REMEZ-TYPE INEQUALITY FOR NON-DENSE
MÜNTZ SPACES WITH EXPLICIT BOUND
Peter Borwein and Tamás Erdélyi
Abstract. Let ΛP
:= (λk )∞
k=0 be a sequence of distinct nonnegative real numbers
∞
with λ0 := 0 and
1/λ
k < ∞. Let ̺ ∈ (0, 1) and ǫ ∈ (0, 1 − ̺) be fixed. An
k=1
earlier work of the first two authors shows that
C(Λ, ǫ, ̺)
n
o
:= sup kpk[0,̺] : p ∈ span{xλ0 , xλ1 , . . . }, m{x ∈ [̺, 1] : |p(x)| ≤ 1} ≥ ǫ
is finite. In this paper an explicit upper bound for C(Λ, ǫ, ̺) is given. In the special
case λk := k α , α > 1, our bounds are essentially sharp.
1. Introduction
In this paper Λ := (λk )∞
k=0 always denotes a sequence of real numbers satisfying
0 = λ0 < λ1 < λ2 < · · · .
In [1] a Remez-type inequality for Müntz polynomials:
p(x) =
n
X
ak xλk
k=0
or equivalently for Dirichlet sums:
P (t) =
n
X
ak e−λk t
k=0
is established. The most common
form of this inequality states that for every
P∞
satisfying
1/λ
sequence (λk )∞
k < ∞, there exists a constant C(Λ, ǫ) dek=0
k=0
pending only on Λ and ǫ (and not on n, ̺, or A) so that
kpk[0,̺] ≤ C(Λ, ǫ)kpkA
1991 Mathematics Subject Classification. Primary: 41A17, Secondary: 30B10, 26D15.
Key words and phrases. Remez-type inequality, Müntz-type theorems.
Research of the first author supported, in part, by NSERC of Canada. Research of the second
author supported, in part, by NSF under Grant No. DMS-9024901 and conducted while an
NSERC International Fellow at Simon Fraser University.
Typeset by AMS-TEX
1
2
PETER BORWEIN AND TAMÁS ERDÉLYI
for every Müntz polynomial p, as above, associated with the sequence (λk )∞
k=0 , and
for every set A ⊂ [̺, 1] of Lebesgue measure at least ǫ > 0. Throughout this paper
k · kA denotes the uniform norm on A ⊂ R.
Using this Remez-type inequality, we resolved two reasonably long standing conjectures in [1]. In this paper we give an explicit upper bound for the best possible
C(Λ, ǫ) in the above Remez-type inequality for non-dense Müntz spaces. Theorem
2.3 extends an inequality of Schwartz [4] in two directions. Theorem 2.1 offers a
more explicit bound for the sequences Λ := (k α )∞
k=0 , α > 1. The sharpness of the
Remez-type inequality of Theorem 2.1 is shown by Theorem 2.2.
2. Results
Theorem 2.1. Let λk := k α , k = 0, 1, . . . , α > 1. Let ̺ ∈ (0, 1), ǫ ∈ (0, 1 − ̺),
and ǫ ≤ 1/2. There exists a constant cα > 0 depending only on α so that
kpk[0,̺] ≤ exp cα ǫ1/(1−α) kpkA
for every p ∈ span{xλ0 , xλ1 , . . . } and for every set A ⊂ [̺, 1] of Lebesgue measure
at least ǫ > 0.
The next theorem shows that the inequality of Theorem 2.1 is essentially the
best possible.
Theorem 2.2. Let λk := k α , k = 0, 1, . . . , α > 1. For every α > 1 and ǫ ∈
(0, 1/2], there exists a constant cα > 0 depending only on α and Müntz polynomials
0 6= p = pα,ǫ ∈ span{xλ0 , xλ1 , . . . }
depending only on α and ǫ so that
|p(0)| ≥ exp cα ǫ1/(1−α) kpk[1−ǫ,1] .
Theorem 2.1 is a special case of the following more general, but less explicit
result.
P∞
Theorem 2.3. Suppose 0 = λ0 < λ1 < λ2 < · · · and
k=0 1/λk < ∞. Let
̺ ∈ (0, 1) and ǫ ∈ (0, 1 − ̺). Let δ := − 21 log(1 − ǫ). Let N ∈ N be chosen so that
∞
X
k=N +1
Let
σk := Aλk
1
δ
≤ .
λk
3
with
A :=
δ
.
3N
Then, with c := kt−1 sin tkL2 (R) ,
kpk[0,̺]
N 1
3c Y
kpkA
2+
≤
δ
σk
k=1
for every p ∈ span{xλ0 , xλ1 , . . . } and for every set A ⊂ [̺, 1] of Lebesque measure
at least ǫ > 0.
REMEZ-TYPE INEQUALITY FOR NON-DENSE MÜNTZ SPACES
3
3. Lemmas
Our first lemma shows that C(Λ, ǫ) in the Remez-type inequality is related to a
much simpler (Chebyshev-type) extremal problem. This is proved in both [1] and
[2].
Lemma 3.1. Suppose 0 = λ0 < λ1 < λ2 < · · · , ρ ∈ (0, 1), and ǫ ∈ (0, 1 − ρ). Then
sup kpk[0,̺] : p ∈ span{xλ0 , xλ1 , . . . }, m{x ∈ [̺, 1] : |p(x)| ≤ 1} ≥ ǫ
|p(0)|
: p ∈ span{xλ0 , xλ1 , . . . } .
= sup
kpk[1−ǫ,1]
Our key lemma is the following.
Lemma 3.2. Suppose 0 = λ0 < λ1 < λ2 < · · · and
δ ∈ (0, 1), let N ∈ N be chosen so that
∞
X
k=N +1
Let
σk := Aλk
Then
|P (∞)| ≤
P∞
k=0
1/λk < ∞. Given
1
δ
≤ .
λk
3
with
A :=
δ
.
3N
N 1
3c Y
kP k[−δ,δ]
2+
δ
σk
k=1
for every P ∈ span{e
−λ0 t
,e
−λ1 t
, . . . } with c := kt−1 sin tkL2 (R) .
In the proof of Lemma 3.2 we will need the following observation.
Lemma 3.3. Let 0 = λ0 < λ1 < λ2 < · · · . Suppose
(1) F ∈ E δ ∩ L2 (R) ;
(2) F (iλk ) = 0 , k = 1, 2, . . . (i is the imaginary unit) ;
(3) F (0) = 1 .
Then
|P (∞)| ≤ kF kL2 (R) kP kL2 [−δ,δ]
for every P ∈ span{e
−λ0 t
, e−λ1 t , . . . }.
An entire function f is called a function of exponential type δ if there exists a
constant c depending only on f so that
|f (z)| ≤ c exp(δ|z|) ,
z ∈ C.
The collection of all such entire functions of exponential type δ is denoted be E δ .
The Paley-Wiener Theorem (see, for example, [3]) characterizes the functions F
which can be written as the Fourier transform of some function f ∈ L2 [−δ, δ]. We
will need it in the proof of Lemma 3.3.
4
PETER BORWEIN AND TAMÁS ERDÉLYI
Theorem (Paley-Wiener). Let δ ∈ (0, ∞). Then f ∈ E δ ∩ L2 (R) if and only
if there exists an f ∈ L2 [−δ, δ] so that
F (z) =
δ
Z
f (t)eitz dt .
−δ
The following comparison theorem for Müntz polynomials is proved in [2]. We
will need it in the proof of Theorem 2.3.
∞
Lemma 3.4. Let Λ := (λk )∞
k=0 and Γ := (γk )k=0 be increasing sequences of nonnegative real numbers with λ0 = 0, γ0 = 0, and λk ≤ γk for each k. Let 0 < a < b.
Then
|p(0)|
λn
λ1
λ0
: p ∈ span{x , x , . . . , x }
max
kpk[a,b]
|p(0)|
γn
γ1
γ0
≥ max
: p ∈ span{x , x , . . . , x } .
kpk[a,b]
4. Proofs
Proof of Lemma 3.3. By the Paley-Wiener Theorem
F (z) =
Z
δ
f (t)eitz dt
−δ
for some f ∈ L2 [−δ, δ]. Now if
P (t) = a0 +
n
X
ak e−λk t ,
k=1
then
Z
δ
f (t)P (t) dt = a0
−δ
Z
δ
f (t) dt +
−δ
= a0 F (0) +
n
X
k=1
n
X
ak
Z
δ
f (t)e−λk t dt
−δ
ak F (iλk ) = a0 = P (∞) .
k=1
Hence by the Cauchy-Schwartz Inequality and the L2 inversion theorem of Fourier
transforms, we obtain
|P (∞)| ≤ kf kL2 [−δ,δ] kP kL2 [−δ,δ] ≤ kF kL2 (R) kP kL2[−δ,δ]
and the lemma is proved. REMEZ-TYPE INEQUALITY FOR NON-DENSE MÜNTZ SPACES
5
Proof of Lemma 3.2. We define
N sin(σk z/λk )
z
sin(δz/3) Y
1−
F (z) :=
δz/3
iλk
σk z/λk
∞
Y
k=1
1−
k=N +1
sin(z/λk )
sin i
4 !
,
where i is the imaginary unit. It follows easily that
F ∈ Eδ ,
F (0) = 1 ,
F (iλk ) = 0 ,
k = 1, 2, . . .
and
|F (t)| ≤
N sin(δt/3) Y
1
,
2+
(δt/3
σk
t ∈ R.
k=1
Hence Lemma 3.3 implies that
|P (∞)| ≤
N 1
3c Y
kP k[−δ,δ]
2+
δ
σk
k=1
for every P ∈ span{e−λ0 t , e−λ1 t , . . . } with c := kt−1 sin tkL2 (R) . Proof of Theorem 2.3. When A = [1 − ǫ, 1], the theorem follows from Lemma 3.2
by the substitution x = e−δ e−t . The general case follows from Lemma 3.1. Proof of Theorem 2.1. Let
(4.1)
1
δ := − log(1 − ǫ) .
2
Observe that N in Theorem 2.1 can be chosen so that
$
1/(1−α) %
δ(α − 1)
(4.2)
N :=
+1.
3
Also, σk in Lemma 3.2 is of the form
σk =
δk α
.
3N
Let M + 1 be the smallest value of k ∈ N for which
1
< 1,
σk
that is ,
3N
≤ 1.
kα δ
Note that
M :=
$
3N
δ
1/α %
.
6
PETER BORWEIN AND TAMÁS ERDÉLYI
If 0 < M < N , then
Y
N N Y
3N
1
=
2+ α
2+
σk
δk
k=1
k=1
!
! M −αM
N
M
Y
Y
M
9N
9N
3N −M
3 =
≤
δk α
δ
e
k=M+1
k=1
α M
9e N
=
M −αM 3N −M
δ
α M
1/α !−αM
9e N
1 3N
≤
3N −M
δ
2
δ
≤ (3(2e)α )M 3N −M ≤ (3(2e)α )N ,
and the theorem follows by (4.1), (4.2), and Theorem 2.1.
If N ≤ M , then
Y
N N Y
3N
1
=
2+ α
2+
σk
δk
k=1
k=1
! N −αN
N
Y
N
9N
9N
=
≤
δk α
δ
e
k=1
N α N 1/(1−α) !(1−α)N
δ(α − 1)
9eα N (1−α)
9e
=
≤
δ
δ
3
N
α N δ(α − 1)
9e
N
≤ (3eα (α − 1)) ,
≤
δ
3
and the theorem follows by (4.1), (4.2), and Theorem 2.1.
If M = 0, then
Y
N N
Y
1
2+
≤
3 = 3N ,
σk
k=1
k=1
and the theorem follows by (4.1), (4.2), and Theorem 2.1.
Proof of Theorem 2.2. Let n ∈ N be a fixed. We define γk := knα−1 , k = 0.1, . . . .
n
Let Tn (x) := 12 (x − 1) and
!n
α−1
α−1
1 + (1 − ǫ)n
2xn
−
∈ span{xγ0 , xγ1 , . . . xγn }.
Qn (x) := Tn
1 − (1 − ǫ)nα−1
1 − (1 − ǫ)nα−1
Then, by Lemma 3.4,
|p(0)|
|Qn (0)|
λ1
λ0
sup
: p ∈ span{x , x , . . . } ≥
= |Qn (0)|
kpk[1−ǫ,1]
kQn k[1−ǫ,1]
n
1
.
=
1 − (1 − ǫ)nα−1
REMEZ-TYPE INEQUALITY FOR NON-DENSE MÜNTZ SPACES
7
Now let n be the smallest integer satisfying nα−1 ≥ ǫ−1 . Since (1 − ǫ)1/ǫ is bounded
away from 0 on (0, 1/2], the result follows. Acknowledgements. The authors thank Gábor Halász for suggesting Theorem
2.3 to establish Remez-type inequalities for non-dense Müntz spaces with good
explicit bounds. We also thank him for helping us to close the gap between the
upper bound of Theorem 2.1 and the lower bound of Therem 2.2.
References
1. P. B. Borwein and T. Erdélyi, Generalizations of Müntz’s theorem via a Remez-type
inequality for Müntz spaces, manuscript.
2. P. B. Borwein and T. Erdélyi, Polynomials and Polynomials Inequalities, SpringerVerlag, New York, N.Y., 1995.
3. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, N.Y., 1987.
4. L. Schwartz, Etude des Sommes d’Exponentielles, Hermann, Paris, 1959.
Department of Mathematics and Statistics, Simon Fraser University, Burnaby,
B.C., Canada V5A 1S6 (P. Borwein)
Department of Mathematics, Texas A&M University, College Station, Texas
77843 (T. Erdélyi)