BEST APPROXIMATIONS IN L1 ARE NEAR BEST IN Lp , p < 1
LAWRENCE G. BROWN AND BRADLEY J. LUCIER
(Communicated by J. Marshall Ash)
Abstract. We show that any best L1 polynomial approximation to a function f
in Lp , 0 < p < 1, is near best in Lp .
Let I = [0, 1]d and let Pr be the set of all polynomials in d variables of total
degree less than r. It is known that for each f in L1 (I) a (not necessarily unique)
best approximation E1 f to f exists in Pr , and that E1 f = q if and only if
Z
Z
Z
s(x) dx −
s(x) dx ≤
|s(x)| dx for all s in Pr ,
(1)
E+
E−
E0
where E+ = {x ∈ I | q(x) > f (x)}, E− = {x ∈ I | q(x) < f (x)}, and E0 =
{x ∈ I | q(x) = f (x)}. Condition (1) makes sense even if f is not in L1 (I), and
when (1) is satisfied, we call q a best L1 (I) approximation to f and denote q
by E1 f . It is easy to show that for constant approximations ( r = 1 ) the extended
E1 , which is the median operator, is defined for all measurable f and bounded
on Lp (I) for any p > 0 (see [2] for a discussion of medians). It is also easy to
show that the similarly extended L2 best projection operator onto polynomials is
bounded on Lp for p ≥ 1 and any r > 0. These facts motivate us to prove the
following theorem.
Theorem. For each f in Lp (I), 0 < p < 1, and for all r > 0, a best L1 (I)
approximation E1 f exists in Pr . Moreover, for all choices of E1 f ,
kf − E1 f kp ≤ (1 + 2K)1/p inf kf − qkp ,
q∈Pr
where
K = sup
q∈Pr
kqk∞
.
kqk1
This theorem provides a method to find near-best polynomial approximations in
Lp (I) for 0 < p < 1. Such approximations are useful in atomic decompositions of
Received by the editors October 29, 1990 and, in revised form, May 28, 1991.
1991 Mathematics Subject Classification. 41A50, 41A10, 46E30.
Key words and phrases. Best approximation, Lp spaces.
The second author was supported in part by the National Science Foundation (grants DMS8802734 and DMS-9006219), the Office of Naval Research (contract N00014-91-J-1152), and by
the Army High Performance Computing Research Center.
Typeset by AMS-TEX
1
2
L. G. BROWN AND B. J. LUCIER
the Besov spaces Bpα (Lp (I)), p < 1, which are the regularity spaces for nonlinear
approximation in Lq (I), q −1 = p−1 − α/d, by wavelets and free-knot splines.
Theory and applications to image and surface compression can be found in [2–6].
Of course, such near-best approximations are known to exist; the new thing here is
that L1 (I) projections provide them.
We make several remarks about the theorem. First, a more careful argument
shows that
1
kf − E1 f kp ≤ (2K) p −1 inf kf − qkp
q∈Pr
for 0 < p < 1; see [1]. Second, our proof extends to approximation by any finitedimensional subpace of L1 (Ω) ∩ L∞ (Ω) for any finite measure space (Ω, dµ). (A
measure space with atoms must first be embedded into a continuous measure space.)
Generalizations to the behavior of the best Lp (I) approximation in Lp−1 (I) for
p > 1, and the fact that the best L1 (I) approximation is defined for any measurable
f are contained in [1].
We first prove the following lemma, which contains the main argument.
Lemma. If there exists a best L1 (I) approximation E1 f ∈ Pr to f ∈ Lp (I),
then
1/p
kqk∞
kf kp .
kE1 f kp ≤ 2 sup
q∈Pr kqk1
Proof. Condition (1) implies, in particular, that for q = E1 f ,
Z
Z
Z
q(x) dx −
q(x) dx ≤
|q(x)| dx.
(2)
E+
E−
E0
Our approach is to bound from below kf kp among all f satisfying (2) for a
particular q.
We introduce the function

0,
x ∈ E+ , q(x) > 0,




q(x),
x
∈ E+ , q(x) ≤ 0,


x ∈ E− , q(x) < 0,
g(x) = 0,



q(x), x ∈ E− , q(x) ≥ 0,



q(x), x ∈ E0 .
Clearly, kgkp ≤ kf kp and the sets Ẽ+ ⊂ E+ , Ẽ− ⊂ E− , and Ẽ0 ⊃ E0 for g and
q satisfy
Z
Z
Z
q(x) dx −
q(x) dx ≤
|q(x)| dx.
(3)
Ẽ+
Ẽ−
Ẽ0
R
R
Now I |g|p = Ẽ0 |q|p , and to further bound kf kp from below we minimize
R
|q(x)|p dx among all partitions (Ẽ+ , Ẽ− , Ẽ0 ) with q > 0 on Ẽ+ , q < 0 on
Ẽ0
Ẽ− , and (3) holding. These conditions imply that
Z
Z
|q(x)| dx ≤
|q(x)| dx;
Ẽ+ ∪Ẽ−
Ẽ0
BEST APPROXIMATIONS IN L1 ARE NEAR BEST IN Lp , p < 1
i.e.,
Z
|q(x)| dx ≥
Ẽ0
3
1
kqk1 .
2
We claim that the best choice of Ẽ0 satisfies
Z
1
|q(x)| dx = kqk1 .
inf |q(x)| ≥ sup |q(x)| and
2
x∈Ẽ0
Ẽ0
x6∈Ẽ0
R
Suppose Ẽ00 is any other choice. We can assume Ẽ 0 |q(x)| dx = 12 kqk1 , because
0
R
otherwise we could make Ẽ00 , and, a fortiori, Ẽ 0 |q(x)|p dx, smaller.
0
Let a be any number between supx6∈Ẽ0 |q(x)| and inf x∈Ẽ0 |q(x)|, and let A =
R
R
Ẽ0 \ Ẽ00 and B = Ẽ00 \ Ẽ0 . Then A |q(x)| dx = B |q(x)| dx and
Z
Z
Z
|q(x)| dx =
|q(x)|p |q(x)|1−p dx ≥ a1−p
|q(x)|p dx,
ZA
ZA
ZA
p
1−p
1−p
|q(x)| dx =
|q(x)| |q(x)|
dx ≤ a
|q(x)|p dx.
B
Therefore,
B
B
Z
Z
|q(x)| dx ≤
|q(x)|p dx,
p
A
B
Z
Z
and
|q(x)|p dx ≤
Ẽ0
R
Ẽ0
|q(x)| dx = 12 kqk1 , we have
kqk∞ |Ẽ0 | ≥
Now
|q(x)|p dx.
R
R
|q(x)|p dx
|q(x)|p dx
I
RI
R
≤
.
p
|q(x)|p dx
I |f (x)| dx
Ẽ0
So
Since
Ẽ00
1
kqk1 ,
2
or
|Ẽ0 | ≥
1 kqk1
.
2 kqk∞
Z
|q(x)|p dx ≤ ap (1 − |Ẽ0 |)
I\Ẽ0
and
Z
|q(x)|p dx ≥ ap |Ẽ0 |.
Ẽ0
So
R
R
|q(x)|p dx
|q(x)|p dx
1 − |Ẽ0 |
1
kqk∞
I
R
RI\Ẽ0
=
1
+
≤1+
=
≤2
.
p
p
kqk1
|q(x)| dx
|q(x)| dx
|Ẽ0 |
|Ẽ0 |
Ẽ0
Ẽ0
Therefore, kqkp ≤ (2K)1/p kf kp , where K = supq∈Pr (kqk∞ /kqk1 ). 4
L. G. BROWN AND B. J. LUCIER
Corollary. For each f in Lp (I), 0 < p < 1, a best L1 (I) polynomial approximation E1 f exists.
Proof. For each positive integer n, define fn by

f (x) > n,

 n,
fn (x) = f (x), |f (x)| ≤ n,


−n,
f (x) < −n.
Then fn ∈ L∞ (I) and kfn kp ≤ kf kp . Best L1 (I) approximations qn to fn exist
for all n, and
kqn kp ≤ Ckfn kp ≤ Ckf kp .
However, for each p and r there is a constant C1 such that for all q ∈ Pr ,
kqk∞ ≤ C1 kqkp . Therefore for all n we have kqn k∞ ≤ CC1 kf kp , so that for some
n we have kqn k∞ < n. This implies that we can choose E1 f = qn because the
sets E+ , E− , and E0 are the same for f as for fn . Proof of the main theorem. The corollary shows that E1 f exists. Now E1 is linear
with respect to addition of polynomials in Pr : If we let g = f + q, q ∈ Pr , and
E1 g = E1 f + q, then clearly condition (1) is satisfied because E+ , E− , and E0
are the same for g and E1 g as for f and E1 f . So E1 f + q is a best L1 (I)
approximation to f + q according to our definition.
Finally, because kf + gkpp ≤ kf kpp + kgkpp , we have for all q ∈ Pr ,
kf − E1 f kpp ≤ kf − qkpp + kq − E1 f kpp
= kf − qkpp + kE1 (q − f )kpp
≤ (1 + 2K)kf − qkpp
by the lemma.
References
Lp
1. L. G. Brown,
best approximation operators are bounded on Lp−1 , in preparation.
2. R. A. DeVore, B. Jawerth, and B. J. Lucier, Image compression through wavelet transform
coding, IEEE Trans. Information Theory 38 (1992), 719–746.
, Surface compression, Computer Aided Geometric Design 9 (1992), 219–239.
3.
4. R. A. DeVore, B. Jawerth, and V. A. Popov, Compression of wavelet decompositions, Amer.
J. Math. 114 (1992), 737–785.
5. R. A. DeVore and V. A. Popov, Free multivariate splines, Constr. Approx. 3 (1987), 239–248.
, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305 (1988), 397–414.
6.
Department of Mathematics, Purdue University, West Lafayette, IN 47907
E-mail address: [email protected], [email protected]