A Class of Best Simultaneous Approximation Problems - CiteSeerX

Computers Math. Applic. Vol. 31, No. 10, pp. 45-53, 1996

Pergamon

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A Class of B e s t S i m u l t a n e o u s Approximation Problems CHONG LI Department of Mathematics, Hangzhou Institute of Commerce Hangzhou, P.R. China G. A. WATSON Department of Mathematics and Computer Science, University of Dundee Dundee DD1 4HN, Scotland

(Received January 1996; accepted February 1996) A b s t r a c t - - F o r a general class of best simultaneous approximation problems, characterization and uniqueness results are established. K e y w o r d s - - B e s t approximation, Simultaneous approximation, Characterization, Uniqueness.

1. I N T R O D U C T I O N There has recently been increased interest in vector valued approximation, and in particular, in the special case of best approximation of a finite number of functions simultaneously [1-6]. Specifically in [3], characterization and uniqueness results are given for a class of problems involving Lp norms. In this paper we give a general treatment of a class of problems which includes these problems as special cases. In the first section, characterization results are proved, and in the following section uniqueness is considered. For simplicity we consider the simultaneous approximation of two functions: the extension to the case of any finite number of functions is straightforward. The setting for the problems considered here is as follows. Let (X, ]I.H) be a normed linear space, and let H-HA be a monotonic norm on II~2: I[.[[A is a monotonic norm if for any a, b, c, d in ]R with [a[ < [cl, [bl , (y*,s>)>A : s • S } .

Then (4) shows that s* • P s ( F ) if and only if fs* E SI is a best Chebyshev approximation to f F from S f . For (x, y) • X x X, let G ( x , y ) = {(b*,x*,y*) e ~ : _ O,

for all s E S. Using the Krein-Milman theorem for b*, then for x* and then for y*, the result (1) follows. | We now restrict attention to approximation from a linear subset of X, so that

where y l , . . . , y n are linearly independent elements of X. The coefficients c i may be allowed to take any value in R or they may have restrictions placed upon them. To take into account the latter possibility, we consider a general class of approximation problems based on the concept of an RS-set: this name seems to have originated in [9], motivated by work of Rozema and Smith in [10]. DEFINITION 2. A n n-dimensional subspace S of X is called an interpolating subspace if no nontrivial linear combination of n linearly independent extreme points of the dual ball W annihilates S.

DEFINITION 3. Let Yl , . . . , Y,~ be n linearly independent elements of X . We call

:1

an RS-set if each Ji is a subset of R of one of the following types:

(I) the whole of R, (II) a nontrivial proper closed (bounded or unbounded) interval of R, (III) a singleton,

48

C. LI

AND

G. A. WATSON

and in addition every subset of { Y l , . . . , Y,,) consisting o f all yi with Ji of t y p e (I) and s o m e yi with Ji o f type (II) spans an interpolating subspace. For Ji o f type (II) define ai = inf Ji, ~3i = sup Ji. n

.

For s* = ~ i = 1 ci Yi, define

{i: c i

J+(s*)

J - ( s * ) -- {i : ci•

=

~}

g(s*) = J+ (s*) U J - (s*) . Define ai (s*) = l

if i E J+ (s*)

and

-1

if i E J_ (s*) .

Finally, let P = s =

ciyi : ci = 0 for Ji of type

.

k i=1

THEOREM 2. Let S be an RS-set of X , and let F = ( x , y ) , where x , y E X . T h e n s* E P s ( F ) , ff and only if there exist (b~,x*,y~) E E ( x - s * , y - s*), i = 1 , . . . , l and j l + l , . . . , J m E J(s*), m with 1 < m 0 , . . . ,%n > O, ~-~i=l"Yi = 1 such that for any n s = ~ i = 1 ciyi E P, l

rn

0.

Therefore,

max

(b* ,x* ,y*)EE(x-x*,y-y*)

This implies t h a t s* = 0 E

II so

d(F, X) < d(F, S).

Ps(F). -

/b*, ((x*, s), (y*, s)))~ > 0,

for all s E S.

But for c = t 2 - 1/2 E C [ - 1 , 1]

tl, Ily - cll Ill = 1 < II ,ix - s*ll, II , -

*ll II1,

Now take s~ = k E S , - 1 / 2 < k < 1/2. T h e n

IIx- s~ll = I1 - kl,

t l y - s~ll = I1 + kl,

II I1~- s~ll, I l y - s~ll II, -- 2. Thus, we have s~ E Ps(F), so t h a t the solution is not unique. For the special cases when II.IIA is the 11 n o r m on R 2, and X = L 1 or X = l~(t), it is possible to give a more direct approach to the problem [14]. Let C be a set, (C, tt) a measure space and X = L i ( C , # ) as in [3]. Let Y = {1,2} and #1(1) = #1(2) = 1, with A = # i x # the p r o d u c t measure on Y x C. It is easy to see t h a t s* E S C L i ( C , # ) is a best 11 simultaneous a p p r o x i m a t i o n to x and y in L i ( C , #) if and only if fs* E S I is a best L 1 approximation to fF in L I ( Y x C, A) where

fF(i,t)

S x(t), i = 1 , y(t),

i = 2,

for all (i, t) E Y x C, and

Sf = {f~(i,t) = s(t), N e x t let X =

l~(t) as in rt

[31, and define 121,~(t-) as follows for

Now for x = ( x l , . . . , x ~ ) ,

t = (tl,... ,t~,tl,...

,t,~):

2n

Itxiit = E laiiti + E i=1

for all s e S } .

lailti-n'

forall

(al,...,a2n) E l~,(t).

i=n+l

y = (Yl,...,Y~) E

ln(t)

f~ = (xl,...,zn,ul

define

.... ,y,) E l~n(t),

S f = { f s = ( 8 1 , . . . , 8rt, 8 1 , . . . , 8 n ) , for all s = ( s l , . . . , sn) E S } .

T h e n s* E S is a best l I simultaneous approximation to x and y in l~(t) if and only if f~. E S f is a best l 1 approximation to fF E l~n(t-). T h u s using results (for example [15]) on best L 1 or 11 a p p r o x i m a t i o n to an element in L 1 or l~n(t-), we can easily get the results of Sections 3 and 4 in [3]. In particular, T h e o r e m s 3.1 and 3.3 in [3] are T h e o r e m 2.1 and Corollary 2.5, respectively, in [15] for LI(C, A); T h e o r e m s 4.4 and 4.1 in [3] are Theorems 6.5 and 6.1, respectively, in [15] for l~n(t-). T h e only difference is t h a t S in [3] is convex, while it is linear of dimension n in [15]. Note t h a t the best lp approximation of an infinite n u m b e r of elements of X is considered in [5].

Best Simultaneous Approximation Problems

53

REFERENCES 1. A. Pinkus, Uniqueness in vector valued approximation, J. Approx. Theory 73, 17-92 (1993). 2. G.A. Watson, A characterization of best simultaneous approximations, J. Approx. Theory 75, 175-182 (1993). 3. J. Shi and R. Huotari, Simultaneous approximation from convex sets, Computers Math. Applic. 30, 197-206 (1995). 4. C. Li and G.A. Watson, On best simultaneous approximation, Preprint (1995). 5. C. Li, Best simultaneous approximation by RS-sets, Num. Math. (JCU) 15, 62-71 (1993). 6. S. Tanimoto, A characterization of best simultaneous approximations, J. Approx. Theory 59,359 361 (1989). 7. F.L. Bauer, J. Stoer and C. Witzgall, Absolute and monotonic norms, Numer. Math. 3, 257-264 (1961). 8. D. Braess, Nonlinear Approximation Theory, Springer-Verlag, New York, (1986). 9. D. Amir, Uniqueness of best simultaneous approximations and strictly interpolating snbspaces, J. Approx. Theory 40, 196-201 (1984). 10. E.R.. Rozema and P.W. Smith, Global approximation with bounded coefficients, J. Approx. Theory 16, 162-174 (1976). 11. B.L. Chalmers and G.D. Taylor, A unified theory of strong uniqueness in uniform approximation with constraints, J. Approx. Theory 37, 29-43 (1983). 12. I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, New York, (1970). 13. G.A. Watson, Approximation Theory and Numerical Methods, John Wiley, Chichester, (1980). 14. C. Li, Nonlinear simultaneous Lp approximation, J. of Zhejiang Normal University 14, 14-22 (1987). 15. A. Pinkus, On L1-Approximation, Cambridge University Press, (1989).