L1-APPROXIMATION OF DIFFERENTIABLE FUNCTIONS. ANDRÃS KROÃ, MANFRED SOMMER AND HANS STRAUSS. (Communicated by J. Marshall Ash).
PROCEEDINGS OF THE AMERICAN MATHEMATICALSOCIETY Volume 106, Number 4, August 1989
ON STRONG UNIQUENESS IN ONE-SIDED L1-APPROXIMATION OF DIFFERENTIABLE FUNCTIONS ANDRÁSKROÓ, MANFRED SOMMERAND HANS STRAUSS (Communicated by J. Marshall Ash) Abstract. We consider the problem of one-sided Z.1-approximation in C ' [0,1 ]. A sharp estimate for the rate of strong uniqueness for arbitrary unicity subspaces of C'[0,1] is given.
Introduction Let Un — span[p,, ■■■,Pn] be an «-dimensional subspace of the space C[0,1] of real continuous functions. Then uf e Un is called a best one-sided
L1-approximation of /e
C[0,1] if Uj-< f on [0,1] and \\f - uÁ\ < \\f - u\\
for all u e Un satisfying u < f on [0,1]. (Here || • || denotes the usual L'-norm on [0,1]). In order the ensure existence of best approximants we shall assume throughout the paper without further reference that Un contains a strictly positive function on [0,1]. It is well known that for the uniqueness of best one-sided L1-approximants it is necessary to restrict the consideration to the space C [0,1] of continuously differentiable functions (see [4] for details.) We shall say that Un c Cl [0,1 ] is a unicity subspace of C'[0,1] if every f e C [0,1] has a unique best one-sided L -approximant
in Un . Let Un be a unicity subspace of C [0, If and assume that O is the best approximant of a given / e C'[0,1]. (This can be always achieved by subtracting from a given function its best approximant.) Then the best approximant is called strongly unique of order y(y > 1) if for all u e Un such that u < f
on [0,1] and \\f - u\\ < \\f\\ + 1 we have
(i)
11/-«H>11/11 +*>li',
where Kf > 0 depends only on / and Un. (The reason why we consider only m's G Un satisfying ||/-w|| 1 and ||w|| —► oo.) If y = 1 then we simply say that the best approximant is strongly unique. More generally, the problem of strong uniqueness consists in finding a positive continuous function q>AS)—»0, S—►+0 such that
\\f-u\\>\\f\\
+ \\u\\
11/11 +c1||M||i//-1(c2||M||),
where c,, c2 > 0 depend only on f and Un .
If, in particular, HAS) = 0(Sa), 0 < a < 1, i.e. f and derivatives of elements of Un are Lip a functions, then it follows from (2) that strong uniqueness or order (a + I)/a takes place. Our second statement shows that the order of strong uniqueness given above is, in general, the best possible. Theorem 2. Let Un(n > 2) be a unicity subspace of Cl[0,1]. Then there exists a geC [0,1] with O being its best one-sided Ll-approximant and a sequence of functions uk 6 Un, uk < g(k =1,2,...), uk —► 0 ( 0 independent of k
(3)
\\g-ukll. Preliminary
results
We shall need several known results from the theory of one-sided L -approxi-
mation. For /€C'[0,1] X€(0,1)}.
set Z,(/) = {x € [0,1]: f(x) = 0, and f{x) = 0 if
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ONE-SIDED L1-APPROXIMATION OF DIFFERENTIABLE FUNCTIONS
Proposition 1 [1, p. 80-82]. Let Un C c'[0,1],
/eC'[0,l],
1013
In order that
O be a best one-sided L -approximant of f in Un it is necessary and sufficient that / > 0 and there exist points x{, ... , xk e Z¡ (/) ( 1 < k < n) and positive
numbers Aj,... ,Xk such that /•l
(4)
k
/ m = ^A/m(xi),
J°
i=i
Proposition 2 [4]. Leí Í7n c C'[0,1]. alent:
ueUn.
Then the following statements are equiv-
(i) Un is a unicity subspace of C1 [0,1 ] ;
(ii) there do not exist a u e Un\{0}, points {jc(}/=1 in ZAu*) (1 < k < n) and positive numbers {A,},=1 so that (4) holds for all ue Un.
The next proposition shows the existence of positive quadrature formulas with less than n active points. Proposition 3 [3]. Let Un be an n-dimensional subspace of C[0,1] (n > 2) containing a strictly positive function. Then there exist points x{, ... ,xk e
[0,1],
1 < k < n —1, and positive numbers k{, ... ,Xk so that (4) holds for
every ue Un.
Proof of Theorem 1. Let Un,f and u e i^„\{0} satisfy the assumptions of the theorem and set e — \\f - u\\ - \\f\\ ; then 0 < e < 1. Since O is the best one-sided L¡-approximant of / by Proposition 1 there exist points {xi}i=l in Zx{f), 1 < k < n , and positive numbers {A,};=1 such that (4) holds for every element of Un . In particular,
(5) e=||/- «H - 11/11 =./o¡\f-u)- Jof f=-fuJo =-¿¿..«(x,.). ;=1 Furthermore,
u(x¿) < f(x¡) = 0,
1 < / < k, hence applying this in (5) we
obtain (6) \u(xi)\ 0
if x € [0,1], x ¿ x¡,0 < i < kQ+ 1 and bAy/(ö)< co(g' ,S) < b^{S).
By
Proposition 1 O is the best one-sided L -approximant of g. Since 1 < k0 < n - 1, there exists ape Un\{0} such that p(x¡) = 0, 1 < i < kQ. Set
(11)
Mk= max (±p(x) - g(xf) = ±p(ik) - g{Zk),
0 < Çk< 1.
Proposition 2 implies that among x¡ 's, 1 < i < kQ, there should be at least one Xj e (0,1) for which p'(Xj) ^ 0. Since x¡ e Z{(g), I < i < k0, it follows that Mk > 0 for every k = 1,2, ... . Furthermore, min0(•*,)= ° »hence
/•' (14)
1 y1 , , H-\bJk)
Jo
K Jo
K
H1\bjk^ "°7 k Furthermore (13) implies that uk(x) = p(x)/k + o(l/k) for k large enough. This and (14) yields
.,
that is ||uj| > b%/k
, H-\bJk) 2 f^J\x) = 0, 2 < j < r, whenever x e Z, (/). (This can be proved analogously to Theorem 1). References 1. R. Hettich and P. Zencke, Numerische methoden der approximation und semi-infinite Optimierung, Teubner-Verlag, Stuttgart, 1982. 2. A. Kroó, On strong unicity of Lvapproximation, Proc. Amer. Math. Soc. 83 (1981), 725-729. 3. G. Nürenberger, Unicity in one-sided L¡-approximation and quadrature formulae, J. Approx.
Theory 45 (1985), 271-279. 4. A. Pinkus and H. Strauss, One-sided Lx-approximation to dijferentiable functions, J. Approx.
Theory Appl. 3 (1987), 81-96. Mathematical Institute of the Reáltanoda u. 13-15, H-1053 Hungary
Hungarian
Mathematisch-Geographische Fakultät, Eichstätt, Federal Republic of Germany Institut für Angewandte Mathematik, Erlangen, Federal Republic of Germany
Academy
Katholische Universität
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of
Universität
Sciences,
Eichstätt,
Erlangen-Nürnberg,
Budapest,
D-8078 D-8520
