AN INEQUALITY FOR BI-ORTHOGONAL PAIRS
CHRISTOPHER MEANEY
Bi-orthogonal Pairs
Department of Mathematics
Faculty of Science
Macquarie University
North Ryde NSW 2109, Australia
vol. 10, iss. 4, art. 94, 2009
EMail: chrism@maths.mq.edu.au
Title Page
Received:
26 November, 2009
Accepted:
14 December, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
42C15, 46C05.
Key words:
Bi-orthogonal pair, Bessel’s inequality, Orthogonal expansion, Lebesgue constants.
Christopher Meaney
Contents
JJ
II
J
I
Page 1 of 16
Abstract:
We use Salem’s method [13, 14] to prove an inequality of Kwapień and
Pełczyński concerning a lower bound for partial sums of series of bi-orthogonal
vectors in a Hilbert space, or the dual vectors. This is applied to some lower
bounds on L1 norms for orthogonal expansions.
Go Back
Full Screen
Close
Contents
1
Introduction
3
2
The Kwapień-Pełczyński Inequality
4
3
Applications
3.1 L1 estimates . . . . . . . . . . . . . . . . . . .
3.2 Salem’s Approach to the Littlewood Conjecture
3.3 Linearly Independent Sequences . . . . . . . .
3.4 Matrices . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
8
8
11
13
13
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page
Contents
JJ
II
J
I
Page 2 of 16
Go Back
Full Screen
Close
1.
Introduction
Suppose that H is a Hilbert space, n ∈ N, and that J = {1, . . . , n} or J = N. A pair
of sets {vj : j ∈ J} and {wj : j ∈ J} in H are said to be a bi-orthogonal pair
when
hvj , wk iH = δjk ,
∀j, k ∈ J.
The inequality in Theorem 2.1 below comes from Section 6 of [6], where it was
proved using Grothendieck’s inequality, absolutely summing operators, and estimates on the Hilbert matrix. Here we present an alternate proof, based on earlier
ideas from Salem [13, 14], where Bessel’s inequality is combined with a result
of Menshov [10]. Following the proof of Theorem 2.1, we will describe Salem’s
method of using L2 inequalities to produce L1 estimates on maximal functions. Such
estimates are related to the stronger results of Olevskiı̆ [11], Kashin and Szarek [4],
and Bochkarev [1]. We conclude with an observation about the statement of Theorem 2.1 in a linear algebra setting. Some of these results were discussed in [9], where
it was shown that Salem’s methods emphasized the universality of the RademacherMenshov Theorem.
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page
Contents
JJ
II
J
I
Page 3 of 16
Go Back
Full Screen
Close
2.
The Kwapień-Pełczyński Inequality
Theorem 2.1. There is a positive constant c with the following property. For every n ≥ 1, every Hilbert space H, and every bi-orthogonal pair {v1 , . . . , vn } and
{w1 , . . . , wn } in H,
k
X
(2.1)
log n ≤ c max kwm kH max vj .
1≤m≤n
1≤k≤n j=1
H
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Proof. Equip [0, 1] with a Lebesgue measure λ and let V = L2 ([0, 1], H) be the
space of H-valued square integrable functions on [0, 1], with inner product
Z 1
hF, GiV =
hF (x), G(x)iH dx
Title Page
Contents
0
and norm
Z
kF kV =
1
kF (x)k2H dx .
JJ
II
J
I
0
Suppose that {F1 , . . . , Fn } is an orthonormal set in L2 ([0, 1]) and define vectors
p1 , . . . , pn in V by
pk (x) = Fk (x)wk ,
Then
1 ≤ k ≤ n, x ∈ [0, 1].
hpk (x), pj (x)iH = Fk (x)Fj (x) hwk , wj iH ,
1 ≤ j, k ≤ n,
and so {p1 , . . . , pn } is an orthogonal set in V . For every P ∈ V , Bessel’s inequality
states that
n
X
|hP, pk iV |2
(2.2)
≤ kP k2V .
2
kwk kH
k=1
Page 4 of 16
Go Back
Full Screen
Close
Note that here
1
Z
hP, pk iV =
hP (x), wk iH Fk (x)dx,
1 ≤ k ≤ n.
0
Now consider a decreasing sequence f1 ≥ f2 ≥ · · · ≥ fn ≥ fn+1 = 0 of
characteristic functions of measurable subsets of [0, 1]. For each scalar-valued G ∈
L2 ([0, 1]) define an element of V by setting
PG (x) = G(x)
n
X
Bi-orthogonal Pairs
Christopher Meaney
fj (x)vj .
vol. 10, iss. 4, art. 94, 2009
j=1
The Abel transformation shows that
PG (x) = G(x)
n
X
Title Page
∆fk (x)σk ,
Contents
k=1
Pk
where ∆fk = fk − fk+1 and σk =
j=1 vj , for 1 ≤ k ≤ n. The functions
∆f1 , . . . , ∆fn are characteristic functions of mutually disjoint subsets of [0, 1] and
for each 0 ≤ x ≤ 1 at most one of the values ∆fk (x) is non-zero. Notice that
n
X
2
2
kPG (x)kH = |G(x)|
∆fk (x) kσk k2H .
JJ
II
J
I
Page 5 of 16
Go Back
k=1
Full Screen
Integrating over [0, 1] gives
Close
kPG k2V ≤ kGk22 max kσk k2H .
1≤k≤n
Note that
hPG (x), pk (x)iH = G(x)fk (x)Fk (x) hvk , wk iH ,
and
1 ≤ k ≤ n,
Z
hPG , pk iV =
1
G(x)fk (x)Fk (x)dx hvk , wk iH ,
1 ≤ k ≤ n.
0
Combining this with Bessel’s inequality (2.2), we arrive at the inequality
2
n Z
X
1
2
(2.3)
Gf
F
dλ
max kσk k2H .
k
k
kw k2 ≤ kGk2 1≤k≤n
[0,1]
k H
k=1
This implies that
n Z
X
(2.4)
k=1
[0,1]
Bi-orthogonal Pairs
2 ! 2
2
2
Gfk Fk dλ ≤ max kwk kH kGk2 max kσk kH .
1≤j≤n
1≤k≤n
We now concentrate on the case where the functions F1 , . . . , Fn are given by Menshov’s result (Lemma 1 on page 255 of Kashin and Saakyan [3]). There is a constant
c0 > 0, independent of n, so that
j
(
)!
X
√
1
(2.5)
λ
x ∈ [0, 1] : max ≥ .
Fk (x) > c0 log(n) n
1≤j≤n 4
k=1
Let us use M(x) to denote the maximal function
j
X
M(x) = max Fk (x) ,
1≤j≤n Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page
Contents
JJ
II
J
I
Page 6 of 16
Go Back
0 ≤ x ≤ 1.
Full Screen
k=1
Define an integer-valued function m(x) on [0, 1] by
(
)
m
X
m(x) = min m : Fk (x) = M(x) .
k=1
Furthermore, let fk be the characteristic function of the subset
Close
{x ∈ [0, 1] : m(x) ≥ k} .
Then
n
X
m(x)
fk (x)Fk (x) = Sm(x) (x) =
X
Fk (x),
∀0 ≤ x ≤ 1.
k=1
k=1
For an arbitrary G ∈ L2 ([0, 1]) we have
Z 1
n Z
X
G(x)Sm(x) (x)dx =
0
k=1
1
Bi-orthogonal Pairs
G(x)fk (x)Fk (x)dx.
0
Using the Cauchy-Schwarz inequality on the right hand side, we have
2 !1/2
Z 1
n Z 1
X
√
(2.6)
G(x)Sm(x) (x)dx ≤ n
Gfk Fk dλ
,
0
k=1
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page
Contents
0
for all G ∈ L2 ([0, 1]). We will use the function G which has |G(x)| = 1 everywhere
on [0, 1], with
G(x)Sm(x) (x) = M(x),
∀0 ≤ x ≤ 1.
In this case, the left hand side of (2.6) is
√
c0
kMk1 ≥ log(n) n,
4
because of (2.5). Combining this with (2.6) we have
2 !1/2
n Z 1
X
√
√
c0
log(n) n ≤ n
Gfk Fk dλ
.
4
0
k=1
This can be put back into (2.4) to obtain (2.1). Notice that kGk2 = 1 on the right
hand side of (2.3).
JJ
II
J
I
Page 7 of 16
Go Back
Full Screen
Close
3.
3.1.
Applications
L1 estimates
In this section we use H = L2 (X, µ), for a positive measure space (X, µ). Suppose
2
we are given an orthonormal sequence of functions (hn )∞
n=1 in L (X, µ), and suppose that each of the functions hn is essentially bounded on X. Let (an )∞
n=1 be a
sequence of non-zero complex numbers and set
k
X
Mn = max khj k∞ and Sn∗ (x) = max aj hj (x) , for x ∈ X, n ≥ 1.
1≤j≤n
1≤k≤n Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
j=1
Lemma 3.1. For a set of functions {h1 , . . . , hn } ⊂ L2 (X, µ) ∩ L∞ (X, µ) and maximal function
k
X
Sn∗ (x) = max aj hj (x) ,
1≤k≤n j=1
we have
|aj hj (x)| ≤ 2Sn∗ (x),
and
∀x ∈ X, 1 ≤ j ≤ n,
Title Page
Contents
JJ
II
J
I
Page 8 of 16
Go Back
P
k
j=1 aj hj (x)
Sn∗ (x)
≤ 1,
∀1 ≤ k ≤ n and x where Sn∗ (x) 6= 0.
Full Screen
Close
Proof. The first inequality follows from the triangle inequality and the fact that
aj hj (x) =
j
X
k=1
ak hk (x) −
j−1
X
k=1
ak hk (x)
for 2 ≤ j ≤ n. The second inequality is a consequence of the definition of Sn∗ .
Fix n ≥ 1 and let
1/2
∗
vj (x) = aj hj (x) (Sn∗ (x))−1/2 and wj (x) = a−1
j hj (x) (Sn (x))
for all x ∈ X where Sn∗ (x) 6= 0 and 1 ≤ j ≤ n. From their definition,
{v1 , . . . , vn } and {w1 , . . . , wn }
are a bi-orthogonal pair in L2 (X, µ). The conditions we have placed on the functions
hj give:
Z
Mn2
−2
2
∗
|hj |2 (Sn∗ ) dµ ≤
kwj k2 = |aj |
2 kSn k1
min
|a
|
X
1≤k≤n k
and
2 Z
2
k
k
k
X
X
X
1
v
=
a
h
dµ
≤
a
h
j
j j
j j .
∗
X (Sn ) j=1
j=1
j=1
2
1
We can put these estimates into (2.1) and find that
log n ≤ c
Mn
1/2
min1≤k≤n |ak |
kSn∗ k1
1/2
k
X
max aj hj .
1≤k≤n j=1
We could also say that
k
X
max aj hj ≤ kSn∗ k1
1≤k≤n j=1
and so
log(n) ≤ c
1
Mn
min1≤k≤n |ak |
kSn∗ k1 .
1
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page
Contents
JJ
II
J
I
Page 9 of 16
Go Back
Full Screen
Close
2
Corollary 3.2. Suppose that (hn )∞
n=1 is an orthonormal sequence in L (X, µ) con∞
sisting of essentially bounded functions. For each sequence (an )n=1 of complex numbers and each n ≥ 1,
k
k
2
2 X
X
min |ak | log n ≤ c max khk k∞ max aj hj max aj hj 1≤k≤n
1≤k≤n
1≤k≤n 1≤k≤n j=1
and
j=1
1
1
Bi-orthogonal Pairs
k
X
min |ak | log n ≤ c max khk k∞ max aj hj .
1≤k≤n
1≤k≤n
1≤k≤n j=1
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
1
The constant c is independent of n, and the sequences involved here.
Title Page
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].
Contents
2
Corollary 3.3. Suppose that (hn )∞
n=1 is an orthonormal sequence in L (X, µ) consisting of essentially bounded functions with khn k∞ ≤ M for all n ≥ 1. For each
decreasing sequence (an )∞
n=1 of positive numbers and each n ≥ 1,
k
k
X
X
2
2
(an log n) ≤ cM max aj hj max aj hj 1≤k≤n 1≤k≤n j=1
and
j=1
1
k
X
an log n ≤ cM max aj hj .
1≤k≤n j=1
1
1
JJ
II
J
I
Page 10 of 16
Go Back
Full Screen
Close
In particular, if (an log n)∞
n=1 is unbounded then

∞
k
X

aj hj 
is unbounded.
max 1≤k≤n j=1
1
n=1
The constant c is independent of n, and the sequences involved here.
3.2.
Salem’s Approach to the Littlewood Conjecture
2
Bi-orthogonal Pairs
Christopher Meaney
We concentrate on the case where H = L (T) and the orthonormal sequence is a
subset of {einx : n ∈ N}. Let
vol. 10, iss. 4, art. 94, 2009
m1 < m 2 < m 3 < · · ·
Title Page
be an increasing sequence of natural numbers and let
Contents
imk x
hk (x) = e
for all k ≥ 1 and x ∈ T. In addition, let
Dm (x) =
m
X
eikx
k=−m
be the mth Dirichlet kernel. For all N ≥ m ≥ 1, there is the partial sum
!
X
X
ak hk (x) = Dm ∗
ak hk (x).
mk ≤m
mk ≤N
It is a fact that Dm is an even function which satisfies the inequalities:
(
2m + 1 for all x,
(3.1)
|Dm (x)| ≤
1
1
1/|x|
for 2m+1
< x < 2π − 2m+1
.
JJ
II
J
I
Page 11 of 16
Go Back
Full Screen
Close
Lemma 3.4. If p is a trigonometric polynomial of degree N, then the maximal function of its Fourier partial sums
S ∗ p(x) = sup |Dm ∗ p(x)|
m≥1
satisfies
kS ∗ pk1 ≤ c log (2N + 1) kpk1 .
Proof. For such a trigonometric polynomial p, the partial sums are all partial sums
of p ∗ DN , and all the Dirichlet kernels Dm for 1 ≤ m ≤ N are dominated by a
function whose L1 norm is of the order of log(2N + 1).
We can combine this with the inequalities in Corollary 3.2, since
k
m
X
X
aj hj ≤ c log (2mn + 1) aj hj .
max 1≤k≤n j=1
j=1
1
1
We then arrive at the main result in [14].
Corollary 3.5. For an increasing sequence (mn )∞
n=1 of natural numbers and a se∞
quence of non-zero complex numbers (an )n=1 the partial sums of the trigonometric
series
∞
X
ak eimk x
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page
Contents
JJ
II
J
I
Page 12 of 16
Go Back
Full Screen
k=1
Close
satisfy
k
X
log n
imj (·) min |ak | p
≤ c max aj e
.
1≤k≤n
1≤k≤n log(2mn + 1)
j=1
1
This was Salem’s attempt at Littlewood’s conjecture, which was subsequently
settled in [5] and [8].
3.3.
Linearly Independent Sequences
Notice that if {v1 , . . . , vn } is an arbitrary linearly independent subset of H then there
is a unique subset
n
wj : 1 ≤ j ≤ n ⊆ span ({v1 , . . . , vn })
so that {v1 , . . . , vn } and {w1n , . . . , wnn } 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 {v1 , . . . , vn } in an
inner-product space H, with dual basis {w1n , . . . , wnn },
k
X
log n ≤ c max kwkn kH max vj 1≤k≤n
1≤k≤n j=1
and
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page
H
k
X
n
log n ≤ c max kvk kH max wj .
1≤k≤n
1≤k≤n j=1
Bi-orthogonal Pairs
Contents
JJ
II
J
I
H
The constant c > 0 is independent of n, H, and the sets of vectors.
Page 13 of 16
Go Back
3.4.
Matrices
Full Screen
Suppose that A is an invertible n × n matrix with complex entries and columns
a1 , . . . , a n ∈ C n .
Let b1 , . . . , bn be the rows of A−1 . From their definition
n
X
j=1
bij ajk = δik
Close
and so the two sets of vectors
n
o
T
T
b1 , . . . , bn
and
{a1 , . . . , an }
are a bi-orthogonal pair in Cn . Theorem 2.1 then says that
k
X
log(n) ≤ c max kbk k max aj .
1≤k≤n
1≤k≤n j=1
2
The norm here is the finite dimensional ` norm. This brings us back to the material
in [6]. Note that [4] has logarithmic lower bounds for `1 -norms of column vectors of
orthogonal matrices.
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page
Contents
JJ
II
J
I
Page 14 of 16
Go Back
Full Screen
Close
References
[1] S.V. BOCHKAREV, A generalization of Kolmogorov’s theorem to biorthogonal systems, Proceedings of the Steklov Institute of Mathematics, 260 (2008),
37–49.
[2] K. HOFFMAN AND R.A. KUNZE, Linear Algebra, Second ed., Prentice Hall,
1971.
[3] B.S. KASHIN AND A.A. SAAKYAN, Orthogonal Series, Translations of
Mathematical Monographs, vol. 75, American Mathematical Society, Providence, RI, 1989.
[4] B.S. KASHIN, A.A. SAAKYAN AND S.J. SZAREK, Logarithmic growth of
the L1 -norm of the majorant of partial sums of an orthogonal series, Math.
Notes, 58(2) (1995), 824–832.
[5] S.V. KONYAGIN, On the Littlewood problem, Izv. Akad. Nauk SSSR Ser. Mat.,
45(2) (1981), 243–265, 463.
[6] S. KWAPIEŃ AND A. PEŁCZYŃSKI, The main triangle projection in matrix
spaces and its applications, Studia Math., 34 (1970), 43–68. MR 0270118 (42
#5011)
[7] S. KWAPIEŃ, A. PEŁCZYŃSKI AND S.J. SZAREK, An estimation of the
Lebesgue functions of biorthogonal systems with an application to the nonexistence of some bases in C and L1 , Studia Math., 66(2) (1979), 185–200.
[8] O.C. McGEHEE, L. PIGNO AND B. SMITH, Hardy’s inequality and the L1
norm of exponential sums, Ann. of Math. (2), 113(3) (1981), 613–618.
Bi-orthogonal Pairs
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page
Contents
JJ
II
J
I
Page 15 of 16
Go Back
Full Screen
Close
[9] C. MEANEY, Remarks on the Rademacher-Menshov theorem, CMA/AMSI
Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis,
and Related Topics”, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 42,
Austral. Nat. Univ., Canberra, 2007, pp. 100–110.
[10] D. MENCHOFF, Sur les séries de fonctions orthogonales, (Premiére Partie. La
convergence.), Fundamenta Math., 4 (1923), 82–105.
Bi-orthogonal Pairs
[11] A.M. OLEVSKIĬ, Fourier series with respect to general orthogonal systems.
Translated from the Russian by B. P. Marshall and H. J. Christoffers., Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 86. Berlin-Heidelberg-New
York: Springer-Verlag., 1975.
[12] M.A. PINSKY, Introduction to Fourier Analysis and Wavelets, Brooks/Cole,
2002.
[13] R. SALEM, A new proof of a theorem of Menchoff, Duke Math. J., 8 (1941),
269–272.
[14] R. SALEM, On a problem of Littlewood, Amer. J. Math., 77 (1955), 535–540.
Christopher Meaney
vol. 10, iss. 4, art. 94, 2009
Title Page
Contents
JJ
II
J
I
Page 16 of 16
Go Back
Full Screen
Close