Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
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3.1. L
1
estimates
In this section we use H = L
2
X, µ, for a positive measure space X, µ. Suppose we are given an orthonormal sequence of functions
h
n ∞
n=1
in L
2
X, µ, and sup- pose that each of the functions
h
n
is essentially bounded on X. Let a
n ∞
n=1
be a sequence of non-zero complex numbers and set
M
n
= max
1≤j≤n
kh
j
k
∞
and S
∗ n
x = max
1≤k≤n k
X
j=1
a
j
h
j
x ,
for x ∈ X, n ≥ 1.
Lemma 3.1. For a set of functions
{h
1
, . . . , h
n
} ⊂ L
2
X, µ ∩ L
∞
X, µ and max- imal function
S
∗ n
x = max
1≤k≤n k
X
j=1
a
j
h
j
x ,
we have |a
j
h
j
x| ≤ 2S
∗ n
x, ∀x ∈ X, 1 ≤ j ≤ n,
and P
k j=1
a
j
h
j
x S
∗ n
x ≤ 1,
∀1 ≤ k ≤ n and x where S
∗ n
x 6= 0. Proof. The first inequality follows from the triangle inequality and the fact that
a
j
h
j
x =
j
X
k=1
a
k
h
k
x −
j−1
X
k=1
a
k
h
k
x
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
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v
j
x = a
j
h
j
x S
∗ n
x
− 12
and w
j
x = a
− 1
j
h
j
x S
∗ n
x
12
for all x ∈ X where S
∗ n
x 6= 0 and 1 ≤ j ≤ n. From their definition, {v
1
, . . . , v
n
} and {w
1
, . . . , w
n
} are a bi-orthogonal pair in
L
2
X, µ. The conditions we have placed on the functions h
j
give: kw
j
k
2 2
= |a
j
|
− 2
Z
X
|h
j
|
2
S
∗ n
dµ ≤ M
2 n
min
1≤k≤n
|a
k
|
2
kS
∗ n
k
1
and
k
X
j=1
v
j 2
2
= Z
X
1 S
∗ n
k
X
j=1
a
j
h
j 2
dµ ≤
k
X
j=1
a
j
h
j 1
. We can put these estimates into
2.1 and find that
log n ≤ c M
n
min
1≤k≤n
|a
k
| kS
∗ n
k
12 1
max
1≤k≤n k
X
j=1
a
j
h
j 12
1
. We could also say that
max
1≤k≤n k
X
j=1
a
j
h
j 1
≤ kS
∗ n
k
1
and so logn ≤ c
M
n
min
1≤k≤n
|a
k
| kS
∗ n
k
1
.
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
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n ∞
bers and each n ≥ 1,
min
1≤k≤n
|a
k
| log n
2
≤ c max
1≤k≤n
kh
k
k
∞ 2
max
1≤k≤n k
X
j=1
a
j
h
j 1
max
1≤k≤n k
X
j=1
a
j
h
j 1
and min
1≤k≤n
|a
k
| log n ≤ c max
1≤k≤n
kh
k
k
∞
max
1≤k≤n k
X
j=1
a
j
h
j 1
. The constant
c is independent of n, and the sequences involved here. As observed in [
4 ], this can also be obtained as a consequence of [
11 ]. In addition,
see [ 7
]. The following is a paraphrase of the last page of [
13 ]. For the special case of
Fourier series on the unit circle, see Proposition 1.6.9 in [ 12
].
Corollary 3.3. Suppose that
h
n ∞
n=1
is an orthonormal sequence in L
2
X, µ con- sisting of essentially bounded functions with
kh
n
k
∞
≤ M for all n ≥ 1. For each decreasing sequence
a
n ∞
n=1
of positive numbers and each n ≥ 1,
a
n
log n
2
≤ cM
2
max
1≤k≤n k
X
j=1
a
j
h
j 1
max
1≤k≤n k
X
j=1
a
j
h
j 1
and a
n
log n ≤ cM max
1≤k≤n k
X
j=1
a
j
h
j 1
.
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page Contents
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max
1≤k≤n k
X
j=1
a
j
h
j 1
∞ n=1
is unbounded. The constant
c is independent of n, and the sequences involved here.
3.2. Salem’s Approach to the Littlewood Conjecture
