ABSTRACT -. We give DeVore-Gopengauz-type inequalities for Boolean sum modifica- tions of convolution-type operators which are based upon arbitrary ...
A P P R O X I M A T I O N BY B O O L E A N S U M S O F P O S I T I V E L I N E A R O P E R A T O R S V: O N T H E C O N S T A N T S IN D E V O R E - G O P E N G A U Z - T Y P E I N E Q U A L I T I E S (l) A. Boos (2), JIA-DING CAO (3), H. H. GONSKA(4)
ABSTRACT We give DeVore-Gopengauz-type inequalities for Boolean sum modifications of convolution-type operators which are based upon arbitrary positive trigonometric kernels. Special emphasis is on the magnitude of the constants in front of the second order modulus of continuity. We derive upper bounds for the best possible values of these, as well as for some asymptotic constants which turn out to be rather small. -
1991 Mathematics Subject Classification: 41A25, 41A17, 41A10, 41A35, 41A29, 41A36, 41A44. KEy WORDS: degree of approximation, positive linear operators, pointwise estimates, second order modulus of continuity, approximation with constraints, convolutiontype operators, asymptotic constants. 1. I n t r o d u c t i o n
Let IN = {1, 2, ...} be the set of natural numbers. For f e C[a,b] (real-valued and continuous functions on the compact interval [a,b]), let II f II = max {] f(t) [:
(1) code of (2) (3) (4) many.
Received: 27 July 1993. This is a project supported by the Natural Science Foundation of China, project: 19171020. European Business School, D-65375 Oestrich-Winkel, Germany. Dept. of Mathematics, Fudan University, Shanghai 200433, PRC. Dept. of Mathematics, University of Duisburg, D-47048 Duisburg, Ger-
A. Boos -JIA-DING CAO H. H. GONSKA:Approximation by Boolean Sums of
290
-
a ~< t ~< b} denote the Ceby]ev norm of f. Furthermore, let /7, be the set of algebraic polynomials of degree ~< n. For f a C[a,b], the first order modulus of continuity is defined by o~, (f,5) = sup([ f(Xl) -- F(X2) l, X,, X 2 E [ a , b ] , I x , -- x 2 I "~< (~}, 0 -~< (~ ~ b - a .
The second order modulus of continuity o92(f,6) is given by
=
sup([ f ( x - h ) - 2 f ( x )
+ fix+h)I,
x, x + h
~ [a,b], 0 ~< h ~< 6}, 0 ~
(h,-1)]
299
t
and d d 1 dx G+(n)(h'x) = ~xx Gm(n)(h,x) + -~- [h(1) -- Om(n)(h,1)] -(3.1) 1 -- ~ [ h ( - 1 ) - Gm(n)(h,-1)].
I f - 1 < x < 1, then from (1.1) it follows that d dx Gm(n)(h,x) (3.2)
= --~ = --~
h'[cos(arccos x+v)] sin(arccos x+v) X/ 1 - x 2
,~ h'[cos(arccos x+v)]
(cos v +
~
x
Km(n)(V) dv
)
sin v Km(n)(V ) dv.
From (1.1), we also have
(3.3)
G~(~)(h, 1) = ~ - i
Gm(n)(h,-1) = ~ - t (3.4)
= ~-I
h(cos v) Km(n)(V) dv,
h[cos ~ -t- v)] Km(n)(V) dv,
h ( - cos v) Km(,)(v) dv. Y~
By Taylor's theorem,
h(t) = h ( - 1 ) + h ' ( - 1 ) ( t + l )
Hence
1 + - - h"(02)(t+l) 2, where - 1 < 02 < 1. 2
300
A. Boos
-JIA-DING CAO -
H. H. GONSKA: Approximation by Boolean Sums of
Gm(n)(h,-1) = h ( - 1 ) + h ' ( - 1 ) Gm(n)(t+l,-1) + 1
+ - - Gm(n)[h"(02)(t+l)2,-1]. 2 From (3.4), we have Gm(n)(t+l,-1)
= ~-l
( - cos v + l ) Km(n)(V ) dv = --Ol,m(n)+l.
/~
Furthermore, 1
1
[ h ( - 1 ) - Gm(n)(h,-1)] =
[Gm(n)(h,-1)
1
- -
h(--1)]
1
2 h ' ( - 1 ) Gm(n)(t+l,-1 ) + - - Gm(n)[h"(0~)(t+l)2,-1] 4 1
(3.5)
1
2 h'(--1)(-Ol'm(n) + 1) + - - G m ( n ) [ h " ( t ~ 2 ) ( t + l ) 2 , - 1 ] 4
Likewise, 1
h(t) = h(1) + h'(1) + - - h"(03)(t-1) 2, where - 1 < t~a < 1. 2 Thus 1
Gm(n)[h(t),l] = h(1) + h'(1) Gm(n)(t-l,1) + --~ Gm(n)[h"[03)(t-1)2,1]. From (3.3), we have Gm(n)(t-l,1) = ~-I
j'[
(cos v - l ) Km(n)(V) dv = ~l,m(n/ --1, and
1
~
[h(1) - Gm(n)(h,1)]
Positive Linear Operators V." on the Constants in Devore-Gopengauz-Type Inequalities
(3.6)
1
1
2 h'(1)Gm(.)(t-l,1)-
4 Gm(")[h"(03)(t-1)2'l]
1
1
2 [1--01'm(n)] h ' ( 1 ) -
4 Gm(n)[h"(O3)(t-1)2'l]"
301
Combining (3.5) and (3.6) gives 1
1
-2- [h(l) - Gm(n)(h,1)] - -~- [ h ( - 1 ) - Gm(n)(h,-1)] 1
1
= -~[1--0t,r.,(.) ] h'(1) + T [1--OLm(~)] h'(--1) 1
(3.7)
1
4 Gm(')[h"(03)(t-1)2'l] + --4 Gm(n)[h"(02)(t+l)2'-l]" Since sin v Km(n)(v) is an odd function, one has
h'(x)
( cos v +
x
)
sin v K,~(,)(v) dv
M 1-x 2 = h'(x) " - -
(3.8)
cos v Km(n)(V) dv + 0
= Ql,m(n)h'(x).
Combining (3.1), (3.2), (3.8) and (3.7), we have d dx G m(n)(h,x) -- h'(x)
1/;
= -zr
,~
[h'(cos(arccos x+v)) - h'(x)]
(cos v +
x V 1-x 2
)
sin v Km(.)(v) dv
+ [h'(x) Ql,m(n) -- h'(x) + __1 [l__Ql,m(n) ] h'(1) + ~ l [1--~l,m(n)] h'(--1)] 2 2 1
1
4 Gm(n)[h ( 3)(t-I)2,1] + ~ G m ( n ) [ h " ( 0 2 ) ( t + l ) 2 , - 1 ]
302
A. Boos
-JIA-DING CAO -
H. H. GONSKA: Approximation by Boolean Sums of
= :El + Ez+ E3+ E4.
9
LEMMA 5. Let h e C2[ - 1, 11. Then [ E2 [ ~< C~ 4 " II h" II, where r. is given as in L e m m a 3. PROOF.
E2 = h'(x)
=
1
1 - ~ - h ' ( - 1 ) + - - h'(1) - h'(x) 2
[1--~Ol,m(n)]
= [l--Ql,m(n)]
1
h'(x) + --h'(1)[1--Pl,m(n)] + - - h ' ( - 1 ) [ 1 - - P l , m ( n ) ] 2 2
~l,m(.) --
[Iy
= [l--0"m(n)]
1
[ h ' ( - 1 ) - h'(x)] + -~. [h'(1) - h'(x)]
1
_
]
21 h"(O4) (x+ l ) + - h " ( O s ) ( 1 - x ) 2 , where --1
~ , then z~,(x) = r, ~/ 1 - x 2 (4.1) then implies
I g(x) - W.(g,x) I ~< D~ ~. (1-x 2) 9 II g" II. Hence for n ~ IN and l x I ~< 1, we have
I gIx) - Wn(g,x) I ~ D3" ~. (1-x 2) 9 II g" II, where D3 : = max (D],D2), i.e., (4.4)
I g(x) - G+~(,)(g,x) I ~< D3 ~ ( 1 - x 2) 9 II g" II. This completes the proof. As compared to L e m m a 1 1, a somewhat simplified estimate is given in
LEMMA 12.
L e t m ( n ) ~> 2, Km(n)(V) I> 0, I x l ~< 1 a n d g e C 2 [
9
- 1, 1]. T h e n
I G+(,)(g, x) - g(x) I ~< D~ ~ ( l - x 2) 9 II g" I[, where D~ : = max (D/,D~),D{ : = 2C21 + m 3 C4 ' D ~ : = 2~/-2-C, + 4C~ + - - 1 C~. 2 2 PROOF. IfKm(.)(v) I> 0, we have (see [1]) 0 < 1 - O2,m(n) ~< 4 (1 -- OX,m(n)), and thus "k'/l--O2,m(n) ~ 2%/ 1--Pl,m(n) ~ 2 C~ an. Hence we may take CI : = 2 C1 (for the definition Of Cl see condition (ii) in L e m m a 3), so that now D1 = (C21 + - - 1 ~ ) 4 D2__2C~+%/-~-
+ - - 3 C~ = 2 C 2 + - - 3 C~ = : Ol,' and 2 2
67+--I C~
~l + _ l 2
2
Positive Linear Operators V: on the Constants in Devore-Gopengauz- Type Inequalities
= 2C~ + 2 ~
C1 + 2 C ~ + - -
1
311
C4
2 = 2 ~
C1 +4C21 + - -
1
C~ = D~.
2 In order to obtain the main results of this section, we shall use the following lemma which was recently obtained in [14] using new Steklov means of Zuk [19]. LEMMA 13. Let Hn : C[a,b] --~ C[a,b] be a sequence of linear operators satisfying the following conditions: (i) [[ Hnf[[ < N, 9 11fl[ for all f e C[a,b], 1 (ii) f o r 0 ~ < e~(x) ~ < - - - ( b 2
a) one has
[ Hn(g,x) - g(x) [ ~< N2" ~ ( x ) 9 It g" [[ for all g a C2[a,b]. 1 Then for all f ~ C[a,b] and 0 < h ~ - - (b - a), 2
I Hn(f,x) - f(x) l ~
2, Km(n)(V ) ~ 0, I x [ ~ 1, and
V 1-01,re(n) ~ CI an,
V-:
-201,m(.) + ~ - Q2,m(.)
r. = max (an,fin), 0 < v. ~< 1. Then forf~C[-
1, 1],
m
Positive Linear Operators
V: on
the Constants in Devore-Gopengauz-Type Inequalities
313
I G+(~)(f,x) - f(x) I ~< M/ o.~2( f,v. 1V-i--Z~-x2 ), where M { ' -~
-4
---max 2
2C~+
C~,2. V~-C,
+4.C2
+__C 4 " 2
PROOF. We apply Lemmas 12 and 13.
9
As an immediate consequence of Theorem 2 we have COROLLARY 1.
(i)
L e t m(n) I> 2, Km(n)(V ) ~ 0, 0 < En ~
1, a n d let
1 -- ~)l,m(n) = O(E2n),
3 (ii) - 2-
1 - 20x,.,(.) + -~- Q2,m(n) = 0(e4~).
T h e n for f e C I - 1, 1],
, G+ .,(.~(f,,,)
(
- f(x) I < C - ~
f, ~,- ~ ,
)
I • ] < 1.
Here the constant C is independent of f, x, and n. REMARK 1.
1
For the case e, = - - , Corollary 1 generalizes T h e o r e m 5.4 in our n
paper [2] in the sense that the assumption cn ~< m(n) ~< ~n made there (with two absolute constants c and ~) can be dropped. 9 For the operators G +s,-s (which are based upon the Jackson kernels of higher orders), Corollary 1 implies the following COROLLARY 2. (see [2, Theorem 5.5]) Let n I> 2 and s >t 3. Then for f ~ C I - 1, 1],
x) - f(x) l ~< C(s) " +2 ( f, x/ t -- x 2 ),lxl 3,
n
where M3 = 9 9 M2 < 134. PROOF. (i)
Using T h e o r e m 1, we now take Cl = 1, Cl = 1 n
For the constant D3 from L e m m a 1 1, we get
40
f iT
' C2=
~
30 iT
316
A. Boos -Jm-DINo CAo - H. H. GONSaA: Approximation by Boolean Sums of
(4.10)
( 12 + 1 40 3 30 --"-+ - - " - - 2" 12 + D3 = max \ 4 11 2 11 '
(66 ~-~ m a x
1
4o
1
3o)
2
11
2
11
57 --
"i'i''ll
~10) +
2~,G-.
~
57 = --
11
~10 +
2V'U..--
11 .
(ii) Obviously, +
+
Gs([./sl+ll-3 = Gain/31 : C [ - 1, 1] --* H~. I f n I> 3, then [n/3] + 1 i> 2. From (4.8), we thus have IG~%/s] ( f , x ) - f(x)[ ~< M2w2 (f, V ' l - x 2 ) [n/3] + 1 M2"o)2
~ 9 M20)2 f, ~/" l - x 2
f,-- ~ n
),n~>3"
n
This implies (ii).
5. Two asymptotic inequalities for
+
G sn--$
The results of this section are motivated by a theorem of Esseen concerning the classical Bernstein operators Bn. In his article [9], Esseen determined the asymptotic constant. ~, -- lim "--'~*
[ B,(f,x)-f(x)[ sup max r~c[o,u\rt0 O~x~r+ n
~gs--2r d~ ~
2), we also
d~ = (n--l) z,
whence
/53/
~12/n/ ~ 0 (--:), and t~us~l/n/-- 0 ( ~ ) Combining (5.2) and (5.3) now implies L e m m a 15. We define the following auxiliary quantities /to(S) =
~-2~
~2(s) =
~2-2~
J0
sin -~-
sin -~-
T
d~, s ~ 1,
d~, s ~ 2,
d~' s ~> 3"
These will be relevant for the formulation of the final results of this note. Evaluations of the above integrals and some explicit values of the constants M4(s) figuring in Theorem 4 below are given in an appendix.
Positive Linear Operators V: on the Constants in Devore-Gopengauz-Type Inequalities LEMMA 16.
Let s I> 2, then
lira n 2 . (1 -- Ol,sn-s) "~|
PRooF. 1 Bn~$ ~
321
--
/
--
~2(s'-~-) 2~o(s)
t nvt2s t nvt2s sin -2
2 dv = - -
V
sin - 2
9l"
"
sin
sin
2
dv.
V
2
Substituting v = ~ , we obtain n
0)2~d~
sin -~-
( sin "~-~ ) 2s
,n, 2s--, or nl-2S'Bns-= Z n -2s /o
sin ~ 2sd~.
Using Lemma 15 (with r = 0), we obtain lim nl_2s. Bns = - -2" 22s j~o~ __2s( sin @ ) 2s d~ 2 =__.---4s.f~(s).
(5.4) We also observe that
(1-c~
sin -~ 2 /0 nv t 2s z--_2_v dv= ~ (1-cosy) t
sin
--
2 nv v sin 2
t 2sdv
A. Boos
322
4s
=__
H. H. GONSKA:Approximation by Boolean Sums of
-JIA-DINO CAO -
nv sin - 2
sin2 v
2
/ 2s dv
V
sin - 2
25
= ~4 f0"=sin 2 ~ a-n 2n
(
t sin ~2 t sin
~
d~ substituting v = - - . n
2n
Using Lemma 15 (with r = 1) gives
1 r~ n 3-2s.m / _
lim
-- ~zc4. n2_2s
(l-cosv) t
sin - 2 nv V sin - 2
~
(sin
sin
2--2s
4 f ~-2s(sin~-O)2sd~
.,r 2-2s " o s
(5.5)
= -- 9
s I> 2.
From (5.1), (5.4) and (5.5), we have lim n 2 (1 -
01,sn-s)
t
2s
25
d~
dv
/sins,n___n!t
Positive Linear Operators V: on the Constants in Devore-Gopengauz-Type Inequalities
323
2s
n3_2~ " __1
(1 -- c o s v )
dv
--- l i m
rl-.-~ oa
nt-2s . Bn,s $
-
~2(s)
/g
~2(s)
2 ---4~
2~(s) 9
,g
LEMMA 17.
~__,~
L e t s ~ 3. T h e n
"
-
2Ol,s.-s
+
~
O~,s.-s
-
4/7~o(s)
2
PROOF.
We have 25
sin -~-
(1-cos ,Tg"
v)2
~
sin
dv = ~
_
v
--~
(1-cos v) 2
nvI
sin - 2
=~
8s
sin sin4 v 2
z
nv 2
sin
v 2
=_
8/o~
sin 4
ma
I
sin - -
2n
sin - 2n
t
2s
dv
I 2s substituting v =
~ ).
n
2s
324
A. Boos
- J I A - D I N G CAO -
H. H. GONSKA: Approximation by Boolean Sums of
By L e m m a 15 (with r = 2), we now get
1
lim n 5-25" - n--*m
~
t nvt sin
2
]
( 1 - c o s v) 2
1_at
sin
v
2 nat
= 8 " n 4 - 2 5 a r fo
_
8
9~24 - 2 s
~0
4--25
(sin~n)
~-25
sin
y
~ . 25
-(sin-~-)
d~
d~ (s t> 3)
5
(5.6)
= --
9 ~4(s).
2~
From [1] and (5.4), (5.6), we have
.mn4 (3
n---~,=
'
T
= lira n 4. 1"~ c~
- - 201'sn--s -[- - 2 - 02,5n--5
~
at
)
(1 -- cosv) 2Ksn_5(v) dv
tnvt 25
sin
n5_25 " __1 / / j~-
at
(1 - cos v) 2
dv
sin
v --
2
= lim
n--~ oo
n l - 2 s . Bn, s 5
2-~ ~ ( s )
2 - - - 45 9 ~ ( s ) Jg
~4(s)
4/~o(s)
25
dv
Positive Linear Operators V: on the Constants in Devore-Gopengauz-Type Inequalities
325
We are n o w in a position to prove the m a i n result o f this section. THEOREM 4.
(5.7)
Let s I> 3. T h e the following asymptotic inequality holds:
lira ~'-*|
sup r~c[-l,l]
I
sup Ixl 0 be given. Choose n so that [n/s] + 1 I> max(K2(e), 2). Then it follows from (5.8) that
I G+E.m(f,x) - f(x) l = [ G+([,/s]+,)-,(f, x) - f(x) J ~< M4(s, e) 02 ( f,
X,/ l _ x 2 ) [n/s] + 1
X/ 1-x 2 ) n/s
n
~< s':'M4(s, e)o~2 (f, X/ 1 - x 2 ) n This implies for n as given above sup
sup
f~cl-,,1] ,x,
