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OF APPROXIMATION

THEORY

58,

358-360

( 1989)

Note A Note on Strong

Uniqueness

Constants

R. GIKHHMANN Karholische Unicersircil, Eichsralr 8078, Wesr Germany Communicated hv E. W. Chenev

Received November 12, 1987

We expand a result of Blatt, concerning the strong uniqueness constants of c) lY8Y Academic Press. Inc. uniform best approximations on [ - 1, I].

Let X be compact, f~ C(X), and V be a subspace of C(X), the space of all real-valued continuous functions on X. Let u E I/ be a best uniform approximation to f‘~ C(X), i.e., a best approximation with respect to the uniform norm

11~11 = llrllx := y:; Irb)l. u is then called a strongly unique best approximation with

IIf‘-wil 3 Ilf-ull +Y I!+-VII

for all

to f; if there is a y > 0 WE V.

(1)

The largest constant satisfying (1) is called the strong uniqueness constant

r(f) and (seePI) Y(f) =

inf SUP (w H’Ev, :w1= I xE,c(r)

r(x)) w(x),

where r =,f - v and E(r) := (xE X: [r(x)1 = llrll }. Following

[5] we define for a Banach space E and a subspace U c E i-( lJ, E) := inf{ llpll: p: E--f lJ is a projection}.

358 0021-9045/89$3.00 Copyrighl 8) 1989 by Academic Press, Inc. All rights of reproduction in any form rrwrved.

(2)

359

STRONG UNIQUENESSCONSTANTS THEOREM

1. LLI v be a strongly unique best approximation &f-‘)~

Prooj:

4 VI Ecrj, W(r))) 4v C(W)

to jl Then

.

By (2) r(f)

d

inf

,“~V.‘Wi7, IIWIIL(,)

= (sup{ IIwII: IIM’IIE(r)<1, )t’E V})-. I. Let T: C(E(r)) -+ Vls(rI be any projection. Define 4: V/E(r) + V by fj(wIE(r))=w. 4 is well defined, since otherwise there is a WE V with IIHJI(= 1 and w I E,r,= 0, contradicting y(.f’) > 0. Define for f~ C(X)

Then

4 K CW < IISII< II41 liTI GUT’

IITII.

Thus

4K C(W)

1.

4VIf+,t

C(E(r))).

1

Let V=n,,

X=[a,b],

and IE(r)l=m+n+l.

Then

l+J;;; 7P r(f) = 7”(f) G log(n+ 1)‘4’ ProoJ Since dim Z7,I E(,I=n+ from [S, p. 3413

Wni.,r,~ C(E(r))!G n::: (

1 and dimC(E(r))=nim+

1+

Jtn+m)(n+ n+m+

It is a well-known result (see, e.g., [6]) that i-( V, C(X)) 2 -$ log(n + 1).

Thus the result follows from Theorem 1. 1

1). 1

1, we get

360

R.GROTHMANN

This is sharper than a result of Blatt [l]. Furthermore, the proof is easier, since we used the functional analytic result of KGnig et al. in Theorem 1. COROLLARY 2. Let f~ C” [a, b], and amume that [email protected]+l) has o(log(n)2) zeros in [a, b] (as n + cc ). Then

lim inf y,(f) = 0. n-+m

(3)

ProoJ: With the equioscillation theorem [6] and multiple application of Rolle’s theorem, one can see that

IW,)l = 12+ 410g(n)2), where Y, denotes the error function of the best approximation with respect to fin. The result follows now from Corollary 1. j Corollary 2 gives a partial answer to a question which was raised by Poreda [7]. It has been conjectured in [4], that all nonpolynomial f~ C(X) satisfy (3). A further result in this direction can be found in [3].

REFERENCES 1. H.-P. BLATT, “Exchange Algorithms, Error Estimations and Strong Unicity Constants,” NATO AS1 series, Series C, Math. and Phys. Sciences,Vol. 136, Dordrecht, 1984. 2. M. W. BARTELTAND H. W. MCLAUGHLIN,Characterizations of strong unicity constants in approximation theory, J. Approx. Theory 9 (1973) 255-260. 3. R. GROTHMANN,On the Real CF-method and strong uniqueness constants, J. Approx. Theory 55 (1988), 86103. 4. M. S. HENRY AND J. A. ROULIER,Lipschitz and strong unicity constants for changing dimension, J. Approx. Theory 22 (1978), 8684. 5. H. KOENIG,D. R. LEWIS,AND P.-K. LIN, Finite dimensional projection constants, Studiu Math. 75 (1983), 341-358. 6. G. MEINARDUS,“Approximation of Functions,” Springer-Verlag, New York/Berlin, 1967. 7. S. J. POREDA,Counterexamples in best approximation, Proc. Amer. Math. Sot. 56 (1976), 167-171.