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On Positive and Copositive Polynomial and Spline Approximation inLp[−1, 1], 0<p<∞
Journal of Approximation Theory AT2984 journal of approximation theory 86, 320334 (1996) article no. 0073
On Positive and Copositive Polynomial and Spline Approximation in L p[&1, 1], 0
K. A. Kopotun* University of Alberta
and X. M. Yu Southwest Missouri State University Communicated by Dany Leviatan Received January 9, 1995; accepted in revised form October 18, 1995
For a function f # L p[&1, 1], 0< p< , with finitely many sign changes, we construct a sequence of polynomials P n # 6 n which are copositive with f and such that & f &P n & p C| .( f , (n+1) &1 ) p , where | .( f , t) p denotes the DitzianTotik modulus of continuity in L p metric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, then | . cannot be replaced by | 2 if 1< p< . In fact, we show that even for positive approximation and all 0< p< the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained. 1996 Academic Press, Inc.
1. INTRODUCTION Let L p[a, b] be the set of all measurable functions on [a, b] such that & f & Lp[a, b] < , where
& f & Lp[a, b] :=
{
\|
b
| f (x)| p dx a
ess sup | f (x)|,
+
1p
,
0< p< , p= .
x # [a, b]
Let C k[a, b] denote the set of k-times continuously differentiable functions on [a, b], and C[a, b] the set of all continuous functions. We also denote * Partially supported by the Izaak Walton Killam Memorial Scholarship.
320 0021-904596 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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by 6 n the set of all polynomials of degree n, by N the set of natural numbers and N0 :=N _ [0]. Throughout this paper C denote constants which are independent of f and n, and are not necessarily the same even if they occur in the same line. The notation C=C(+ 1 , ..., + & ) is used to emphasize that C depends only on + 1 , ..., +& and is independent of everything else. For Y s :=[ y 1 , ..., y s | y 0 :=&1< y 1 < y 2 < } } } < y s <1=: y s+1 ], we denote by 2 0(Y s ) the set of all functions f # L p[&1, 1] such that (&1) s&k f (x) 0 for x # [ y k , y k+1 ], k=0, ..., s, i.e., every f # 2 0(Y s ) has 0 s< sign changes at the points in Y s and is nonnegative near 1. In particular, if s=0, then 2 0 :=2 0(Y 0 ) denotes the set of all nonnegative functions on [&1, 1]. A function g is said to be copositive with f if f(x) g(x) 0, for all x # [&1, 1]. We are interested in approximating functions from 2 0(Y s ) and 2 0 by polynomials P n of degree n and splines s n with no more than n (fixed) knots that are copositive with f. If s=0, this is also called positive approximation. For f # L p[&1, 1] let E n( f ) p := inf & f &P n & p Pn # 6n
denote the degree of unconstrained approximation, and let E (0) n ( f , Y s ) p :=
inf
Pn # 6n & 20(Ys )
& f &P n & p
be the degree of copositive approximation to f by algebraic polynomials of degree n, where & } & p :=& } & Lp[&1, 1] . In particular, (0) E (0) n ( f ) p :=E n ( f , Y 0 ) p :=
inf
Pn # 6 n & 20
& f &P n & p
is the degree of positive approximation. Positive approximation of f # C & 2 0 has the same order as that of unconstrained approximation: E n( f ) E (0) n ( f ) 2E n( f ) . For examm &1 ) , n m&1, where ple, E (0) n ( f ) C| ( f , n | m( f , t, [a, b]) p := sup &2 m h ( f , } , [a, b])& Lp[a, b] 0
denotes the usual m th modulus of smoothness of f # L p[a, b], m
m h
2 ( f , x, [a, b]) :=
{
: i=0
m
\ i + (&1) if
0,
x\
m&i
\
f x&
m h+ih , 2
+
m h # [a, b], 2
otherwise.
is the symmetric m th difference and | m( f , t) p :=| m( f , t, [&1, 1]) p .
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At the same time, even if f has only one sign change, following its sign is not so easy and the order of approximation deteriorates. It was shown by S. P. Zhou [21] that there exists f # C 1[&1, 1] & 2 0(Y s ), s 1, such that lim sup n
E (0) n ( f , Ys) = + . 4 | ( f , n &1 )
(1)
Recently, Y. K. Hu and X. M. Yu [9] (see also [7], [8] and [16]) and K. A. Kopotun [13] showed that 3 &1 E (0) ) n ( f , Y s ) C(Y s ) | ( f , n
(2)
3 &1 E (0) ) , n ( f , Y s ) C(Y s ) | .( f , n
(3)
and
respectively, where m |m . ( f , t) p := sup &2 h.( } )( f , } , [&1, 1])& p 0
is the m th DitzianTotik modulus of smoothness with .(x) :=- 1&x 2. Thus, the investigation of copositive approximation of continuous functions in the uniform metric is complete (in the sense of the orders of the moduli of smoothness). At the same time, little is known about copositive approximation of functions in L p & 2 0(Y s ) for 1 p< and s 1, and it seems that nothing is known in the case for 0< p<1. It turns out that things become more complicated in L p , and even positive approximation is no longer trivial. It was shown by Zhou [21] that there exists f # C 1[&1, 1] & 2 0(Y s ), s 1 satisfying lim sup n
E (0) n ( f , Ys) p = + , 2+[1 p] | ( f , n &1 ) p
1 p< ,
and that, in the case s=0, there exists f # C 1[&1, 1] & 2 0 such that lim sup n
(0) n 3+[1 p]
E
|
( f )p = + , ( f , n &1 ) p
1 p< .
Our first theorem shows that it is impossible to obtain the estimate in terms of | 2 for positive polynomial approximation for all 0< p< (for 1 p< this was conjectured by Zhou [21]). The proof of this theorem, as well as those of our other main theorems, will be postponed until later sections.
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323
Theorem 1. For every n # N, 0< p< , 0<= 2 and A>0, there exists a nonnegative function f # C [&1, 1] such that for every polynomial P n # 6 n that is nonnegative at x=1, the following inequality holds: & f &P n & Lp[1&=, 1] >A| 2( f , 1) p .
(4)
Now that the order of | 2 is impossible, we seek the next best rate. The theorem below shows that | . is indeed reachable, thus, being the best order of positive polynomial approximation in L p . Theorem 2. If f # L p[&1, 1], 0< p< and f (x) 0, x # [&1, 1], then for every n # N0 &1 E (0) )p , n ( f ) p C| .( f , (n+1)
(5)
where C is an absolute constant in the case 1 p< and C=C( p) if 0
(6)
where C depends on s, $ and also on p in the case for 0
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approximation by splines with equal spacing is at least | 2. Soon after, Hu and Yu [9] proved an analogue of (2) for such splines (in fact, (2) is derived from this result for splines, with the aid of results in [10]). If p< , the rate drops to | 1 for copositive spline approximation, too. We first state our affirmative result as a theorem below. If no continuity is desired (r=1), this result can be easily obtained by piecewise constant functions, see Lemma 3.5. Hu [6] proved the theorem for r=2 and 3, 1 p< and equal spacing. His method can be modified for 0< p<1 and unequal spacing. The general case (for any r 1) can proved by applying Beatson's blending lemma [1] to local constant approximations of f on overlapping subintervals. To obtain such local constant approximations, one can use best constant L p approximation where f does not change its sign, and use 0 where it does, (see the proof of Lemma 3.5 for error estimate). We omit the proof of the theorem. Theorem 4. Let f # L p[&1, 1] & 2 0(Y s ), 0< p< , s 0 and let r 1 be an integer. Let T n :=[z 0 , ..., z n | &1 :=z 0
(7)
where d :=max(z i &z i&1 ) is the mesh size of T n , and C is a constant depending on the maximum ratio \ :=max(z i+2 &z i+1 )(z i+1 &z i ) and on p in the case 0
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2. COUNTEREXAMPLE In this section we construct the counterexample described in Theorem 1. This counterexample is a modification of the one used by the second author in the proof of Theorem 2 of [15]. (Also, as was noted by the referee it is possible to prove Theorem 1 considering a truncated linear function f $(x)=(1&$&x) + whose multi-fold integrals were used by A. S. Shvedov in [20].) Proof of Theorem 1. Let n # N, 0<= 2, A>0 and 0< p< be fixed, and define f (x) :=b(1&x)&ln(1&x+e &b )&ln b, where b 1 is a parameter to be chosen later. Clearly, f # C [&1, 1], and it is easy to check that f assumes its minimum 1&be &b >0 at x=1+e &b &b &1 # (0, 1), thus f (x)>0, x # [&1, 1]. Using the estimate
|
1
|ln(1&x+e &b )| p dx
&1
=
|
2+e&b
|ln x| p dx
e&b
|
3
|ln x| p dx=
0
|
1
(&ln x) p dx+
0
|
3
(ln x) p dx
1
2(ln 3) p +1( p+1)=: M 1p
(8)
we derive the inequality | 2( f , 1) p =| 2(ln(1&x+e &b ), 1) p 4 max[1, 1p] &ln(1&x+e &b )& p 4 max[1, 1p]M 1 =: M 2 .
(9)
Now suppose that the assertion of the theorem is not true, i.e., there exists a polynomial P n(x)=a 0 +a 1(1&x)+ } } } +a n(1&x) n with a 0 0 such that & f &P n & Lp[1&=, 1] A| 2( f , 1) p . Then from (8) and (9), we have &P n(x)&b(1&x)+ln b& Lp[1&=, 1] 2 max[1, 1 p](M 1 +AM 2 )=: M 3 . Therefore (see Lemma 7.3 of [18], for example), &(a 0 +ln b)+(a 1 &b)(1&x)+a 2(1&x) 2 + } } } +a n(1&x) n& C[1&=, 1] CM 3 = &1 p =: M 4 ,
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which gives a 0 +ln b M 4
or
P n(1)=a 0 M 4 &ln b.
By choosing b>e M4 (note that M 4 may depend on n, =, p and A but is independent of b) we get P n(1)<0, thus, obtaining a contradiction. K
3. NOTATION AND AUXILIARY RESULTS The following notation is used in the rest of this paper: 2 n(x) :=
- 1&x 2 1 + 2, n n
$ n(a, b) :=min[2 n(a), 2 n(b)],
x j :=cos I j :=[x j , x j&1 ],
j? , n
0 j n,
h j := |I j | =x j&1 &x j ,
1 j n
and j := j (x) :=
hj . |x&x j | +h j
In the proof of Theorem 2 we need the following two lemmas. Lemma 3.1. For every n # N, 1 j n&1 and + 10 there exist polynomials T j and T j of degree n satisfying for x # [&1, 1]: 0 / j (x)&T j (x) C(+) +j , 0 T j (x)&/ j (x) C(+) +j ,
(10)
where / j (x) :=
1,
{0,
if x x j , otherwise.
Proof. For the special case +=18 polynomials T j and T j are, respectively, Q j and Q j+1 from Lemma 1 of [14]. The general case + 10 is similar (see also [19]). K
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Lemma 3.2. For any function g # L p[&1, 1], 0< p< , the following inequality holds:
\
n
: |( g, h j , Ij ) pp j=1
+
1p
CC 0 | .(g, (n+1) &1 ) p ,
(11)
where, for every j, Ij #I j is such that |Ij | C 0 |I j |, and C depends on p if 0
\
C 1+
|b&a| $ n(a, b) &1 p | .(S n , n &1 ) p, $ n(a, b)
+
0< p< .
(12)
Proof. Let J :=[ j | x j # [a, b]]. Since for every j # J the inequality h j = |I j | $ n(a, b) holds, the interval [a, b] contains not more than 1+[ |b&a|$ n(a, b)] intervals I j . Also, for every j # J we have | .(S n , n &1 ) pp =
sup 0
|
1
|2 h.(x)(S n , x)| p dx
&1
C |S n(x j + )&S n(x j & )| p h j C |S n(x j + )&S n(x j & )| p $ n(a, b). Therefore |S n(b)&S n(a)| : |S n(x j + )&S n(x j & )| j#J
\
C 1+
|b&a| $ (a, b) &1 p | .(S n , n &1 ) p . $ n(a, b) n
+
K
Lemma 3.4. For every y k # Y s =[ y 1 , ..., y s ] and + 2, there exists an increasing polynomial T n( y k , x) # 6 n , copositive with sgn(x& y k ) in [&1, 1] and such that |sgn(x& y k )&T n( y k , x)| C
\
$ n( y k , x) | y k &x| +$ n( y k , x)
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+
+.
(13)
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HU, KOPOTUN, AND YU
Proof. The inequality was proved in [13] for +=2 with 2 n( y k ) instead of $ n( y k , x). The proof is similar for any + 2. Using the inequalities (see [19], for example ) 2 n( y) 2 <42 n(x)( |x& y| +2 n(x))
(14)
(|x& y| +2 n(x))< |x& y| +2 n( y)<2( |x& y| +2 n(x))
(15)
and 1 2
for any x, y # [&1, 1], we have 2 - 2 n(x)( | y k &x| +2 n(x)) 2 n( y k ) =4 1 | y k &x| +2 n( y k ) 2 ( | y k &x| +2 n(x))
2 n(x)
| y &x| +2 (x) . k
n
Therefore, for T n( y k, x) satisfying |sgn(x& y k )&T n( y k , x)| C
\
2 n( y k ) | y k &x| +2 n( y k )
+
2+
,
we have |sgn(x& y k )&T n( y k , x)| C
\
$ n( y k , x) | y k &x| +$ n( y k , x)
+
+.
K
Lemma 3.5. Let f be as in Theorem 4, and T n be such that there are at least two z i's in each ( y j , y j+1 ) for all j. Then there exists a piecewise constant spline s n on T n such that it is copositive with f and satisfies & f &s n & Lp(Ji ) C|( f , |Ji |, Ji ) p
(16)
for each i, where J i :=[z i , z i+1 ] Ji [z i&1 , z i+2 ], and C depends on the ratio \ :=max(z i+2 &z i+1 )(z i+1 &z i ). If 0
(17)
where C depends on $ :=min 0 i s | y i+1 & y i | and also on p in the case of 0< p<1. Proof. We call the interval J i :=[z i , z i+1 ] contaminated if z i < y j z i+1 for some y j # Y s , 1 j s. Then by assumption there is exactly one y j in each of the contaminated intervals J mj , j=1, ..., s, and there is at least one non-contaminated interval J i between J mj and J mj+1 for any 0 j s.
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(Here, for convenience we used the notation m 0 :=&1, m s+1 :=n, and J &1 =J n :=<.) Note that f does not change sign between J mj and J mj+1 . Let c i be a best L p constant approximation of f on J i , i=0, ..., n&1, then & f &c i & Lp(Ji ) C|( f , |J i |, J i ) p .
(18)
If J i is not contaminated, c i has the same sign as f . We define s n :=
0,
{c , i
for for
x # [z mj , z mj +1 ), 1 j s, x # [z i , z i+1 ), i{m j for any 1 j s.
Since c mj &1 and c mj +1 have opposite signs, we have |c mj &1 | |c mj +1 &c mj &1 | = |Jmj | &1p &c mj +1 &c mj &1 & Lp(Jmj ) C |Jmj | &1p (& f &c mj +1 & Lp(Jm ) +& f &c mj &1 & Lp(Jm ) ) j
C |Jmj |
&1p
j
|( f , |Jmj |, Jmj ) p ,
where Jmj :=[z mj &1 , z mj +2 ]. Here in the last step, we have used the fact that a (near) best L p polynomial approximation of f on an interval I is also a near best one on an interval J$I if their sizes are comparable (cf. DeVore and Popov [4, Lemma 3.3]). Hence & f &s n & Lp(Jm ) =& f & Lp(Jm ) C(& f &c mj &1 & Lp(Jm ) j
j
j
+ |c mj &1 | |J mj | 1p ) C|( f , |Jmj |, Jmj ) p ,
(19)
and this, together with (18), gives (16). From the construction, it is obvious that s n is copositive with f. For n>N :=C$ &1 such that there are at least two x i's in each ( y j , y j+1 ) for all j, inequality (17) immediately follows from (16) and Lemma 3.2. K
4. POSITIVE POLYNOMIAL APPROXIMATION Proof of Theorem 2. Throughout this section C denotes absolute constants in the case 1 p< and constants depending only on p if 0< p<1. As usual, these constants are not necessarily the same even if they occur in the same line. First, we approximate f by a piecewise constant function S: n&1
S( f , x) :=s n + : (s j &s j+1 ) / j (x), j=1
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where s j is a best L p constant approximant to f on the interval I j . Since, S( f , x)=s j on I j , then S( f , x) 0, x # [&1, 1]. Also, it is well known (see [3], for example) that & f &s j & Lp(Ij ) C|( f , h j , I j ) p . Therefore, using Lemma 3.2 we have n
& f &S( f )& pp = : j=1
|
| f (x)&s j | p dx
Ij
n
C p : |( f , h j , I j ) pp C p| .( f , (n+1) &1 ) pp .
(20)
j=1
Now we define n&1
P n( f , x) :=s n + : (s j &s j+1 ) j=1
+
\
sgn(s j &s j+1 )+1 T j (x) 2
1&sgn(s j &s j+1 ) T j (x) . 2
+
The polynomial P n( f , x) is nonnegative since n&1
P n( f , x) s n + : (s j &s j+1 ) / j (x)=S( f , x) 0. j=1
Also, choosing +:=1+[10min[1, p]], employing the methods used in [2] (the case for 0
|
1
&1
C p
|
1
&1
n&1
\ \: h
: |s j &s j+1 | +j
j=1
+
p
dx
n&1
&1p j
&s j &s j+1 & Lp(Ij ) +j
j=1
+
p
dx.
Now, using the inequality ( ! i ) p ! ip in the case 0
&P n( f )&S( f )& pp C p : h &1 &s j &s j+1 & Lp p(Ij ) j j=1
|
1
&1
j+ min[1, p] dx
n&1
C p : &s j &s j+1 & Lp p(Ij) , j=1
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POLYNOMIAL AND SPLINE APPROXIMATION
331
since 1&1 :j dx C(:) h j for : 2. Finally, using Minkowski's inequality for p 1 and its analog if 0
&P n( f )&S( f )& pp C p : & f &s j & Lp p(Ij _ Ij&1 ) j=1 n
C p : |( f , h j , I j _ I j&1 ) pp C p| .( f , (n+1) &1 ) pp . j=1
The proof of Theorem 2 is now complete.
K
5. COPOSITIVE POLYNOMIAL AND SPLINE APPROXIMATION Proof of Theorem 3. It is sufficient to prove Theorem 3 for sufficiently large n, say, n C$ &1, since for small n its assertion follows from the fact that & f & p C| .( f , 1) p C| .( f , (n+1) &1 ) p for f # 2 0(Y s ), s 1, and 0 n C$ &1. Here, the first inequality can be proved the same way as (16), using y j in place of z i , and the observation that |( f , 1) p t| .( f , 1) p . We shall prove Theorem 3 by induction on s, the number of sign changes. For s=0 Theorem 2 gives the proof. Now we assume that (6) is valid for every function f in L p[&1, 1] & 2 0(Y s&1 ) with Y s&1 =[ y 1 , ..., y s&1 ]. For f # L p[&1, 1] & 2 0(Y s ), it follows from Lemma 3.5 that there exists a piecewise constant spline S # 2 0(Y s ) with knots [x j ] nj=0 satisfying (17). Let S(x) :=S(x) sgn(x& y s ). Then S # L p[&1, 1] & 2 0(Y s&1 ), and by the assumption, there exists a polynomial P n # 6 n & 2 0(Y s&1 ) such that &S &P n& C| .(S, (n+1) &1 ) p .
(21)
Define P n(x) :=P n(x) T n( y s , x), where T n( y s , x) is the polynomial copositive with sgn(x& y s ) and given in Lemma 3.4 for + 2+4p. It is apparent that P n # 6 2n & 2 0(Y s ). We are going to estimate &S&P n & p and then & f &P n & p . From (21), we have &S&P n & p =&S(x) sgn(x& y s )&P n(x) T n( y s , x)& p C[&S(x)[sgn(x& y s )&T n( y s , x)]& p +&[S(x)&P n(x)] T n( y s , x)& p ] C(I+| .(S, (n+1) &1 ) p ),
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where I :=&S(x)[sgn(x& y s )&T n( y s , x)]& p. Here we have used the fact that |T n( y s , x)| C. Since |S(x)| = |S(x)| and S( y s )=0, we have I p=
|
1
|S(x)&S( y s )| p |sgn(x& y s )&T n( y s , x)| p dx.
&1
It follows from Lemmas 3.3 and 3.4 that I p C p| .(S, (n+1) &1 ) pp
|
1
&1
_$ n( y s , x) &1
\
\
1+
| y s &x| $ n( y s , x)
$ n( y s , x) | y s &x| +$ n( y s , x)
+
+
p
+p
dx
:=C p| .(S, (n+1) &1 ) pp 3. And we have 3 :=
|
1
&1
\
$ n( y s , x) | y s &x| +$ n( y s , x)
+
(+&1) p
$ n( y s , x) &1 dx C.
Indeed, it is easy to check the above inequality with 2 n( y s ) instead of $ n( y s , x) . Then, using (14) and (15), we can prove this inequality in terms of $ n( y s , x) as we did in the proof of Lemma 3.4. Thus, we obtain &S&P n & p C[| .(S, (n+1) &1 ) p +| .(S, (n+1) &1 ) p ]. From (17), we have | .(S, (n+1) &1 ) p C| .( f , (n+1) &1 ) p , and & f &P n & p C(& f &S& p +&S&P n & p ) C(| .( f , (n+1) &1 ) p +| .(S, (n+1) &1 ) p ). To complete the proof, we only need to show that | .(S, (n+1) &1 ) p C| .(S, (n+1) &1 ) p . Indeed, using the definition of S (more precisely, the fact that S coincides with &S on [&1, y s ], |S(x)| = |S(x)|, and S does not change sign on [ y s&1 , 1]) we conclude that for any numbers a, b # [&1, 1] such that |b&a| is sufficiently small ( |b&a| | y s & y s&1 | will do) the following inequality holds |S(b)&S(a)| |S(b)&S(a)|.
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333
This immediately yields &2 h.(x)(S, x, [&1, 1])& pp =
|
|S(x+h.(x)2)&S(x&h.(x)2)| p dx
|
|S(x+h.(x)2)&S(x&h.(x)2)| p dx
[x | x\h.(x)2 # [&1, 1]]
[x | x\h.(x)2 # [&1, 1]]
=&2 h.(x)(S, x, [&1, 1])& pp , and, therefore, completes the proof of Theorem 3.
K
REFERENCES 1. R. K. Beatson, Restricted range approximation by splines and variational inequalities, SIAM J. Numer. Anal. 19 (1982), 372380. 2. R. A. DeVore, D. Leviatan, and X. M. Yu, Polynomial approximation in L p(0< p<1) space, Constr. Approx. 8 (1992), 187201. 3. R. A. DeVore and G. G. Lorentz, ``Constructive Approximation,'' Springer-Verlag, Berlin, 1993. 4. R. A. DeVore and V. A. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305, No. 1 (1988), 397414. 5. Z. Ditzian and V. Totik, ``Moduli of Smoothness,'' Springer-Verlag, Berlin, 1987. 6. Y. K. Hu, Positive and copositive spline approximation in L p[0, 1], Comput. Math. Appl. 30 (1995), 137146. 7. Y. K. Hu, D. Leviatan, and X. M. Yu, Copositive polynomial approximation in C[0, 1], J. Anal. 1 (1993), 8590. 8. Y. K. Hu, D. Leviatan, and X. M. Yu, Copositive polynomial and spline approximation, J. Approx. Theory 80 (1995), 204218. 9. Y. K. Hu and X. M. Yu, The degree of copositive approximation and a computer algorithm, SIAM J. Numer. Anal. 33, No. 1 (1996), 388398. 10. Y. K. Hu and X. M. Yu, Discrete modulus of smoothness of splines with equally spaced knots, SIAM J. Numer. Anal. 32, No. 5 (1995), 14281435. 11. K. G. Ivanov, On a new characteristic of functions. II. Direct and converse theorems for the best algebraic approximation in C[&1, 1] and L p[&1, 1], PLISKA 5 (1983), 151163. 12. K. A. Kopotun, A note on simultaneous approximation in L p[&1, 1] (1 p< ), Analysis 15 (1995), 151158. 13. K. A. Kopotun, On copositive approximation by algebraic polynomials, Anal. Math. 21 (1995), 269283. 14. K. A. Kopotun, Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials, Constr. Approx. 10, No. 2 (1994), 153178. 15. K. A. Kopotun, Uniform estimates of convex approximation of functions by polynomials, Mat. Zametki 51, No. 3 (1992), 3546 [in Russian]; English translation, Math. Notes 51, No. 3 (1992), 245254. 16. D. Leviatan, The degree of copositive approximation by polynomials, Proc. Amer. Math. Soc. 88 (1983), 101105.
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17. D. Leviatan and H. N. Mhaskar, The rate of monotone spline approximation in the L p -norm, SIAM J. Math. Anal. 13, No. 5 (1982), 866874. 18. P. Petrushev and V. Popov, ``Rational Approximation of Real Functions,'' Cambridge Univ. Press, Cambridge, England, 1987. 19. I. A. Shevchuk, ``Approximation by Polynomials and Traces of the Functions Continuous on an Interval,'' Naukova Dumka, 1992, Kiev. [in Russian] 20. A. S. Shvedov, Orders of coapproximation of functions by algebraic polynomials, Mat. Zametki 29, No. 1 (1981), 117130 [in Russian]; English translation, Math. Notes 29, No. 1 (1981), 6370. 21. S. P. Zhou, On copositive approximation, Approx. Theory Appl. 9, No. 2 (1993), 104110.
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On Positive and Copositive Polynomial and Spline Approximation in L p[&1, 1], 0
K. A. Kopotun* University of Alberta
and X. M. Yu Southwest Missouri State University Communicated by Dany Leviatan Received January 9, 1995; accepted in revised form October 18, 1995
For a function f # L p[&1, 1], 0< p< , with finitely many sign changes, we construct a sequence of polynomials P n # 6 n which are copositive with f and such that & f &P n & p C| .( f , (n+1) &1 ) p , where | .( f , t) p denotes the DitzianTotik modulus of continuity in L p metric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, then | . cannot be replaced by | 2 if 1< p< . In fact, we show that even for positive approximation and all 0< p< the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained. 1996 Academic Press, Inc.
1. INTRODUCTION Let L p[a, b] be the set of all measurable functions on [a, b] such that & f & Lp[a, b] < , where
& f & Lp[a, b] :=
{
\|
b
| f (x)| p dx a
ess sup | f (x)|,
+
1p
,
0< p< , p= .
x # [a, b]
Let C k[a, b] denote the set of k-times continuously differentiable functions on [a, b], and C[a, b] the set of all continuous functions. We also denote * Partially supported by the Izaak Walton Killam Memorial Scholarship.
320 0021-904596 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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by 6 n the set of all polynomials of degree n, by N the set of natural numbers and N0 :=N _ [0]. Throughout this paper C denote constants which are independent of f and n, and are not necessarily the same even if they occur in the same line. The notation C=C(+ 1 , ..., + & ) is used to emphasize that C depends only on + 1 , ..., +& and is independent of everything else. For Y s :=[ y 1 , ..., y s | y 0 :=&1< y 1 < y 2 < } } } < y s <1=: y s+1 ], we denote by 2 0(Y s ) the set of all functions f # L p[&1, 1] such that (&1) s&k f (x) 0 for x # [ y k , y k+1 ], k=0, ..., s, i.e., every f # 2 0(Y s ) has 0 s< sign changes at the points in Y s and is nonnegative near 1. In particular, if s=0, then 2 0 :=2 0(Y 0 ) denotes the set of all nonnegative functions on [&1, 1]. A function g is said to be copositive with f if f(x) g(x) 0, for all x # [&1, 1]. We are interested in approximating functions from 2 0(Y s ) and 2 0 by polynomials P n of degree n and splines s n with no more than n (fixed) knots that are copositive with f. If s=0, this is also called positive approximation. For f # L p[&1, 1] let E n( f ) p := inf & f &P n & p Pn # 6n
denote the degree of unconstrained approximation, and let E (0) n ( f , Y s ) p :=
inf
Pn # 6n & 20(Ys )
& f &P n & p
be the degree of copositive approximation to f by algebraic polynomials of degree n, where & } & p :=& } & Lp[&1, 1] . In particular, (0) E (0) n ( f ) p :=E n ( f , Y 0 ) p :=
inf
Pn # 6 n & 20
& f &P n & p
is the degree of positive approximation. Positive approximation of f # C & 2 0 has the same order as that of unconstrained approximation: E n( f ) E (0) n ( f ) 2E n( f ) . For examm &1 ) , n m&1, where ple, E (0) n ( f ) C| ( f , n | m( f , t, [a, b]) p := sup &2 m h ( f , } , [a, b])& Lp[a, b] 0
denotes the usual m th modulus of smoothness of f # L p[a, b], m
m h
2 ( f , x, [a, b]) :=
{
: i=0
m
\ i + (&1) if
0,
x\
m&i
\
f x&
m h+ih , 2
+
m h # [a, b], 2
otherwise.
is the symmetric m th difference and | m( f , t) p :=| m( f , t, [&1, 1]) p .
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At the same time, even if f has only one sign change, following its sign is not so easy and the order of approximation deteriorates. It was shown by S. P. Zhou [21] that there exists f # C 1[&1, 1] & 2 0(Y s ), s 1, such that lim sup n
E (0) n ( f , Ys) = + . 4 | ( f , n &1 )
(1)
Recently, Y. K. Hu and X. M. Yu [9] (see also [7], [8] and [16]) and K. A. Kopotun [13] showed that 3 &1 E (0) ) n ( f , Y s ) C(Y s ) | ( f , n
(2)
3 &1 E (0) ) , n ( f , Y s ) C(Y s ) | .( f , n
(3)
and
respectively, where m |m . ( f , t) p := sup &2 h.( } )( f , } , [&1, 1])& p 0
is the m th DitzianTotik modulus of smoothness with .(x) :=- 1&x 2. Thus, the investigation of copositive approximation of continuous functions in the uniform metric is complete (in the sense of the orders of the moduli of smoothness). At the same time, little is known about copositive approximation of functions in L p & 2 0(Y s ) for 1 p< and s 1, and it seems that nothing is known in the case for 0< p<1. It turns out that things become more complicated in L p , and even positive approximation is no longer trivial. It was shown by Zhou [21] that there exists f # C 1[&1, 1] & 2 0(Y s ), s 1 satisfying lim sup n
E (0) n ( f , Ys) p = + , 2+[1 p] | ( f , n &1 ) p
1 p< ,
and that, in the case s=0, there exists f # C 1[&1, 1] & 2 0 such that lim sup n
(0) n 3+[1 p]
E
|
( f )p = + , ( f , n &1 ) p
1 p< .
Our first theorem shows that it is impossible to obtain the estimate in terms of | 2 for positive polynomial approximation for all 0< p< (for 1 p< this was conjectured by Zhou [21]). The proof of this theorem, as well as those of our other main theorems, will be postponed until later sections.
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323
Theorem 1. For every n # N, 0< p< , 0<= 2 and A>0, there exists a nonnegative function f # C [&1, 1] such that for every polynomial P n # 6 n that is nonnegative at x=1, the following inequality holds: & f &P n & Lp[1&=, 1] >A| 2( f , 1) p .
(4)
Now that the order of | 2 is impossible, we seek the next best rate. The theorem below shows that | . is indeed reachable, thus, being the best order of positive polynomial approximation in L p . Theorem 2. If f # L p[&1, 1], 0< p< and f (x) 0, x # [&1, 1], then for every n # N0 &1 E (0) )p , n ( f ) p C| .( f , (n+1)
(5)
where C is an absolute constant in the case 1 p< and C=C( p) if 0
(6)
where C depends on s, $ and also on p in the case for 0
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approximation by splines with equal spacing is at least | 2. Soon after, Hu and Yu [9] proved an analogue of (2) for such splines (in fact, (2) is derived from this result for splines, with the aid of results in [10]). If p< , the rate drops to | 1 for copositive spline approximation, too. We first state our affirmative result as a theorem below. If no continuity is desired (r=1), this result can be easily obtained by piecewise constant functions, see Lemma 3.5. Hu [6] proved the theorem for r=2 and 3, 1 p< and equal spacing. His method can be modified for 0< p<1 and unequal spacing. The general case (for any r 1) can proved by applying Beatson's blending lemma [1] to local constant approximations of f on overlapping subintervals. To obtain such local constant approximations, one can use best constant L p approximation where f does not change its sign, and use 0 where it does, (see the proof of Lemma 3.5 for error estimate). We omit the proof of the theorem. Theorem 4. Let f # L p[&1, 1] & 2 0(Y s ), 0< p< , s 0 and let r 1 be an integer. Let T n :=[z 0 , ..., z n | &1 :=z 0
(7)
where d :=max(z i &z i&1 ) is the mesh size of T n , and C is a constant depending on the maximum ratio \ :=max(z i+2 &z i+1 )(z i+1 &z i ) and on p in the case 0
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2. COUNTEREXAMPLE In this section we construct the counterexample described in Theorem 1. This counterexample is a modification of the one used by the second author in the proof of Theorem 2 of [15]. (Also, as was noted by the referee it is possible to prove Theorem 1 considering a truncated linear function f $(x)=(1&$&x) + whose multi-fold integrals were used by A. S. Shvedov in [20].) Proof of Theorem 1. Let n # N, 0<= 2, A>0 and 0< p< be fixed, and define f (x) :=b(1&x)&ln(1&x+e &b )&ln b, where b 1 is a parameter to be chosen later. Clearly, f # C [&1, 1], and it is easy to check that f assumes its minimum 1&be &b >0 at x=1+e &b &b &1 # (0, 1), thus f (x)>0, x # [&1, 1]. Using the estimate
|
1
|ln(1&x+e &b )| p dx
&1
=
|
2+e&b
|ln x| p dx
e&b
|
3
|ln x| p dx=
0
|
1
(&ln x) p dx+
0
|
3
(ln x) p dx
1
2(ln 3) p +1( p+1)=: M 1p
(8)
we derive the inequality | 2( f , 1) p =| 2(ln(1&x+e &b ), 1) p 4 max[1, 1p] &ln(1&x+e &b )& p 4 max[1, 1p]M 1 =: M 2 .
(9)
Now suppose that the assertion of the theorem is not true, i.e., there exists a polynomial P n(x)=a 0 +a 1(1&x)+ } } } +a n(1&x) n with a 0 0 such that & f &P n & Lp[1&=, 1] A| 2( f , 1) p . Then from (8) and (9), we have &P n(x)&b(1&x)+ln b& Lp[1&=, 1] 2 max[1, 1 p](M 1 +AM 2 )=: M 3 . Therefore (see Lemma 7.3 of [18], for example), &(a 0 +ln b)+(a 1 &b)(1&x)+a 2(1&x) 2 + } } } +a n(1&x) n& C[1&=, 1] CM 3 = &1 p =: M 4 ,
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which gives a 0 +ln b M 4
or
P n(1)=a 0 M 4 &ln b.
By choosing b>e M4 (note that M 4 may depend on n, =, p and A but is independent of b) we get P n(1)<0, thus, obtaining a contradiction. K
3. NOTATION AND AUXILIARY RESULTS The following notation is used in the rest of this paper: 2 n(x) :=
- 1&x 2 1 + 2, n n
$ n(a, b) :=min[2 n(a), 2 n(b)],
x j :=cos I j :=[x j , x j&1 ],
j? , n
0 j n,
h j := |I j | =x j&1 &x j ,
1 j n
and j := j (x) :=
hj . |x&x j | +h j
In the proof of Theorem 2 we need the following two lemmas. Lemma 3.1. For every n # N, 1 j n&1 and + 10 there exist polynomials T j and T j of degree n satisfying for x # [&1, 1]: 0 / j (x)&T j (x) C(+) +j , 0 T j (x)&/ j (x) C(+) +j ,
(10)
where / j (x) :=
1,
{0,
if x x j , otherwise.
Proof. For the special case +=18 polynomials T j and T j are, respectively, Q j and Q j+1 from Lemma 1 of [14]. The general case + 10 is similar (see also [19]). K
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Lemma 3.2. For any function g # L p[&1, 1], 0< p< , the following inequality holds:
\
n
: |( g, h j , Ij ) pp j=1
+
1p
CC 0 | .(g, (n+1) &1 ) p ,
(11)
where, for every j, Ij #I j is such that |Ij | C 0 |I j |, and C depends on p if 0
\
C 1+
|b&a| $ n(a, b) &1 p | .(S n , n &1 ) p, $ n(a, b)
+
0< p< .
(12)
Proof. Let J :=[ j | x j # [a, b]]. Since for every j # J the inequality h j = |I j | $ n(a, b) holds, the interval [a, b] contains not more than 1+[ |b&a|$ n(a, b)] intervals I j . Also, for every j # J we have | .(S n , n &1 ) pp =
sup 0
|
1
|2 h.(x)(S n , x)| p dx
&1
C |S n(x j + )&S n(x j & )| p h j C |S n(x j + )&S n(x j & )| p $ n(a, b). Therefore |S n(b)&S n(a)| : |S n(x j + )&S n(x j & )| j#J
\
C 1+
|b&a| $ (a, b) &1 p | .(S n , n &1 ) p . $ n(a, b) n
+
K
Lemma 3.4. For every y k # Y s =[ y 1 , ..., y s ] and + 2, there exists an increasing polynomial T n( y k , x) # 6 n , copositive with sgn(x& y k ) in [&1, 1] and such that |sgn(x& y k )&T n( y k , x)| C
\
$ n( y k , x) | y k &x| +$ n( y k , x)
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+
+.
(13)
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HU, KOPOTUN, AND YU
Proof. The inequality was proved in [13] for +=2 with 2 n( y k ) instead of $ n( y k , x). The proof is similar for any + 2. Using the inequalities (see [19], for example ) 2 n( y) 2 <42 n(x)( |x& y| +2 n(x))
(14)
(|x& y| +2 n(x))< |x& y| +2 n( y)<2( |x& y| +2 n(x))
(15)
and 1 2
for any x, y # [&1, 1], we have 2 - 2 n(x)( | y k &x| +2 n(x)) 2 n( y k ) =4 1 | y k &x| +2 n( y k ) 2 ( | y k &x| +2 n(x))
2 n(x)
| y &x| +2 (x) . k
n
Therefore, for T n( y k, x) satisfying |sgn(x& y k )&T n( y k , x)| C
\
2 n( y k ) | y k &x| +2 n( y k )
+
2+
,
we have |sgn(x& y k )&T n( y k , x)| C
\
$ n( y k , x) | y k &x| +$ n( y k , x)
+
+.
K
Lemma 3.5. Let f be as in Theorem 4, and T n be such that there are at least two z i's in each ( y j , y j+1 ) for all j. Then there exists a piecewise constant spline s n on T n such that it is copositive with f and satisfies & f &s n & Lp(Ji ) C|( f , |Ji |, Ji ) p
(16)
for each i, where J i :=[z i , z i+1 ] Ji [z i&1 , z i+2 ], and C depends on the ratio \ :=max(z i+2 &z i+1 )(z i+1 &z i ). If 0
(17)
where C depends on $ :=min 0 i s | y i+1 & y i | and also on p in the case of 0< p<1. Proof. We call the interval J i :=[z i , z i+1 ] contaminated if z i < y j z i+1 for some y j # Y s , 1 j s. Then by assumption there is exactly one y j in each of the contaminated intervals J mj , j=1, ..., s, and there is at least one non-contaminated interval J i between J mj and J mj+1 for any 0 j s.
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(Here, for convenience we used the notation m 0 :=&1, m s+1 :=n, and J &1 =J n :=<.) Note that f does not change sign between J mj and J mj+1 . Let c i be a best L p constant approximation of f on J i , i=0, ..., n&1, then & f &c i & Lp(Ji ) C|( f , |J i |, J i ) p .
(18)
If J i is not contaminated, c i has the same sign as f . We define s n :=
0,
{c , i
for for
x # [z mj , z mj +1 ), 1 j s, x # [z i , z i+1 ), i{m j for any 1 j s.
Since c mj &1 and c mj +1 have opposite signs, we have |c mj &1 | |c mj +1 &c mj &1 | = |Jmj | &1p &c mj +1 &c mj &1 & Lp(Jmj ) C |Jmj | &1p (& f &c mj +1 & Lp(Jm ) +& f &c mj &1 & Lp(Jm ) ) j
C |Jmj |
&1p
j
|( f , |Jmj |, Jmj ) p ,
where Jmj :=[z mj &1 , z mj +2 ]. Here in the last step, we have used the fact that a (near) best L p polynomial approximation of f on an interval I is also a near best one on an interval J$I if their sizes are comparable (cf. DeVore and Popov [4, Lemma 3.3]). Hence & f &s n & Lp(Jm ) =& f & Lp(Jm ) C(& f &c mj &1 & Lp(Jm ) j
j
j
+ |c mj &1 | |J mj | 1p ) C|( f , |Jmj |, Jmj ) p ,
(19)
and this, together with (18), gives (16). From the construction, it is obvious that s n is copositive with f. For n>N :=C$ &1 such that there are at least two x i's in each ( y j , y j+1 ) for all j, inequality (17) immediately follows from (16) and Lemma 3.2. K
4. POSITIVE POLYNOMIAL APPROXIMATION Proof of Theorem 2. Throughout this section C denotes absolute constants in the case 1 p< and constants depending only on p if 0< p<1. As usual, these constants are not necessarily the same even if they occur in the same line. First, we approximate f by a piecewise constant function S: n&1
S( f , x) :=s n + : (s j &s j+1 ) / j (x), j=1
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where s j is a best L p constant approximant to f on the interval I j . Since, S( f , x)=s j on I j , then S( f , x) 0, x # [&1, 1]. Also, it is well known (see [3], for example) that & f &s j & Lp(Ij ) C|( f , h j , I j ) p . Therefore, using Lemma 3.2 we have n
& f &S( f )& pp = : j=1
|
| f (x)&s j | p dx
Ij
n
C p : |( f , h j , I j ) pp C p| .( f , (n+1) &1 ) pp .
(20)
j=1
Now we define n&1
P n( f , x) :=s n + : (s j &s j+1 ) j=1
+
\
sgn(s j &s j+1 )+1 T j (x) 2
1&sgn(s j &s j+1 ) T j (x) . 2
+
The polynomial P n( f , x) is nonnegative since n&1
P n( f , x) s n + : (s j &s j+1 ) / j (x)=S( f , x) 0. j=1
Also, choosing +:=1+[10min[1, p]], employing the methods used in [2] (the case for 0
|
1
&1
C p
|
1
&1
n&1
\ \: h
: |s j &s j+1 | +j
j=1
+
p
dx
n&1
&1p j
&s j &s j+1 & Lp(Ij ) +j
j=1
+
p
dx.
Now, using the inequality ( ! i ) p ! ip in the case 0
&P n( f )&S( f )& pp C p : h &1 &s j &s j+1 & Lp p(Ij ) j j=1
|
1
&1
j+ min[1, p] dx
n&1
C p : &s j &s j+1 & Lp p(Ij) , j=1
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since 1&1 :j dx C(:) h j for : 2. Finally, using Minkowski's inequality for p 1 and its analog if 0
&P n( f )&S( f )& pp C p : & f &s j & Lp p(Ij _ Ij&1 ) j=1 n
C p : |( f , h j , I j _ I j&1 ) pp C p| .( f , (n+1) &1 ) pp . j=1
The proof of Theorem 2 is now complete.
K
5. COPOSITIVE POLYNOMIAL AND SPLINE APPROXIMATION Proof of Theorem 3. It is sufficient to prove Theorem 3 for sufficiently large n, say, n C$ &1, since for small n its assertion follows from the fact that & f & p C| .( f , 1) p C| .( f , (n+1) &1 ) p for f # 2 0(Y s ), s 1, and 0 n C$ &1. Here, the first inequality can be proved the same way as (16), using y j in place of z i , and the observation that |( f , 1) p t| .( f , 1) p . We shall prove Theorem 3 by induction on s, the number of sign changes. For s=0 Theorem 2 gives the proof. Now we assume that (6) is valid for every function f in L p[&1, 1] & 2 0(Y s&1 ) with Y s&1 =[ y 1 , ..., y s&1 ]. For f # L p[&1, 1] & 2 0(Y s ), it follows from Lemma 3.5 that there exists a piecewise constant spline S # 2 0(Y s ) with knots [x j ] nj=0 satisfying (17). Let S(x) :=S(x) sgn(x& y s ). Then S # L p[&1, 1] & 2 0(Y s&1 ), and by the assumption, there exists a polynomial P n # 6 n & 2 0(Y s&1 ) such that &S &P n& C| .(S, (n+1) &1 ) p .
(21)
Define P n(x) :=P n(x) T n( y s , x), where T n( y s , x) is the polynomial copositive with sgn(x& y s ) and given in Lemma 3.4 for + 2+4p. It is apparent that P n # 6 2n & 2 0(Y s ). We are going to estimate &S&P n & p and then & f &P n & p . From (21), we have &S&P n & p =&S(x) sgn(x& y s )&P n(x) T n( y s , x)& p C[&S(x)[sgn(x& y s )&T n( y s , x)]& p +&[S(x)&P n(x)] T n( y s , x)& p ] C(I+| .(S, (n+1) &1 ) p ),
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where I :=&S(x)[sgn(x& y s )&T n( y s , x)]& p. Here we have used the fact that |T n( y s , x)| C. Since |S(x)| = |S(x)| and S( y s )=0, we have I p=
|
1
|S(x)&S( y s )| p |sgn(x& y s )&T n( y s , x)| p dx.
&1
It follows from Lemmas 3.3 and 3.4 that I p C p| .(S, (n+1) &1 ) pp
|
1
&1
_$ n( y s , x) &1
\
\
1+
| y s &x| $ n( y s , x)
$ n( y s , x) | y s &x| +$ n( y s , x)
+
+
p
+p
dx
:=C p| .(S, (n+1) &1 ) pp 3. And we have 3 :=
|
1
&1
\
$ n( y s , x) | y s &x| +$ n( y s , x)
+
(+&1) p
$ n( y s , x) &1 dx C.
Indeed, it is easy to check the above inequality with 2 n( y s ) instead of $ n( y s , x) . Then, using (14) and (15), we can prove this inequality in terms of $ n( y s , x) as we did in the proof of Lemma 3.4. Thus, we obtain &S&P n & p C[| .(S, (n+1) &1 ) p +| .(S, (n+1) &1 ) p ]. From (17), we have | .(S, (n+1) &1 ) p C| .( f , (n+1) &1 ) p , and & f &P n & p C(& f &S& p +&S&P n & p ) C(| .( f , (n+1) &1 ) p +| .(S, (n+1) &1 ) p ). To complete the proof, we only need to show that | .(S, (n+1) &1 ) p C| .(S, (n+1) &1 ) p . Indeed, using the definition of S (more precisely, the fact that S coincides with &S on [&1, y s ], |S(x)| = |S(x)|, and S does not change sign on [ y s&1 , 1]) we conclude that for any numbers a, b # [&1, 1] such that |b&a| is sufficiently small ( |b&a| | y s & y s&1 | will do) the following inequality holds |S(b)&S(a)| |S(b)&S(a)|.
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This immediately yields &2 h.(x)(S, x, [&1, 1])& pp =
|
|S(x+h.(x)2)&S(x&h.(x)2)| p dx
|
|S(x+h.(x)2)&S(x&h.(x)2)| p dx
[x | x\h.(x)2 # [&1, 1]]
[x | x\h.(x)2 # [&1, 1]]
=&2 h.(x)(S, x, [&1, 1])& pp , and, therefore, completes the proof of Theorem 3.
K
REFERENCES 1. R. K. Beatson, Restricted range approximation by splines and variational inequalities, SIAM J. Numer. Anal. 19 (1982), 372380. 2. R. A. DeVore, D. Leviatan, and X. M. Yu, Polynomial approximation in L p(0< p<1) space, Constr. Approx. 8 (1992), 187201. 3. R. A. DeVore and G. G. Lorentz, ``Constructive Approximation,'' Springer-Verlag, Berlin, 1993. 4. R. A. DeVore and V. A. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305, No. 1 (1988), 397414. 5. Z. Ditzian and V. Totik, ``Moduli of Smoothness,'' Springer-Verlag, Berlin, 1987. 6. Y. K. Hu, Positive and copositive spline approximation in L p[0, 1], Comput. Math. Appl. 30 (1995), 137146. 7. Y. K. Hu, D. Leviatan, and X. M. Yu, Copositive polynomial approximation in C[0, 1], J. Anal. 1 (1993), 8590. 8. Y. K. Hu, D. Leviatan, and X. M. Yu, Copositive polynomial and spline approximation, J. Approx. Theory 80 (1995), 204218. 9. Y. K. Hu and X. M. Yu, The degree of copositive approximation and a computer algorithm, SIAM J. Numer. Anal. 33, No. 1 (1996), 388398. 10. Y. K. Hu and X. M. Yu, Discrete modulus of smoothness of splines with equally spaced knots, SIAM J. Numer. Anal. 32, No. 5 (1995), 14281435. 11. K. G. Ivanov, On a new characteristic of functions. II. Direct and converse theorems for the best algebraic approximation in C[&1, 1] and L p[&1, 1], PLISKA 5 (1983), 151163. 12. K. A. Kopotun, A note on simultaneous approximation in L p[&1, 1] (1 p< ), Analysis 15 (1995), 151158. 13. K. A. Kopotun, On copositive approximation by algebraic polynomials, Anal. Math. 21 (1995), 269283. 14. K. A. Kopotun, Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials, Constr. Approx. 10, No. 2 (1994), 153178. 15. K. A. Kopotun, Uniform estimates of convex approximation of functions by polynomials, Mat. Zametki 51, No. 3 (1992), 3546 [in Russian]; English translation, Math. Notes 51, No. 3 (1992), 245254. 16. D. Leviatan, The degree of copositive approximation by polynomials, Proc. Amer. Math. Soc. 88 (1983), 101105.
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17. D. Leviatan and H. N. Mhaskar, The rate of monotone spline approximation in the L p -norm, SIAM J. Math. Anal. 13, No. 5 (1982), 866874. 18. P. Petrushev and V. Popov, ``Rational Approximation of Real Functions,'' Cambridge Univ. Press, Cambridge, England, 1987. 19. I. A. Shevchuk, ``Approximation by Polynomials and Traces of the Functions Continuous on an Interval,'' Naukova Dumka, 1992, Kiev. [in Russian] 20. A. S. Shvedov, Orders of coapproximation of functions by algebraic polynomials, Mat. Zametki 29, No. 1 (1981), 117130 [in Russian]; English translation, Math. Notes 29, No. 1 (1981), 6370. 21. S. P. Zhou, On copositive approximation, Approx. Theory Appl. 9, No. 2 (1993), 104110.
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