Journal of Inequalities in Pure and
Applied Mathematics
RATE OF CONVERGENCE OF THE DISCRETE POLYA ALGORITHM
FROM CONVEX SETS. A PARTICULAR CASE
volume 3, issue 4, article 50,
2002.
M. MARANO, J. NAVAS AND J.M. QUESADA
Departamento de Matemáticas
Universidad de Jaén
Paraje las Lagunillas
Campus Universitario
23071 Jaén, Spain
Received 4 December, 2001;
accepted 28 May, 2002.
Communicated by: A. Babenko
EMail: mmarano@ujaen.es
EMail: jnavas@ujaen.es
Abstract
EMail: jquesada@ujaen.es
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
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 − fk ≤ ρ} at h∗ − f.
2000 Mathematics Subject Classification: 26D15
Key words: Best uniform approximation, Rate of convergence, Polya Algorithm,
Strong uniqueness.
The authors gratefully acknowledge the many helpful suggestions of the referees
during the revision of this paper.
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
Title Page
Contents
Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Relation Between Strong Uniqueness and Rate of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1
The Particular Case |h∗ (j)| = ρ > 0, j = 1, 2, . . . , n 7
2.2
A Numerical Example in Isotonic Approximation . . . 14
References
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1.
Introduction
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
kxkp,w :=
n
X
! p1
wj |x(j)|p
,
1 ≤ p < ∞,
and
j=1
kxk := max |x(j)|.
1≤j≤n
P
Also we define N := nj=1 wj .
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
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
Title Page
kf − hp,w kp,w ≤ kf − hkp,w
∀ h ∈ K.
The existence of at least one best `np,w −approximation 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
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∀ 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.
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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,
(1.1)
n
X
wj (hp,w (j) − h(j))|hp,w (j)|p−1 sgn(hp,w (j)) ≤ 0,
if p > 1.
j=1
(1.2)
X
wj (h1,w (j) − h(j)) sgn(h1,w (j)) ≤
j∈R(h1,w )
X
wj |h(j)|,
if p = 1,
j∈Z(h1,w )
where, if g ∈ Rn , Z(g) := {j ∈ In : g(j) = 0} and R(g) := In \ Z(g).
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
It is also known [1, 6, 7] that if K is an affine subspace, then
(1.3)
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lim hp,w = h∗ ,
p→∞
Contents
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
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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 .
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
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2.
Relation Between Strong Uniqueness and Rate
of Convergence
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.
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).
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
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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
M. Marano, J. Navas and
J.M. Quesada
From (2.3) we finally conclude that
p khp,w − h∗ k ≤
γ(N − m)kh∗ k
m
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
for all p ≥ 1.
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The above inequality improves the proposal in [4] and [5].
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The Particular Case |h∗ (j)| = ρ > 0, j = 1, 2, . . . , n
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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 – is valid in this particular case. Since {x ∈ Rn : kxk ≤
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ρ}
= ρ}, it is easy to see that
o there is a hyperplane
n ∩ K = {h ∈ K : khk
Pn
∗
(x(1), x(2), . . . , x(n)) : j=1 aj sgn(h (j)) x(j) = ρ , with 0 ≤ aj ≤ 1, all
Pn
n
j ∈ In , and
j=1 aj = 1, that separates K from the ball {x ∈ R : kxk ≤ ρ}
P
n
at h∗ , i.e., j=1 aj sgn(h∗ (j)) h(j) ≥ ρ for all h ∈ K.
Definition 2.2. We will say that
(
π :=
(x(1), x(2), . . . , x(n)) :
n
X
)
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
M. Marano, J. Navas and
J.M. Quesada
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and
(2.5)
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
n
X
∗
aj sgn(h (j)) h(j) ≥ ρ
∀ h ∈ K.
j=1
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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
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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
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
Dividing by khpk ,w kppkk ,w , for every pk we obtain
n
X
(2.6)
wj
j=1
hpk ,w (j)
khpk ,w kpk ,w
pk
h(j)
≥ 1 ∀ h ∈ K.
hpk ,w (j)
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Keeping in mind that
wj hppkk ,w (j) ≤ khpk ,w kppkk ,w ≤ kh∗ kppkk ,w = N,
Contents
j ∈ In ,
we can suppose that hppkk ,w (j),
and after passage to a subsequence,
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 →∞
M. Marano, J. Navas and
J.M. Quesada
pk →∞
pk →∞
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Writing
aj = lim wj
all j ∈ In .
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hpk ,w (j)
khpk ,w kpk ,w
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pk
,
j ∈ In ,
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we therefore deduce that 0 < aj < 1, all j ∈ In , and
limits as pk → ∞ in (2.6), we finally conclude that
n
X
Pn
j=1
aj = 1. Taking
aj h(j) ≥ 1 ∀ h ∈ K.
j=1
n
o
P
Then (x(1), x(2), . . . , x(n)) := nj=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.
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
(b) p khp,w − h∗ k is bounded for all p ≥ 1.
(c) pk khpk ,w − h∗ k is bounded for a sequence pk → ∞.
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(d) There exists a strongly separating hyperplane at h∗ .
Contents
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
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implies
aj (1 − h(j)) ≤
X
ai (h(i) − 1)
i∈In ,i6=j
≤
X
ai (h(i) − 1)
i∈In+
≤ (khk − 1)
X
ai
i∈In+
≤ (khk − 1)(1 − aj ).
Thus, for all j ∈ In we have
1
|h(j) − 1| ≤
− 1 (khk − 1) := γ(khk − 1),
mini∈In ai
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, +∞)
t→∞
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
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denote its inverse function, that will 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
= εpβ(εp )
(1 + δp )p−1
and therefore
(2.7)
lim
p→∞
p )/(p−1)
εβ(ε
p
1 − εp
= lim
= 1.
p→∞ 1 + δp
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,
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
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(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
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
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 .
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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.
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
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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
|
{z
} |
{z
}
r
n−r
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of the nondecreasing 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
Contents
∗
• h = h − f = (−1, . . . , −1, 1, . . . , 1).
| {z } | {z }
r
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J
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
M. Marano, J. Navas and
J.M. Quesada
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• fe = (0, 0, . . . , 0);
e∗
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
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P
(p1) 0 < aj < 1, all j ∈ In , and n1 aj = 1;
P
e
h(j) ≥ 1 ∀ e
h ∈ K.
(p2) nj=1 σj aj e
Proposition 2.6. Let S :=
r
P
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
r
X
X
wj
1 1
wj
aj =
+
= + = 1.
2 S j=r+1 2(N − S)
2 2
j=1
j=1
n
X
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)
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
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n
X
j=1
σj aj h(j) ≥ 0 ∀ h ∈ K.
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But if h is a nondecreasing vector, then (2.10) is immediate because
r
X
wj h(j) ≤ h(r)
r
X
j=1
and
n
X
j=1
h(j) ≥ h(r)
j=r+1
wj = S h(r)
n
X
wj = (N − S)h(r),
j=r+1
and therefore
r
n
X
1
1 X
wj h(j) +
wj h(j)
σj aj h(j) = −
2 S j=1
2(N − S) j=r+1
j=1
n
X
≥−
1
1
S h(r) +
(N − S)h(r) = 0.
2S
2(N − S)
This concludes the proof.
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
Title Page
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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
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is a strongly separating hyperplane at e
h∗ , and from Theorem 2.4 this is equiva∗
lent 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 . From Proposition 2.6 and The-
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Page 17 of 21
p
orems 2.4 and 2.5 we only need to show that (2.11) is not a strongly separating
J. Ineq. Pure and Appl. Math. 3(4) Art. 50, 2002
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hyperplane at e
h∗ . This last assertion is true since (2.10), with aj = wj /N , all
j ∈ In , implies
n
X
(2.12)
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
n
`p,w −approximations to f from K, namely, hp,w = (xp,w , xp,w , . . . , xp,w ). It is
easy to check that
xp,w =
a−1+a
S
N −S
1+
p1
S
N −S
+
p1
S
N −S
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
p1
,
1 < p < ∞.
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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,
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1/p
( NS−S ) −1
1/p
1+( N S
hp,w (j) − h∗ (j)
−S )
lim
= lim
p→∞
p→∞
1/p
1/p
p1
S
−1
1
N −S
= lim
2 p→∞
1/p
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1
= ln
2
The rate of convergence is exactly O p1 .
S
N −S
.
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
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References
[1] J. DESCLOUX, Approximations in Lp and Chebychev approximations, J.
Soc. Ind. Appl. Math., 11 (1963), 1017–1026.
[2] A. EGGER AND R. HUOTARI, The Pólya algorithm on convex sets, J.
Approx. Theory, 56(2) (1989), 212–216.
[3] A. EGGER AND R. HUOTARI, Rate of convergence of the discrete Pólya
algorithm, J. Approx. Theory, 60 (1990), 24–30.
[4] R. FLETCHER, J. GRANT AND M. HEBDEN, Linear minimax approximation as the limit of best Lp -approximation, SIAM J. Numer. Anal., 11
(1974), 123–136.
[5] M.D. HEBDEN, A bound on the difference between the Chebyshev norm
and the Hölder norms of a function, SIAM J. Numer. Anal., 2(8) (1971),
270–279.
[6] R. HUOTARI, D. LEGG AND D. TOWNSEND, The Pólya algorithm on
cylindrical sets, J. Approx. Theory, 53 (1988), 335–349.
[7] M. MARANO, Strict approximation on closed convex sets, Approx. Theory and its Appl., 6 (1990), 99–109.
[8] M. MARANO AND J. NAVAS, The linear discrete Pólya algorithm, Appl.
Math. Letter, 8(6) (1995), 25–28.
[9] M. MARANO AND R. HUOTARI, The Pólya algorithm on tubular sets,
Journal of Computational and Applied Mathematics, 54 (1994), 151–157.
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
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[10] D. J. NEWMAN AND H.S. SHAPIRO, Some theorems on Cebysev approximation, Duke Math. J., 30 (1963), 673–682.
[11] G. PÓLYA, Sur un algorithme toujours convergent pour obtenir les polynomes de meilleure approximation de Tchebycheff pour une function continue quelconque, C. R. Acad. Sci. París, 157 (1913), 840–843.
[12] J.R. RICE, Tchebycheff approximation in a compact metric space, Bull.
Amer. Math. Soc., 68 (1962), 405–410.
[13] J.M. QUESADA AND J. NAVAS, Rate of convergence of the linear discrete Polya algorithm, J. Aprox. Theory, 110-1 (2001), 109–119.
[14] I. SINGER, Best Approximation in Normed Linear Spaces by Elements of
Linear Subspaces, Springer Verlag, Berlin, 1970.
Rate of Convergence of the
Discrete Polya Algorithm from
Convex Sets. A Particular Case
M. Marano, J. Navas and
J.M. Quesada
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