Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 4, Article 50, 2002
RATE OF CONVERGENCE OF THE DISCRETE POLYA ALGORITHM FROM
CONVEX SETS. A PARTICULAR CASE
M. MARANO, J. NAVAS, AND J.M. QUESADA
D EPARTAMENTO DE M ATEMÁTICAS
U NIVERSIDAD DE JAÉN
PARAJE LAS L AGUNILLAS
C AMPUS U NIVERSITARIO
23071 JAÉN , S PAIN
mmarano@ujaen.es
jnavas@ujaen.es
jquesada@ujaen.es
Received 4 December, 2001; accepted 28 May, 2002
Communicated by A. Babenko
A BSTRACT. In this work we deal with best approximation in `np , 1 < p ≤ ∞, n ≥ 2. For
1 < p < ∞, let hp denote the best `np -approximation to f ∈ Rn from a closed, convex subset
K of Rn , f 6∈ K, and let h∗ be a best uniform approximation to f from K. In case that h∗ − f
= (ρ1 , ρ2 , · · · , ρn ), |ρj | = ρ for j = 1, 2, · · · , n, we show that the behavior of khp − h∗ k as
p → ∞ depends on a property of separation of the set K from the `n∞ -ball {x ∈ Rn : kx − f k ≤
ρ} at h∗ − f .
Key words and phrases: Best uniform approximation, Rate of convergence, Polya Algorithm, Strong uniqueness.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
Let (w1 , w2 , . . . , wn ) be a fixed vector in Rn , with wj > 0, j ∈ In := {1, 2, . . . , n}, n ≥ 2.
For x = (x(1), x(2), . . . , x(n)) ∈ Rn we define
! p1
n
X
, 1 ≤ p < ∞, and
kxkp,w :=
wj |x(j)|p
j=1
kxk := max |x(j)|.
1≤j≤n
Pn
Also we define N := j=1 wj .
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
This work was partially supported by Junta de Andalucía, Research Group 0268.
The authors gratefully acknowledge the many helpful suggestions of the referees during the revision of this paper.
086-01
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M. M ARANO , J. NAVAS ,
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Throughout the paper, K will always be a nonempty, closed, convex subset of Rn . For
f ∈ Rn \K, we will say that hp,w ∈ K, 1 ≤ p < ∞, is a best `np,w -approximation to f from K if
kf − hp,w kp,w ≤ kf − hkp,w
∀ h ∈ K.
`np,w −approximation
The existence of at least one best
to f from K is a known fact for 1 ≤
p < ∞. Likewise, there always exists a best uniform approximation to f from K, i.e., an
h∗ ∈ K that satisfies
kf − h∗ k ≤ kf − hk ∀ h ∈ K.
We will henceforth assume f = 0 and 0 6∈ K. This causes no loss of generality, since all relevant properties are translation invariant. If 1 < p < ∞, there is a unique best `np,w −approximation.
In this case, the next theorem [14] characterizes the best `np,w −approximation to 0 from K.
Theorem 1.1 (Characterization of the best `np,w -approximation). Let K be a closed, convex
subset of Rn , 0 6∈ K. Then hp,w , 1 ≤ p < ∞, is a best `np,w −approximation to 0 from K if and
only if for all h ∈ K,
n
X
(1.1)
wj (hp,w (j) − h(j))|hp,w (j)|p−1 sgn(hp,w (j)) ≤ 0, if p > 1.
j=1
(1.2)
X
X
wj (h1,w (j) − h(j)) sgn(h1,w (j)) ≤
j∈R(h1,w )
wj |h(j)|,
if p = 1,
j∈Z(h1,w )
n
where, if g ∈ R , Z(g) := {j ∈ In : g(j) = 0} and R(g) := In \ Z(g).
It is also known [1, 6, 7] that if K is an affine subspace, then
lim hp,w = h∗ ,
(1.3)
p→∞
where in this case h∗ is a particular best uniform approximation to 0 from K, called strict
uniform approximation [12, 7] and whose definition is also valid in any closed, convex K. In
[3, 8] it is proved that there exists a constant M > 0 such that p khp,w − h∗ k ≤ M for all
p > 1. Moreover, from [13] it is deduced that there are constants M1 , M2 > 0 and 0 ≤ a ≤ 1,
depending on K, such that
M1 ap ≤ p khp,w − h∗ k ≤ M2 ap
for all p > 1.
In [2, 7] it is shown that if K is not an affine subspace, then hp,w does not necessarily converge
to the strict uniform approximation, though (1.3) is always valid whenever h∗ is the unique best
uniform approximation to 0 from K. In [6, 7] we can find sufficient conditions on K under
which (1.3) is satisfied. In any case, the convergence of hp,w as p → ∞ to a best uniform
approximation is known as the Polya algorithm [11]. The purpose of this paper is to study the
behavior of khp,w − h∗ k as p → ∞ when h∗ is a best uniform approximation to 0 from K and
h∗ satisfies |h∗ (j)| = ρ > 0 ∀ j ∈ In .
2. R ELATION B ETWEEN S TRONG U NIQUENESS AND R ATE
OF
C ONVERGENCE
A useful concept in order to get a first general result on the rate of convergence of the Polya
algorithm is strong uniqueness. It was established in 1963 by Newman and Shapiro [10] in the
context of the uniform approximation to continuous functions by means of elements of a Haar
space, although we could define it in any normed space.
Definition 2.1. Let h∗ ∈ K be a best uniform approximation to 0 ∈ Rn from K. We say that
h∗ is strongly unique if there exists γ > 0 such that
(2.1)
kh − h∗ k ≤ γ(khk − kh∗ k) ∀ h ∈ K.
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It is obvious that if h∗ is strongly unique, then h∗ is the unique best uniform approximation to
0 ∈ Rn from K.
Theorem 2.1. If the best uniform approximation h∗ to 0 from K is strongly unique, then
p khp,w − h∗ k is bounded for all p ≥ 1.
Proof. We first note that for every h ∈ K,
(2.2)
1
1
m p khk ≤ khkp,w ≤ N p khk,
where m := minj∈In {wj }.
Let γ > 0 satisfy (2.1). Then for any p ≥ 1,
(2.3)
khp,w − h∗ k ≤ γ(khp,w k − kh∗ k).
Applying (2.2) and the definition of best `np,w -approximation, we have
khp,w k − kh∗ k ≤
1
m
1
1
p
khp,w kp,w − kh∗ k
kh∗ kp,w − kh∗ k
"m 1
#
N p
≤
− 1 kh∗ k
m
≤
≤
1
p
(N − m)kh∗ k
.
mp
From (2.3) we finally conclude that
γ(N − m)kh∗ k
p khp,w − h k ≤
m
∗
for all p ≥ 1.
The above inequality improves the proposal in [4] and [5].
2.1. The Particular Case |h∗ (j)| = ρ > 0, j = 1, 2, . . . , n. We henceforth suppose that
h∗ ∈ K is a best uniform approximation to 0 from K, where |h∗ (j)| = ρ > 0 for all j ∈ In .
Under these conditions we will analyze the behaviour of khp,w −h∗ k as p → ∞. In Theorem 2.3,
our main result, we will prove that the converse of Theorem 2.1 – which is generally not true –
n
is valid in this particular case. Since
K = {h ∈ K : khk = ρ}, itois easy
n {x ∈ R : kxk ≤ ρ} ∩P
to see that there is a hyperplane (x(1), x(2), . . . , x(n)) : nj=1 aj sgn(h∗ (j)) x(j) = ρ , with
P
0 ≤ aj ≤ 1, all j ∈ In , and nj=1 aj = 1, that separates K from the ball {x ∈ Rn : kxk ≤ ρ}
P
at h∗ , i.e., nj=1 aj sgn(h∗ (j)) h(j) ≥ ρ for all h ∈ K.
Definition 2.2. We will say that
(
)
n
X
π := (x(1), x(2), . . . , x(n)) :
aj sgn(h∗ (j)) x(j) = ρ
j=1
is a hyperplane that strongly separates K from the ball {x ∈ Rn : kxk ≤ ρ} at h∗ , or equivalently, that π is a strongly separating hyperplane at h∗ , if
(2.4)
0 < aj < 1, all j ∈ In ,
n
X
aj = 1
j=1
J. Inequal. Pure and Appl. Math., 3(4) Art. 50, 2002
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4
M. M ARANO , J. NAVAS ,
AND
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and
n
X
(2.5)
aj sgn(h∗ (j)) h(j) ≥ ρ
∀ h ∈ K.
j=1
In the proofs of Lemma 2.2 and Theorems 2.3 and 2.4 we will assume h∗ (j) = 1 for all
j ∈ In . This causes no loss of generality, since we can replace K by the closed, convex set
1
n
∗
h̃ ∈ R : h̃(j) = h(j) sgn(h (j)), j ∈ In , h ∈ K .
ρ
Lemma 2.2. If pk khpk ,w − h∗ k is bounded for pk → ∞, then there exists a strongly separating
hyperplane at h∗ .
Proof. Since limpk →∞ hpk ,w (j) = h∗ (j) = 1, all j ∈ In , we can suppose hpk ,w (j) > 0, all
j ∈ In and, without loss of generality, all pk . Then, for every pk the formula of characterization
(1.1) can be expressed in the form
n
X
wj (hpk ,w (j) − h(j))hppkk −1
,w (j) ≤ 0 ∀ h ∈ K.
j=1
Dividing by
khpk ,w kppkk ,w ,
for every pk we obtain
pk
n
X
h(j)
hpk ,w (j)
≥ 1 ∀ h ∈ K.
wj
kh
k
h
(j)
p
,w
p
,w
p
,w
k
k
k
j=1
(2.6)
Keeping in mind that
wj hppkk ,w (j) ≤ khpk ,w kppkk ,w ≤ kh∗ kppkk ,w = N,
j ∈ In ,
and after passage to a subsequence, we can suppose that hppkk ,w (j), all j ∈ In , and khpk ,w kppkk ,w
are convergent. Now, by hypothesis, pk |hpk ,w (j) − 1| is bounded for all j ∈ In and all pk .
Hence we get
lim hppkk ,w (j) = lim Exp (pk (hpk ,w (j) − 1)) > 0,
pk →∞
pk →∞
all j ∈ In .
Writing
pk
hpk ,w (j)
aj = lim wj
, j ∈ In ,
pk →∞
khpk ,w kpk ,w
P
we therefore deduce that 0 < aj < 1, all j ∈ In , and nj=1 aj = 1. Taking limits as pk → ∞ in
(2.6), we finally conclude that
n
X
aj h(j) ≥ 1 ∀ h ∈ K.
j=1
n
o
Pn
Then (x(1), x(2), . . . , x(n)) := j=1 aj x(j) = 1 is a strongly separating hyperplane at h∗ .
Theorem 2.3. The following statements are equivalent:
(a) The best uniform approximation to 0 from K, h∗ , is strongly unique.
(b) p khp,w − h∗ k is bounded for all p ≥ 1.
(c) pk khpk ,w − h∗ k is bounded for a sequence pk → ∞.
(d) There exists a strongly separating hyperplane at h∗ .
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P OLYA A LGORITHM
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Proof. (a) ⇒ (b) is Theorem 2.1. (b) ⇒ (c) is obvious. (c) ⇒ (d) is Lemma 2.2. To complete the
theorem, we now prove (d) ⇒ (a). Suppose that there is a strongly separating hyperplane π at
h∗ = (1, 1, . . . , 1). Let h ∈ K. Observe that khk ≥ 1. Let In+ denote the subset of indices j in In
such that h(j) ≥ 1, and let In− := In \In+ . For all j ∈ In+ we have |h(j)−1| = h(j)−1 ≤ khk−1.
On the other hand, if j ∈ In− , then |h(j) − 1| = 1 − h(j). Moreover, the inequality
X
ai (h(i) − 1) ≥ 0
i∈In
implies
aj (1 − h(j)) ≤
X
ai (h(i) − 1)
i∈In ,i6=j
≤
X
ai (h(i) − 1)
i∈In+
X
≤ (khk − 1)
ai
i∈In+
≤ (khk − 1)(1 − aj ).
Thus, for all j ∈ In we have
|h(j) − 1| ≤
1
mini∈In ai
− 1 (khk − 1) := γ(khk − 1),
and so kh − h∗ k ≤ γ(khk − kh∗ k).
Our goal now is to show that, under the conditions of Theorem 2.3, either hp,w = h∗ for all p
or there exist constants M1 , M2 > 0 such that
M1 ≤ p khp,w − h∗ k ≤ M2 for all p ≥ 1.
On the other hand, if there exists no strongly separating hyperplane at h∗ , then the following
example in R2 , where limp→∞ hp,w = h∗ , shows that the rate of convergence is as slow as we
want.
Example 2.1. Let α : [1, +∞) → (0, 1] be a continuous strictly decreasing function such that
α(1) = 1 and lim α(t) = 0 and let β : (0, 1] → [1, +∞) denote its inverse function, that will
t→∞
also be a strictly decreasing function. We define
Z 1
f (x) := 1 +
(1 − t)β(1−t) dt, 0 ≤ x ≤ 1,
x
and let K be the convex hull of the set {(x, y) ∈ R2 : y = f (x), x ∈ [0, 1]}.
Observe that h∗ = (1, 1) is the unique best uniform approximation to (0, 0) from K. Moreover, the function f is smooth, convex and f 0 (1) = 0. This implies that the strongly separating
hyperplane at h∗ does not exist.
Let hp = (1 − εp , 1 + δp ) be the best p-approximation to (0, 0) from K, with εp , δp ↓ 0 as
p → ∞. Since the slopes of the curve y = f (x) and the `p -ball coincide at hp , we have
(1 − εp )p−1
p)
= εβ(ε
p
p−1
(1 + δp )
and therefore
(2.7)
1 − εp
= 1.
p→∞ 1 + δp
p )/(p−1)
lim εβ(ε
= lim
p
p→∞
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M. M ARANO , J. NAVAS ,
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If εp ≤ α(p), then β(εp ) ≥ β(α(p)) = p, which contradicts (2.7). Then, for p large, we have
εp > α(p). This shows that the rate of convergence of hp to h∗ as p → ∞ can be as slow as we
want.
Theorem 2.4. The following conditions are equivalent
(a) hp,w = h∗ for all p ≥ 1,
(b) hp0 ,w = h∗ for some p0 ≥ 1,
(c) the hyperplane
(
(2.8)
π :=
(x(1), x(2), . . . , x(n)) :
n
X
j=1
wj
sgn (h∗ (j)) x(j) = ρ
N
)
is a strongly separating hyperplane at h∗ .
Proof. (a) ⇒ (b) is obvious. (b) ⇒ (c) follows immediately from Theorem 1.1. Indeed, if
hp0 ,w = h∗ for some p0 ≥ 1, then from (1.1) if p0 > 1 or (1.2) if p0 = 1, we have
(2.9)
n
X
wj (h∗ (j) − h(j)) sgn(h∗ (j)) ≤ 0 ∀h ∈ K,
j=1
which is equivalent to the fact that π is a strongly separating hyperplane at h∗ . Also from (1.1)
and (1.2), the inequality (2.9) implies that hp,w = h∗ for all p ≥ 1 and so (c) ⇒ (a).
Theorem 2.5. Suppose that hp,w 6= h∗ for some p ≥ 1 and there exists a strongly separating
hyperplane at h∗ . Then there are constants M1 , M2 > 0 such that
M1 ≤ p khp,w − h∗ k ≤ M2
for all p ≥ 1.
Proof. Assume that there exists a strongly separating hyperplane at h∗ , where h∗ (j) = 1 for all
j ∈ In . From Theorem 2.3, there is a constant M2 > 0 such that
p khp,w − h∗ k ≤ M2 .
Therefore, to prove the theorem it is sufficient to show that inf p≥1 {p khp,w − h∗ k} > 0. Suppose the contrary. In order to get a contradiction, we only need to consider the two following
exhaustive cases:
(1) There exists a sequence pk → ∞ such that limpk →∞ pk khpk − h∗ k = 0. In this case
limpk →∞ pk |hpk ,w (j) − 1| = 0 for all j ∈ In . This implies that hppkk ,w (j) → 1 as pk → ∞
and
pk
hpk ,w (j)
wj
∗
aj = lim wj
=
,
j = 1, 2, . . . , n,
pk →∞
khpk ,w kpk ,w
N
which means (see the proof of Lemma 2.2) that the hyperplane (2.8), with h∗ (j) =
ρ = 1 for all j ∈ In , is a strongly separating hyperplane at h∗ . From Theorem 2.4 (c),
hp,w = h∗ for all p ≥ 1, which contradicts the hypothesis of the theorem.
(2) There exists a sequence pk → p0 , 1 ≤ p0 < ∞, such that limpk →p0 pk khpk ,w − h∗ k = 0.
Since hpk ,w → hp0 ,w , we deduce that khp0 ,w − h∗ k = limpk →p0 khpk ,w − h∗ k = 0 and so
hp0 ,w = h∗ . Now, using the statement (b) of Theorem 2.4, we conclude that hp,w = h∗ ,
for all p ≥ 1. A contradiction.
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P OLYA A LGORITHM
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2.2. A Numerical Example in Isotonic Approximation.
Let f = (a + 1, . . . , a + 1, a − 1, . . . , a − 1) ∈ Rn , and let K be the convex set of the nonde|
{z
} |
{z
}
r
n−r
creasing vectors in Rn , i.e.
K = {h ∈ Rn : h(i) ≤ h(j) ∀ i, j ∈ In , i < j} .
In this case, the (unique) best uniform approximation to f from K is the element h∗ = (a, a, . . . , a).
Thus hp,w → h∗ as p → ∞. Furthermore, it is easy to see that
hp,w = (xp,w , xp,w , . . . , xp,w ) ∈ Rn ,
1 < p < ∞,
for some xp,w satisfying a − 1 ≤ xp,w ≤ a + 1.
In order to translate h∗ to a vertex of the `n∞ -ball, we consider the closed, convex set
e = {e
K
h ∈ Rn : e
h(j) = h(j) − f (j) , j ∈ In , h ∈ K}.
In this way we obtain
• fe = (0, 0, . . . , 0);
•e
h∗ = h∗ − f = (−1, . . . , −1, 1, . . . , 1).
| {z } | {z }
r
n−r
To simplify the notation, we will write σj = sgn(e
h∗ (j)), j ∈ In . Now, we are interested in
obtaining a strongly separating hyperplane at e
h∗ , i.e., a hyperplane
(
)
n
X
π := (x(1), x(2), . . . , x(n)) :
aj σj x(j) = 1
j=1
such that
P
(p1) 0 < aj < 1, all j ∈ In , and n1 aj = 1;
P
e
(p2) nj=1 σj aj e
h(j) ≥ 1 ∀ e
h ∈ K.
r
P
Proposition 2.6. Let S :=
wj . Then the above hyperplane π, with
j=1
aj =
wj
if 1 ≤ j ≤ r,
2S
aj =
wj
if r + 1 ≤ j ≤ n,
2(N − S)
satisfies (p1) and (p2), and therefore it is a strongly separating hyperplane at e
h∗ .
Proof. By definition, 0 < aj < 1 for all j ∈ In . Furthermore,
n
X
σj aj e
h∗ (j) =
j=1
n
X
r
n
X
X
wj
wj
1 1
aj =
+
= + = 1.
2 S j=r+1 2(N − S)
2 2
j=1
j=1
Then (p1) holds.
Since
n
X
r
n
X
X
wj
wj
σj aj f (j) = −(a + 1)
+ (a − 1)
= −1,
2S
2(N − S)
j=1
j=1
j=r+1
(p2) is equivalent to
(2.10)
n
X
σj aj h(j) ≥ 0 ∀ h ∈ K.
j=1
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M. M ARANO , J. NAVAS ,
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But if h is a nondecreasing vector, then (2.10) is immediate because
r
X
wj h(j) ≤ h(r)
j=1
r
X
wj = S h(r) and
j=1
n
X
n
X
h(j) ≥ h(r)
j=r+1
wj = (N − S)h(r),
j=r+1
and therefore
n
X
σj aj h(j) = −
j=1
≥ −
r
n
X
1
1 X
wj h(j) +
wj h(j)
2 S j=1
2(N − S) j=r+1
1
1
S h(r) +
(N − S)h(r) = 0.
2S
2(N − S)
This concludes the proof.
From Proposition 2.6 we deduce that if S = N/2, then
(
)
n
X
wj
(2.11)
(x(1), x(2), . . . , x(n)) :
σj
x(j) = 1
N
j=1
is a strongly separating hyperplane at e
h∗ , and from Theorem 2.4 this is equivalent to e
hp,w = e
h∗
for all 1 ≤ p < ∞. In the case that S 6= N/2, we claim that e
hp,w → e
h∗ as p → ∞ exactly
at a rate O
1
p
. From Proposition 2.6 and Theorems 2.4 and 2.5 we only need to show that
(2.11) is not a strongly separating hyperplane at e
h∗ . This last assertion is true since (2.10), with
aj = wj /N , all j ∈ In , implies
(2.12)
n
X
wj h(j) ≥
j=r+1
r
X
wj h(j) ∀ h ∈ K.
j=1
On the other hand, if S < N/2 then h = (−1, −1, . . . , −1) ∈ K does not satisfy (2.12), and an
analogous conclusion is valid for h = (1, 1, . . . , 1) ∈ K if S > N/2. This proves the claim.
In what follows we obtain these same results calculating directly the best `np,w −approximations to f from K, namely, hp,w = (xp,w , xp,w , . . . , xp,w ). It is easy to check that
1
1
a − 1 + a N S−S p + N S−S p
xp,w =
, 1 < p < ∞.
p1
S
1 + N −S
Then we immediately conclude that if S = N/2, then hp,w = h∗ for p > 1, and if S 6= N/2,
then hp,w → h∗ as p → ∞. Moreover, we can calculate the rate of convergence. Indeed,
1/p
( NS−S ) −1
1/p
1+( N S
hp,w (j) − h∗ (j)
1
−S )
lim
= lim
= lim
p→∞
p→∞
1/p
1/p
2 p→∞
The rate of convergence is exactly O p1 .
S
N −S
p1
1/p
−1
1
= ln
2
S
N −S
.
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