Almost Copositive Polynomial
Approximation in L [− 1,1], p < 1
p
Dr. Eman Samir Bhaya, Department of Mathematics, College of Education,
Babylon University, Babil, Hilla, Iraq.
E.mail: emanbhaya@yahoo.com
Abstract. For a function f ∈ L p [−1,1],0 < p < 1 with finitely many sign changes, Hu, Kopotun and Yu
[5] construct a sequence of polynomials pn ∈ Pn which are copositive with f
f − pp
p
(
≤ c( p )ωϕ f , n −1
)
p
(
, where ωϕ f , n −1
)
p
and such that
denotes the Ditzian-Totik modulus of continuity in L p
metric. Also it was shown that this estimate is exact in the sense that if f has at least one sign change
then ωϕ can not be replaced by ω 2 if 0 < p < 1 . In this paper we first show that almost copositive
(
approximation improves the rate to ωϕ2 f , n −1
)
p
(
from ωϕ f , n −1
)
p
the rate for the ordinary copositive
approximation. Our second theorem shows that it is impossible to obtain the first estimate interims of
ωϕ4 .
First conference on mathematical sciences (2006) Zarqa Private University. Jordan
ISSN:1818-8532. pp. 52-56.
1
1. Introduction and definitions
Let L p [a, b] be the set of all measurable functions on [a, b] such that
f
L p [a ,b ]
< ∞ , where
p
b

f L [a ,b ] :=  ∫ f ( x ) dx 
p
a

1/ p
,0 < p < 1 .(see [1] p. 19-
24)
Let Pn denote the set of all polynomials of degree ≤ n , by N the set
of natural numbers. Thought this paper the notation C = C (a, b ) denote
constants that are depend only on a, b and is independent of every thing
else, and are not necessarily the same even if they occur in the same line.
For Ys ; = {y1 , y 2 ,K, y s , y 0 = −1 < y1 < K < y s < 1 = y s +1} . We denote by
∆ 0 (Ys ) the set of all functions f ∈ L p [− 1,1] such that (− 1)
s−k
f ( x ) ≥ 0 for
x ∈ [ y k , yk +1 ], k = 0,1,2, K , s , it mean every f ∈ ∆ 0 (Ys ) has 0 ≤ s < ∞ sign
changes at the points in Ys and is nonnegative near I . 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 coapproximation function from ∆0 (Ys ) by
polynomials pn of degree ≤ n that are copositive with f . For f ∈ L p [− 1,1]
let
2
En ( f ) p := inf f − pn p ,
pn∈Pn
denote the degree of unconstrained approximation and let
En0 ( f , Ys ) p :=
inf
pn ∈Pn ∩∆0 (Ys )
f − pn p ,
be the degree of copositive approximation to f by algebraic polynomials
of degree ≤ n , where
f
p
= f
L p [ a ,b ]
. The degree of intertwining
polynomial approximation of functions f ∈ L p [− 1,1] with respect to Ys is
given by
{
~
En ( f , Ys ) p := inf P − Q p : P, Q ∈ Pn , P − f ∈ ∆0 (Ys ) and f − Q ∈ ∆0 (Ys )},
We call {P, Q} an intertwining pair of polynomials for f with respect to Ys
if P − f , f − p ∈ ∆0 (Ys ) .For more details see [7].
We
j = 0,1,K , s + 1
[
]
denote
On (Ys ,∈) = ∪ sj =1 J j (n,∈)
J j (n, ∈) = y j − ∆ n ( y j )n∈ , y j + ∆ n ( y j )n∈ ∩ [− 1,1],
denote
and
and
On∗ (Ys ,∈) = ∪ sj+=10 J j (n,∈) . If ∈= 0 we shall also use the simpler notation
J j = J j (n ,0 ), On (Ys ) = On (Ys ,0 ) and On∗ (Ys ) = On∗ (Ys ,0 ) . Functions f and g
are called copositive on J ⊂ I := [− 1,1] if f ( x )g ( x ) ≥ 0∀x ∈ J . Function f
and g are called almost copositive on I with respect to Ys if they are
copositive on I − On∗ (Ys ) . We say that f and g are strongly (weakly)
almost copositive on
I
with respect to Ys if they are copositive on
3
I − On (Ys ,∈) where ε < 0(ε > 0 ) . In particular, if ε = −∞ , then strongly
almost copositive functions are just copositive. Define a function class
(ε − alm∆ )0n (Ys ) := {f : (− 1)s−k f (x ) ≥ 0
for x ∈ I − On∗ (Ys ,∈) .
If s = 0 it becomes:
(ε − alm∆ )0n := (ε − alm∆ )0n (Y0 )
[
]
:= { f : f ( x ) ≥ 0 for x ∈ − 1 + n −2+ε ,1 − n −2+ε },
the set of all strongly (weakly) almost nonnegative functions on I if
ε < 0(ε > 0) . Again if ε = 0 we omit the letter ε in the notation and use
(alm∆ )0n (Ys )
and
(alm∆ )0n
the latter is the set of almost nonnegative
functions on I . If ε = −∞ , strongly almost nonnegative functions are just
nonnegative.
We define a function class:
(alm∆ )0n (Ys ) = {f : (− 1)s−k f (x ) ≥ 0 for x ∈ I − On∗ (Ys )}.
The degree of almost
copositive polynomial approximation of f ∈ L p [− 1,1] ∩ ∆0 (Ys ) is
{
}
E n(0 ) ( f , almYs ) p := inf f − p p : p ∈ Pn ∩ (alm∆ )n (Ys ) .
o
Similarly, we define E n(0 ) ( f , ε − almYs ) p the degree of strongly (weakly)
almost copositive polynomial approximation of f ∈ L p [− 1,1] ∩ ∆0 (Ys ) by
means of p ∈ Pn ∩ (ε − alm∆ )n (Ys ) .
0
4
It was shown by Hu, Kopotun and Yu [5] that if f changes its sign
in (− 1,1) , ωϕ1 being the best order of approximation:
Theorem A. If f ∈ L p [− 1,1] ∩ ∆0 (Ys )0 < p < 1 then for every n ∈ N − {0}
En0 ( f , Ys ) p ≤ c( p )ωϕ ( f , n −1 )p .
Also it was shown that:
One can not replace ωϕ1 ( f , n −1 )p by ωϕ2 ( f , n −1 )p for 0 < p < 1, where
ωϕm ( f , t ) p := sup ∆mhϕ (.) ( f ,⋅, [− 1,1]) p
0< h≤t
is the m th Ditzian Totik modulus of smoothness with ϕ ( x ) = 1 − x 2 , and
 m  m
m
m


m −i 
∆mh ( f , x, [− 1,1]) := ∑  (− 1) f  x − h + ih  if x ± h ∈ [− 1,1] .
2
2



i =0  i 
Little is known about copositive and almost copositive approximation of
functions in L p [− 1,1] ∩ ∆0 (Ys ) 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 .
Now the order ωϕ2 is impossible, we seek for a best rate. Our
theorem below shows that almost copositive approximation in L p ,0 < p < 1
improves the rate to ωϕ2 ( f , n −1 )p from ωϕ1 ( f , n −1 )p :
5
TheoremI. Suppose f ∈ L p [− 1,1] ∩ ∆0 (Ys )0 < p < 1 for any n > c(Ys ) we
have
E n(0 ) ( f , almYs ) p ≤ c( p, s )ωϕ2 ( f , n −1 ) p
(1)
The following theorem and corollary show that (1) is exact for
0<p<1, that is
TheoremII. Let Ys be fixed. For any given A > 0,0 < p < 1 and sufficiently
large n ∈ N , there exists a function f ∈ C [− 1,1] ∩ ∆0 (Ys ) such that for
every
polynomial
p n ∈ Pn
which
is
copositive
with
f
on
1 − ys
1 − ys 

y
+
,
1
−
the following inequality holds
s

3
3 
f − pn
where β <
p
> An β ωϕ4 ( f , n −1 )p
(2)
p
.
p+2
CorolaryIII. Let Ys be fixed. For any given 0 ≤ ε < 1 and sufficiently large
n ∈ N ,there exists f ∈ C [− 1,1] ∩ ∆0 (Ys ) such that
E n(0 ) ( f , ε − almYs ) p > ac( p, s )ωϕ4 ( f , n −1 )p .
2. Weak Copositive Approximation
6
In this section we show in theorem I that weak almost copositive
approximation in L p ,0 < p < 1 improves the rate to ωϕ2 from ωϕ1 . We first
need the following result from [6] :
Lemma1. Let Ys , s ≥ 0 be given m ∈ N , µ ≥ 2m + 30,0 < p < ∞ , and let
S ( x ) be a spline of an odd order r = 2m + 1 on the knot sequence
iπ 

where n > c(Ys ) is such that there are at least 4 knots xi
 xi = cos 
n i∈I n (Ys )

in
each
interval
(y , y ), j = 0,..., s
j
j +1
and
I n (Ys ) = {1,..., n} \
{i, i − 1, xi ≤ y j < xi−1} for some 1 ≤ j ≤ s . Then there exists an intertwining
pair of polynomials {P1 , P2 } ⊂ Pc (r )n for S with respect to Ys such that
P1 − P2
p
p
n −1
≤ c(r , s, min{1, p}) ∑ Er −1
p
i =1
(
)
p
S , Iˆi ∪ Iˆi +1 , if o < p < ∞
p
where Iˆi = [xi , xi −1 ].
Also we need the following assertion in [4]
Lemma 2. for any f ∈ L p (I ),0 < p < 1 and r ∈ N we have
E n ( f ) p ≤ c( p )ωϕr ( f , n −1 ) p .
Proof of theorem I.
Note that
~
p
p
E n(0 ) ( f , ε − almYs ) p ≤ E n ( f , Ys ) p
7
(3)
≤ P1 − P2 p , where P1 , P2 the polynomials defined in
p
Lemma 1. Then Lemma 1 and Lemma 2 imply
En(0 ) ( f , ε − almYs ) p ≤ c( p, s ) ∑ E r −1 (S , I i∗ ∪ I i∗+1 ) .
p
p
p
n −1
p
i =1
For f ∈ L p (I ),0 < p < 1 define a quadratic spline on
k +1
iπ 

Tk −2 s =  xi = cos 
, by S = Tf := ∑ ci N i , ci = d i∗−1 ∫ f , d i∗ is an
k i∈I k (Ys )
i = − r +1

I ∗i
 ∗ d i∗ ∗ d i∗  ∗
absolute constant I i = t i − , t i + , t i is an auxiliary knots in [− 1,1]
2
2 

∗
(see [3], p. 223), to get
E n(0 ) ( f , ε − almYs ) p ≤ c( p, s ) ∑ E 2 (S , I i∗ ∪ I i∗+1 )
p
p
p
k −1
p
i =1
k −1
(
≤ c( p, s )∑ ωϕ2 f , I i∗ ∪ I i∗+1 , I i∗ ∪ I i∗+1
i =1
)
p
p
≤ c( p, s )ωϕ2 ( f , n −1 ) p . ☻
p
3. The counter example
In this section we construct the counter example described in
Theorem II. We show that weakly almost copositive approximation doesn’t
8
do better than those Corollary III, in spite of larger intervals in which the
restriction is relaxed.
Proof of Theorem II
  1 + ys  2
 s
1 − ys
Let n ≥ s + 2 , L( x ) =   x −
− b 2  Π (x − y j ) where b <


 j =1
2 
6


is a constant, and let
1 + ys
1 + y s

if x ∉ 
− b,
+ b
6
 6

otherewise

 L( x )
f (x ) = 
0

suppose that (2) is true, it means there exists a polynomial p n ∈ Pn such
1 + ys
1 + y s

that p n ( x ) ≥ 0 for x ∈ 
− b,
+ b  and
6
 6

f − pn
p
≤ An β ωϕ4 ( f , n −1 )p . Let us assume β ≥ 0 . Note that
1/ p
f −L
p
f −L
p
(
 1+2ys +b



p
=  ∫ L( x ) dx 
 1+ ys −b

 2

b<
, and since
= c( p )b 2+1 / p , and
ωϕ4 f , n −1
)
p
(
≤ c( p )ωϕ4 f − L, n −1
≤ c( p ) f − L
p
)
p
(
+ c( p )ωϕ4 L, n −1
+ c( p )n −4 L(4 )
p
≤ c( p )b 2+1 / p + c( p )n −4 .
9
)
p
1 + ys 1 + ys
<
we have
6
2
Also by the well known inequality in [2]
pk
pn − L
L∞ [a ,b ]
≤ c( p, k )(b − a )
−1 / p
pk
L p [ a ,b ]
, for p k ∈ Pk ,
  1 + ys   1 + ys  
≥ c( p )n −1  p n 
 − L
 
  2   2 
p
 1 + ys 
≥ −c( p )n −1 L

 2 
s 1+ y


s
= c( p )n −1b 2 Π 
− yj 
j =1
 2

≥ c( p )n −1b 2 .
Therefore
c( p )n −1b 2 ≤ p n − f
p
+ f −L
p
≤ An β ωϕ4 ( f , n −1 )p + c( p )b 2+1 / p
≤ c( p )n β b 2+1 / p + c( p )n −4 + c( p )b 2+1 / p
≤ c( p )n β b 2+1 / p + c( p )n −4 .
This implies the inequality
c( p )n −1b 2 − n β b 2+1 / p ≤ c( p )n −4 .
Now let b = n
−1−
β
p
, then the last inequality implies
4−3−
n
2β
p
−n

β  1 
β  −1−   2 + 
p
p

and
10


≤ c( p )
4−3−
n
2β
−β
p
≤ c( p ).
But this can not be true for sufficiently large n , since condition on β and
p in the theorem imply 4 > −3 + 2 β + 2 β
1
+β. ☻
p
References
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11
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