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
We concentrate on the case where H = L
2
T and the orthonormal sequence is a subset of
{e
inx
: n ∈ N}. Let m
1
m
2
m
3
· · · be an increasing sequence of natural numbers and let
h
k
x = e
im
k
x
for all k ≥ 1 and x ∈ T. In addition, let
D
m
x =
m
X
k=−m
e
ikx
be the m
th
Dirichlet kernel. For all N ≥ m ≥ 1, there is the partial sum
X
m
k
≤ m
a
k
h
k
x = D
m
∗ X
m
k
≤ N
a
k
h
k
x. It is a fact that
D
m
is an even function which satisfies the inequalities: 3.1
|D
m
x| ≤ 2m + 1
for all x,
1|x| for
1 2m+1
x 2π −
1 2m+1
.
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page Contents
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S
∗
px = sup
m≥1
|D
m
∗ px| satisfies
kS
∗
pk
1
≤ c log 2N + 1 kpk
1
. Proof. For such a trigonometric polynomial
p, the partial sums are all partial sums of
p ∗ D
N
, and all the Dirichlet kernels D
m
for 1 ≤ m ≤ N are dominated by a
function whose L
1
norm is of the order of log2N + 1.
We can combine this with the inequalities in Corollary 3.2
, since max
1≤k≤n k
X
j=1
a
j
h
j 1
≤ c log 2m
n
+ 1
m
X
j=1
a
j
h
j 1
. We then arrive at the main result in [
14 ].
Corollary 3.5. For an increasing sequence
m
n ∞
n=1
of natural numbers and a se- quence of non-zero complex numbers
a
n ∞
n=1
the partial sums of the trigonometric series
∞
X
k=1
a
k
e
im
k
x
satisfy min
1≤k≤n
|a
k
| log n
plog2m
n
+ 1 ≤ c max
1≤k≤n k
X
j=1
a
j
e
im
j
· 1
. This was Salem’s attempt at Littlewood’s conjecture, which was subsequently
settled in [ 5
] and [ 8
].
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page Contents
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Notice that if {v
1
, . . . , v
n
} is an arbitrary linearly independent subset of H then there is a unique subset
w
n j
: 1 ≤ j ≤ n ⊆ span {v
1
, . . . , v
n
} so that
{v
1
, . . . , v
n
} and {w
n 1
, . . . , w
n n
} are a bi-orthogonal pair. See Theorem 15 in Chapter 3 of [
2 ]. We can apply Theorem
2.1 to the pair in either order.
Corollary 3.6. For each
n ≥ 2 and linearly independent subset {v
1
, . . . , v
n
} in an inner-product space
H, with dual basis {w
n 1
, . . . , w
n n
}, log n ≤ c max
1≤k≤n
kw
n k
k
H
max
1≤k≤n k
X
j=1
v
j H
and log n ≤ c max
1≤k≤n
kv
k
k
H
max
1≤k≤n k
X
j=1
w
n j
H
. The constant
c 0 is independent of n, H, and the sets of vectors.
3.4. Matrices
