Uniqueness of simultaneous approximations in continuous function spaces

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: [email protected] (L. Peng), [email protected] (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 Kro´o (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. References [1] D. Amir, Uniqueness of best simultaneous approximation and strictly interpolating subspaces, J. Approx. Theory 40 (1984) 196–201. [2] C. Li, Best simultaneous approximation by RS-sets, Numer. Math. J. Chinese Univ. 15 (1993) 62–71 (in Chinese). [3] X.F. Luo, J.S. He, C. Li, On best simultaneous approximation from suns to an infinite sequence in Banach spaces, Acta Math. Sinica (Chin. Ser.) 45 (2002) 287–294. [4] C. Li, G.A. Watson, On a class of best simultaneous subspace in approximation, Comput. Math. Appl. 31 (1996) 45–53. [5] C. Li, G.A. Watson, On best simultaneous approximation problems, J. Approx. Theory 91 (1997) 332–348. [6] C. Li, G.A. Watson, Best simultaneous approximation of an infinite set of function, Comput. Math. Appl. 37 (1999) 1–9. [7] J. Shi, R. Huotari, Simultaneous approximation from convex sets, Comput. Math. Appl. 30 (1996) 197–206. [8] S. Tanimoto, A characterization of best simultaneous approximation problems, J. Approx. Theory 59 (1989) 359–361. [9] G.A. Watson, A characterization of best simultaneous approximation, J. Approx. Theory 75 (1993) 175–182. [10] A. Pinkus, Uniqueness in vector-valued approximation, J. Approx. Theory 73 (1993) 17–92. [11] A. Kro´o, Best L 1 -approximation with varying weights, Proc. Amer. Math. Soc. 99 (1987) 66–70. [12] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, New York, Berlin, 1970. [13] A. Kro´o, On an L 1 -approximation problem, Proc. Amer. Math. Soc. 94 (1985) 406–410. [14] W. Li, Weak Chebyshev subspaces and A-subspaces of C[a, b], Trans. Amer. Math. Soc. 322 (1990) 583–591. [15] A. Pinkus, Unicity subspaces in L 1 -approximation, J. Approx. Theory 48 (1986) 226–250. [16] A. Pinkus, On L 1 -approximation, in: Cambridge Tracts in Mathematics, vol. 93, Cambridge University Press, Cambridge, 1989. [17] A. Pinkus, B. Wajnryb, Necessary conditions for uniqueness in L 1 -approximation, J. Approx. Theory 53 (1988) 54–66. [18] D. Schmidt, A theorem on weighted L 1 -approximation, Proc. Amer. Math. Soc. 101 (1987) 81–84.