1
i
Y
&,*&,& A .
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Y
340
LI AND WATSON
Thus
&F 0 & f &&F 0 & f *&,
using (3.8),
so that f # P S (F 0 ), a contradiction. This proves necessity.
Now suppose that for some F=(, 1 , ..., , l ) # C(X, Y ) l with
d(F, C)<d(F, S) and f * # P S (F ), f =(,*, ..., ,*), there exists another
f # P S (F ), f =(,, ..., , ). Let
, 0i =2, i &,,
i=1, ..., l,
and let
F 0 =(, 01 , ..., , 0l ).
It follows from the definition of a sunset that f # P S (F 0 ). Now
l
"
"
max : a (, &, ) + max : a (, &,*)
"
"
"
"
&F 0 & f *&= max
: a i (2, i &, &,*)
&a&B =1
A
i=1
l
l
i
&a&B =1
i
i
&a&B =1
A
i=1
i
i=1
=2 &F& f &
=&F 0 & f &.
Thus f * # P S (F 0 ). Now let a # ext(U ), w # ext(W ) such that
l
w, : a i (2, i &, &,*) =&F 0 & f *&,
i=1
which is possible using Theorem 1. Thus for any t # H(F 0, f *),
l
"
"
: a (, &, )(t) + : a (, &,*)(t)
"
" "
"
&F 0 & f *&= : a i (2, i &, &,*)(t)
Y
i=1
l
l
i
i
i=1
i
Y
i
i=1
&F& f &+&F& f *&
=&F 0 & f *&.
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Y
A
341
ON BEST SIMULTANEOUS APPROXIMATION
It follows that
"
l
l
: a i (, i &, )(t)+ : a i (, i &,*)(t)
i=1
i=1
"
l
"
Y
l
= : a i (, i &, )(t)
i=1
" +" : a (, &,*)(t)" .
i
Y
i
i=1
Y
Therefore using the strict convexity of Y,
l
l
: a i (, i &, )(t)= : a i (, i &,*)(t),
i=1
i=1
or equivalently
\
l
+
: a i (,*(t)&,(t))=0,
i=1
for all t # H(F 0, f *).
Since d(F, C)<d(F, S), by Lemma 1 we must have li=1 a i {0, and so
,*(t)=,(t)
for all t # H(F 0, f *).
This proves the sufficiency of the stated conditions. K
4. STRONG UNIQUENESS
It is possible to establish strong uniqueness for the general problem under
a condition which generalizes the Chebyshev set condition for linear best
approximation in the uniform norm. The result hinges on the derivation of
the analogue of the strong Kolomogorov condition for finite dimensional
spaces (see, for example, Wulbert [10] or Nurnberger [5]).
Definition 3 [1]. An n-dimensional subspace S of C(X, Y ) is called an
interpolating subspace if no nontrivial linear combination of n linearly independent extreme points of W annihilates S.
Theorem 5. Let S be an interpolating subspace of C(X, Y ), and let
d(F, C)<d(F, S). Then f *=(,*, ..., ,*) # P S (F) is a strongly unique best
simultaneous approximation to F, that is, there exists #>0 such that
&F& f &&F& f *&+# &,&,*& A
for all f =(,, ..., ,),
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, # S.
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LI AND WATSON
Proof. We can use Theorem 2. Since f * # P S (F), it follows that there
exist a j # ext(U ), w j # ext(W ), : j >0, j=1, ..., r with rj=1 : j =1, and
1rn+1 such that
l
: a ij(w j, , i &,*) =&F& f *&,
j=1, ..., r,
(4.1)
for all , # S.
(4.2)
i=1
r
: :j
j=1
\
l
+
: a ij ( w j, ,*&,) =0
i=1
With no loss of generality, we can assume that the set [w j, j=1, ..., r] is
linearly independent. Because d(F, C)<d(F, S), it follows from Lemma 1
that ; j =: j ( li=1 a ij ){0, j=1, ..., n. Assume that r<n+1. Then we can
take an element , 0 # S such that , 0 {,*, and ( w j, ,*&, 0 ) =; j , j=1, ..., r,
using the fact that S is an interpolating subspace. This means that
r
r
: ; j( w j, ,*&, 0 ) = : ; 2j >0,
j=1
j=1
which is a contradiction. It follows that r=n+1. Now for any , # S, any
(a j, w j ) # ext U_ext W satisfying (4.1) and (4.2) assume that we have
\
l
+
: a ij ( w j, ,*&,) 0,
i=1
j=1, ..., r.
Then it follows from (4.2) that
r
0= : ; j(w j, ,*&,) 0,
j=1
which in turn implies that
( w j, ,*&,) =0,
j=1, ..., r,
or ,*=,, since r=n+1. Thus there exists (a, w) # ext U_ext W with
l
: a i( w, , i &,*) =&F& f *&,
i=1
l
\ : a + ( w, ,*&,) >0.
i
i=1
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ON BEST SIMULTANEOUS APPROXIMATION
Now define
l
{
=
M(F, f *)= (a, w) # ext(U )_ext(W ) : : a i( w, , i &,*) =&F& f *& .
i=1
Then we have
max
(a, w) # M(F, f *)
\
l
+
: a i ( w, ,*&,) >0,
i=1
for all f # S.
Let
#=
inf
max
, # S "[,*] (a, w) # M(F, f *)
{\
l
: ai
i=1
+
w,
,*&,
& f *& f &
= .
Then #>0, since S is finite dimensional. Further for any , # S,
l
l
\
l
+
: a i( w, , i &,) = : a i( w, , i &,*) + : a i ( w, ,*&,).
i=1
i=1
i=1
For any (a, w) # M(F, f *), it follows that
&F& f &&&F& f *&
max
(a, w) # M(F, f *)
\
l
+
: a i ( w, ,*&,)
i=1
# &,*&,& A .
This implies that
&F& f &&F& f *&+# &,&,*& A
and the proof is complete.
for all f # S,
K
Definition 4. For any normed linear space [E, & } &], the modulus of
convexity is defined by
$ E (=)=inf [1& 12 &x+ y& : x, y # E, &x& y&==, &x&=&y&=1],
for 0<=2.
Definition 5 [2, 4, 11]. E is said to be uniformly convex if $ E (=)>0 for
any 0<=2. A uniformly convex space E is p-uniformly convex (or has
modulus of convexity of power type p) if for some c>0, $ E (=)c= p.
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LI AND WATSON
Examples of uniformly convex spaces are Hilbert spaces and the L p
spaces, 1<p<. In fact, L p spaces are 2-uniformly convex if 1<p2, and
p-uniformly convex if p>2.
Remark. Let
d p =inf
{
1
2
&x& p + 12 &y& p && 12 (x+ y)& p
, x, y # E, &x& y&>0 . (4.3)
&x& y& p
=
Then it follows from [2] or [11] that d p >0 if and only if E is p-uniformly
convex.
Theorem 6. Let C(X, Y ) be p-uniformly convex, let S be a convex subset
of C(X, Y ), and let F=(, 1 , ..., , l ) # C(X, Y ) l, with d(F, C)<d(F, S). Then
f *=(,*, ..., ,*) # P C (F) is a strongly unique best simultaneous approximation of order p to F, that is, there exists # p >0 such that
&F& f & p & F& f *& p +# p &,&,*& Ap ,
Proof.
for all
f =(,, ..., ,),
, # S.
Let the stated conditions hold, and let
# p( f )=
&F& f & p &&F& f *& p
.
&,&,*& Ap
Then it is sufficient to prove that
inf [# p( f ) : f =(,, ..., ,), , # S, ,{,*]>0.
Without loss of generality, suppose that C(X, Y ) is complete. Suppose also
that there exists a sequence [ f n ], with f n =( n , ..., n ), n # S, n {,* such
that
# p( f n ) 0,
as n .
(4.4)
We will show that this leads to a contradiction. Now
# p( f n )=
&F& f n & p &&F& f *& p
& n &,*& Ap
| & f n & f *&&&F& f *& |
& n &,*& A
p
&F& f *&
& n &,*& A
p
_
& _
&
&F& f *&
&F& f *&
max : a &
&
_
_& &,*& & .
& &,*& &
l
a#U
&
p
p
i
i=1
n
A
n
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A
ON BEST SIMULTANEOUS APPROXIMATION
345
Thus if [ n ] is unbounded, it follows that
& n &,*& A ,
as n .
This in turn implies that
l
lim # p( f n )max : a i >0,
n
a#U
i=1
which contradicts (4.4). Thus [ n ] is bounded, and so
&F& f n & &F& f *&,
n ,
as
by definition of # p( f ). Thus there exists f =(,, ..., , ), , # clo(S), where clo(S)
denotes the closure of S, and a subsequence of [ f n ] (which we do not
rename) such that n , weakly. Note that C(X, Y ) is reflexive by the
assumption of p-uniform convexity [4].
Now let a=(a 1 , ..., a l ) # U, w # W such that
l
: a i( w, , i &,*) =&F& f *&,
i=1
l
: a i( w, ,*&,) 0,
for all , # S,
(4.5)
i=1
using Theorem 1. Since d(F, C)<d(F, S), by Lemma 1 li=1 a i {0. Then
lim
n
"
l
l
: a i (, i & n )+ : a i (, i &,*)
i=1
i=1
l
"
A
l
w, : a (, & )+ : a (, &,*)
= lim w, : a (, & ) +&F& f *&
= w, : a (, &, ) +&F& f *&,
using (4.5),
w, : a (, &,*) +&F& f *&,
lim
i
n
i
n
i=1
i
i
i=1
l
i
n
i
n
i=1
l
i
i
i
i
i=1
l
i=1
=2 &F& f *&.
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346
LI AND WATSON
Also
lim
n
"
l
: a i (, i & n )
" lim &F& f &=&F& f *&,
A
i=1
n
n
and
l
"
: a i (, i &,*)
i=1
"
=&F& f *&,
A
and so
"
l
l
: a i (, i &, n )& : a i (, i &,*)
i=1
i=1
"
0,
as
n ,
A
using the p-uniform convexity property of C(X, Y ). Since li=1 a i {0, it
follows that
& f n & f *& 0,
n .
as
Now let the sequence [a n =(a n1 , ..., a nl )] # U be such that
"
l
"
: a ni(, i & 12 ( n +,*))
i=1
=&F& 12 ( f n + f *)&.
(4.6)
A
Then
&F& 12 ( f n + f *)& 12
"
l
: a ni(, i & n )
i=1
1
2
"
+ 12
A
"
l
: a ni(, i &,*)
i=1
"
A
1
2
&F& f n &+ &F& f *&.
Also
lim &F& 12 ( f n + f *)&= 12 lim (&F& f n &+&F& f *&)=&F& f *&.
n
n
Thus
lim
n
"
l
"
: a ni(, i & n )
i=1
=&F& f *&.
A
Since li=1 a ni =0 implies that 12 ( f n + f *) # P C (F ), then
d(F, S)&F& 12 ( f n + f *)&=d(F, C),
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(4.7)
347
ON BEST SIMULTANEOUS APPROXIMATION
which contradicts the assumption that d(F, C)<d(F, S). Thus li=1 a ni {0
for any n1. Let
}
l
}
*=inf : a ni .
n
i=1
(4.8)
Assume *=0. Then, going to a subsequence if necessary, we must have
a ni a 0i , i=1, ..., l with li=1 a 0i =0. Since & f n & f *& 0,
"
l
: a 0i (, i &,*)
i=1
"
"
= lim
n
A
l
: a ni(, i & n )
i=1
"
A
=&F& f *&,
using (4.7). This implies that d(F, C)=d(F, S), which is a contradiction.
Thus *>0. Now for any n1,
&F& f *& p &F& 12 ( f n + f *)& p
l
l
p
"_
&" ,
" : a (, & )" + " : a (, &,*)"
&d : a ( &,*)
"
"
1
2
=
: a ni(, i & n )+ : a ni(, i &,*)
i=1
l
l
n
i
1
2
i
n
i
1
2
n
A
i=1
p
i
i=1
l
by (4.6),
A
i=1
A
p
n
i
p
n
A
i=1
p
&F& f n & + &F& f *& p &* p d p & n &,*& Ap ,
1
2
1
2
using the above remark and (4.8). Hence
&F& f *& p &F& f n & 2 &2* p d p & n &,*& Ap ,
and
# p( f n )2* p d p >0.
This is a contradiction and the theorem is proved. K
It is easy to give examples of spaces which satisfy the conditions of this
Theorem. For example, let Y be a p-uniformly convex Banach space, and let
X be a measure space. Then C(X, Y ) is p-uniformly convex with the norm
&,& A =
{|
X
&,(t)& Yp
=
1 p
,
1p<.
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348
LI AND WATSON
ACKNOWLEDGMENT
The work of the first author was supported by the National Nature Science Foundation of
China.
REFERENCES
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2. B. Beauzamy, ``Introduction to Banach Spaces and their Geometry,'' North-Holland,
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3. D. Braess, ``Nonlinear Approximation Theory,'' Springer-Verlag, Berlin, 1986.
4. J. Lindenstrauss and L. Tzafriri, ``Classical Banach Spaces II,'' Springer-Verlag, Berlin,
1977.
5. G. Nurnberger, ``Approximation by Spline Functions,'' Springer-Verlag, Berlin, 1989.
6. A. Pinkus, Uniqueness in vector-valued approximation, J. Approx. Theory 73 (1993),
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7. I. Singer, ``Best Approximations in Normed Linear Spaces by Elements of Linear Subspaces,'' Springer-Verlag, Berlin, 1970.
8. S. Tanimoto, A characterization of best simultaneous approximations, J. Approx. Theory
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9. G. A. Watson, A characterization of best simultaneous approximations, J. Approx. Theory
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manifolds, Amer. J. Math. 18 (1971), 350366.
11. H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. Methods Appl.
16 (1991), 11271138.
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