Approximation of Monotone Functions: A Counter Example R. A. DeVore, D. Leviatan and I. A. Shevchuk Abstract.
When we approximate a continuous nondecreasing function als be nondecreasing. However, this constraint restricts very much the degree of approximation that the polynomials can achieve, namely, only the rate of !2 (f; 1=n). It turns out as we will prove somewhere else that relaxing the monotonicity requirement in intervals of length 1=n2 near the endpoints allows the polynomials to achieve the rate of !3 . On the other hand, we show in this paper, that even when we relax the requirement of monotonicity of the polynomials on sets of measures approaching 0, (no matter how slowly or how fast), !4 is not reachable.
f in [?1; 1], we wish sometimes that also the approximating polynomi-
x1. Introduction Let f 2 C [?1; 1] be nondecreasing on I := [?1; 1]. Then DeVore [1] proved that there exist nondecreasing polynomials such that
kf ? PnkC(I ) c !2(f; 1=n); (1:1) where c is an absolute constant and !k (f ; ) denotes the modulus of smoothness of order k, of f . (See also pointwise estimates by DeVore and Yu [2] and similar estimates involving the second Ditzian- Totik modulus of smoothness by Leviatan [3].) On the other hand it is known (see Shvedov [4]) that in (1.1) one cannot replace !2 by !k with any k 3. It is quite natural to ask whether one can strengthen (1.1) in the sense of being able to replace !2 by moduli of smoothness of higher order, if one is willing to allow Pn not to be monotone on a rather \small" subset of I . This indeed is the case. If we allow the polynomials not to be monotone in intervals of length 1=n2 near the end points, then it is possible to achieve the estimates kf ? pnk c !3(f; 1=n): (1:2) Proceedings of Chamonix 1996 A. Le Mehaute, C. Rabut, and L. L. Schumaker (eds.), pp. 95{102. Copyright c 1997 by Vanderbilt University Press, Nashville, TN. ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.
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R. A. DeVore, D. Leviatan and I. A. Shevchuk
We will prove that as a special case of a more general result (pointwise estimates for comonotone approximation) in another paper. However, even this improvement comes to a halt; it cannot be extended to !4 , and thus not to !k for any k > 3. In order to state our theorem we need some notation. Given > 0 and a nondecreasing function f 2 C [?1; 1], we denote (1)
En
kf ? PnkC(I ); (f ; ) := inf P n
where the in mum is taken over all polynomials Pn of degree not exceeding satisfying meas(fx : Pn0 (x) 0g \ I ) 2 ? : Theorem. For each sequence = fng1n=1, of nonnegative numbers tending to 0, there exists a nondecreasing function f := f 2 C [?1; 1] such that n
n ) lim sup ! (f;(f1;=n ) = 1: n!1 4 (1)
En
(1:3)
Remarks 1. A weaker version of our theorem, where Pn is required to be
monotone in I , that is, under the stronger condition that n = 0, n = 1; 2; : : :, is due to Wu and Zhou [5]. One should note that Wu and Zhou have a single function while Shvedov [4] obtained for an arbitrarily large prescribed c, and for each n, a di erent function fn;c, which violates (1.1). 2. Note that we allow the relaxing of monotonicity (on small sets) anywhere in I , not necessarily near the endpoints, and !4 cannot be had.
x2. A Counter Example (Proof of the Theorem)
While above we have used c as an absolute constant which may di er on di erent occurrences, in this section we will have to keep track of the constants, therefore we denote them by C1; C2; : : :. We begin by recalling some simple properties of the Chebyshev polynomials for the interval [?2; 2]. For > 1, let x t (x) := cos arccos ; x 2 [?2; 2]; 2 denote the Chebyshev polynomial and let zj := 2 cos j , j = 0; : : : ; , be its extrema. Given 0 < b < 21 , we take two points on both sides of zj , (j +b) (j ?b) j = 1; : : : ; ? 1, namely, we set zj;l := 2 cos( ) and zj;r := 2 cos( ). Note that jt (zj;l )j = jt (zj;r )j = cos b; and b b j sin < 4 : (2:1) zj;r ? zj;l = 4 sin
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Approximation of Monotone Functions
We truncate the Chebyshev polynomial by setting 8 < cos b; t (x) > cos b t (x) := t;b (x) := b; t (x) < ? cos b : ?t (cos x); otherwise. Since 2 1 ? cos b = 2 sin2 b < 5b ; (2:2) 2 for any x 2 I , it follows by the monotonicity of the areas as we go away from the origin, and the alternation in sign of these areas, that
j
Zx
(t (u) ? t (u)) duj j
Z2
z[ ];r
z[ ];l
0
2
3
(t (u) ? t (u)) duj < 4 b 5b2 = C1 b ; (2:3)
where we have applied also (2.1). Now, given n 1 and 0 < b < 21 , let := [b 34 n] + 2, where [a] denotes the largest integer not exceeding a. Put t;b := t + cos b; and t~;b := t;b + cos b: Finally, ( ) :=
Zx
( )
t;b u du
T;b x
and
( ) :=
Zx
~ ( )
t;b u du;
fn;b x
x
2 I:
0
0
Obviously fn;b is a nondecreasing function on I and it readily follows by (2.3) that 9 b3 b4 kfn;b ? T;bk C1 C1 n ; (2:4) where we denote by k kJ the max-norm taken on the interval J , and when the norm is on I , we suppress the subscript. If we set z~j;l := 2 cos ( (j+b= 2) ), and z~j;r := 2 cos ( (j?b= 2) ), then we have for all j for which zj 2 I , ~ ? zj = 4 sin
zj;r
and similarly,
(j ? 4b )
sin b 4
>
b
;
(2:5)
(2:50 ) Let j be odd. Since sin b=4 > 3b=4 for b satisfying b=4 < =6, we have 0 (x) = t;b (x) ? cos b=2 + cos b = ?2 sin b=4 sin 3b=4 T;b (2:6) 3b 3b = ? 9b2 ; x 2 [~z ; z~ ]: < ?2 j;l j;r 4 2 4 zj
? z~j;l > b :
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R. A. DeVore, D. Leviatan and I. A. Shevchuk
Then since I [?2; 2], it follows by the Bernstein inequality that 3
C2 (4) (3) kT;b k = kt(3) kt k = C 3: ;b k = kt k (1 ? ( 1 )2 )3=2 [?2;2] 3 2
Hence, by (2.4), (
!4 fn;b ;
1 ) ! (f ? T ; 1 ) + ! (T ; 1 ) 4 n;b ;b 4 ;b n n n 1 (4) 4 2 kf ? T k + kT k n;b
2 C1 4
b9=4 n
;b
+
n4
C3 3 n4
;b
C4
b9=4 n
(2:7)
;
by the relation between and n. Next we need a simple lemma. Lemma 1. There exists a constant C5 such that for any interval J I , we have the following. For any measurable sets E I , if 0 ( ) 0;
Pn x
then
x
2 J n E;
(2:8)
kfn;b ? PnkJ b njJ j ? Cn5 (b9=4 + bjE j + b n ): 5=4
2
(2:9)
Proof: Let J0 denote the middle third of J . We consider two cases. First we assume that J0 contains at most one of the zj 's. Then by the de nition of we get ?3=4 jJ j < C 1 < C b : 6
Hence
6
n
kfn;b ? PnkJ 0 > b njJ j ? C6 bn2
5=4
2
:
(2:10)
On the other hand, if J0 contains at least two extrema zj , then it contains at least 2C7 jJ j extrema, for some constant C7. These extrema satisfy (2.5) and (2.50 ), and about half of them (and at least one) have odd indices, then together with (2.6) we conclude that meas(J0 \ fx
9 b2 0 : T;b(x) < ? g) 1 2b C7 jJ j = C7bjJ j: 4
2
(2:11)
Now, if C7bjJ j jE j, then 2 jE j : kfn;b ? PnkJ 0 b njJ j ? bnC 7
(2:12)
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Approximation of Monotone Functions
Otherwise, which
j j
j j. Then by (2.11) there is a point x0 2 J0 n E , for
C7 b J > E
2 ( ) ? 94b :
0 x0 < T;b
Hence, (2.8) yields,
9b2 P 0 (x ) ? T 0 (x ) 2 p n n 0 ;b 0 4 jJ j 1 ? (1=3)2 kPn ? T;bkJ ; where we have used the Bernstein inequality. Therefore from (2.4),
p j j 3 2 b2 jJ j kP ? T k n ;b J n 4 n kPn ? fn;bkJ + kfn;b ? T;bkJ 9=4 kPn ? fn;bkJ + C1 b n :
b2 J
(2:13)
Taking C5 := maxfC6; C17 ; C1 g, (2.9) now follows by combining (2.10), (2.12) and (2.13). We are now in a position to de ne f := f , for a given sequence = fng. Let bn := (maxf2n; n1 g)2=5 , and set d0 := 1, and dj
:=
9=4
bnj
nj
dj ?1
j 9=4 Y bn
=
=1
n
;
j
1;
where the sequence fn g is de ned by induction as follows. First, we choose 1=8 1 n1 so large that bn1 < 12 (as needed in (2.15) below) and J0 := I . Suppose that fn1 ; : : : ; n?1 g and J?2 J?3 J0, 2, have been de ned. Then put ?1 X dj ?1 fnj ;bn ; F?1 := j j =1
and let J?1 be an interval such that J?1 J?2 and 0 ( ) = 0;
F?1 x
x
2 J?1 :
(2:14)
(The induction process will guarantee the existence of such intervals.) Let be such that jJ?1j b1n=8 ; n N1; ; (2:15) and let !10 (2) k F?1 k : (2:16) N2; := d N1;
?
1
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R. A. DeVore, D. Leviatan and I. A. Shevchuk
Finally, we take
maxfn?1 ; N1; ; N2; g so big that the function fn0 ;bn oscillates a few times inside the interval J?1 and since it vanishes on some interval in each oscillation, that is, inside J?1 , there exists an interval J J?1 as required in (2.14). Now denote 1 X := dj?1 fnj ;bnj ; n >
j =
where the convergence of the series is justi ed by the de nition of the dj 's and the fact that kfn;bn k 2, for all n. In fact 9=4 9=4
9=4
k k 2d?1(1 + n + 1 X 2d?1 2?j = 4d?1 :
bn bn+1 n n+1
bn
+ : : :)
(2:17)
j =0
So we de ne f
:= f :=
1 X j =1
dj ?1 fnj ;bnj ;
and we prove Lemma 2. For each 1 we have !4 (f; 1=n ) C8 d : Proof: First, by (2.17) 4 6 !4 (+1 ; 1=n ) 2 k+1 k 2 d : At the same time, (2.7) yields (
1
!4 d?1 fn ;bn ; =n
Finally,
(
1
!4 F?1 ; =n
) d?1 C4
9=4
bn
n
(2:20)
!9=4
(2:21)
?
1
(2) k F?1 k ?1=10 n =4
d?1
4d ;
1
2=5
n
(2:19)
= C4d :
) 4!2(F?1 ; 1=n ) 4 kF (2) k n2
(2:18)
bn
d
by virtue of (2.16) and the de nitions of bn , d and n . Lemma 2 follows by combining (2.19), (2.20) and (2.21).
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Approximation of Monotone Functions
The last lemma that we need is Lemma 3. There is an absolute constant C9 such that whenever E I is a measurable set satisfying jE j n ; (2:22) and Pn is a polynomial satisfying 0 Pn (x) 0; x 2 I n E; (2:23) then kf ? Pn k (bn?1=8 ? C9)d : (2:24) Proof: Since F?1 is constant on J?1 , we may write (2:25) f (x) = d?1 fn ;bn (x) + +1 (x) + M; x 2 J?1 : Let 1 (P ? M ): Qn := n d ?
1
Then it follows from (2.23),
Q0n x
( ) 0;
x
2 J?1 n E:
Thus by virtue of Lemma 1, 5=4 2 kQn ? fn ;bn kJ?1 bn jnJ?1j ? Cn 5 (b9n=4 + bn jE j + bnn ):
The de nition of n and (2.15) yield,
j
bn J?1 2
=8 j = b17 n
(2:26)
! jJ?1j b17=8 : n 1=8 bn
On the other hand, (2.22) and the de nition of bn imply
j j bn n b9n=4 ;
bn E
and
5=4
bn
Hence (2.26) implies
n
=4 =4 b15 < b9 n n :
9=4 1 bn 17=8 9=4 kQn ? fn ;bn kJ?1 n (bn ? 3C5bn ) = n (bn?1=8 ? 3C5):
In other words, 9=4
kPn ? M ? d?1fn ;bn kJ?1 d?1 n (bn?1=8 ? 3C5) = d (bn?1=8 ? 3C5): bn
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R. A. DeVore, D. Leviatan and I. A. Shevchuk
In view of (2.25), it follows from (2.17) that,
kf ? Pn k kf ? Pn kJ?1 kPn ? M ? d?1 fn ;bn kJ?1 ? k+1k (bn?1=8 ? (3C5 + 4))d ;
and Lemma 3 is proved with C9 := 3C5 + 4. The proof of (1.3) now follows from Lemmas 2 and 3 since (1) (1) En (f ; n ) En (f ; n ) lim sup ! (f; 1=n) lim sup ! (f; 1=n ) n!1 !1 4 4 lim sup C1 (bn?1=8 ? C9) = 1:
!1
8
Acknowledgments. Part of this work was done while the second and third
authors were visiting the University of South Carolina. All three authors acknowledge partial support by ONR grant N00014-91-1076 and by DoD grant N00014-94-1-1163.
References 1. DeVore, R. A., Degree of approximation, in Approximation Theory II, G. G. Lorentz, C. K. Chui, and L. L. Schumaker (eds.), Academic Press, New York, 1976, 117{162. 2. DeVore, R. A. and X. M. Yu, Pointwise estimates for monotone polynomial approximation, Constr. Approx. 1 (1985), 323{331. 3. Leviatan D., Monotone and comonotone approximation revisited, J. Approx. Theory 53 (1988), 1-16. 4. Shvedov, A. S., Orders of coapproximation of functions by algebraic polynomials, Mat. Zametki 29 (1981), 117{130. 5. Wu, X. and S. P. Zhou, A problem on coapproximation of functions by algebraic polynomials in Progress in Approximation Theory, A. Pinkus and P. Nevai (eds.), Academic Press, New York, 1991, 857{866. R. A. DeVore Department of Mathematics University of South Carolina Columbia SC 29208 USA
[email protected]
D. Leviatan School of Mathematical Sciences Sackler Faculty of Exact Sciences Tel Aviv University Tel Aviv 69978, ISRAEL
[email protected]
I. A. Shevchuk Institute of Mathematics National Academy of Sciences of Ukraine Kyiv 252601, UKRAINE
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