AN INEQUALITY FOR BI-ORTHOGONAL PAIRS

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Bi-orthogonal Pairs Christopher Meaney vol. 10, iss. 4, art. 94, 2009

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AN INEQUALITY FOR BI-ORTHOGONAL PAIRS

CHRISTOPHER MEANEY

Department of Mathematics Faculty of Science

Macquarie University

North Ryde NSW 2109, Australia EMail:chrism@maths.mq.edu.au

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 con- stants.

Abstract: We use Salem’s method [13, 14] to prove an inequality of Kwapie´n and Pełczy´nski 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 onL¹norms for orthogonal expansions.

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1. Introduction

Suppose thatHis a Hilbert space,n∈N, and thatJ ={1, . . . , n}orJ =N. A pair of sets {v_j : j ∈J} and {w_j : j ∈J} in H are said to be a bi-orthogonal pair when

hvj, wki_H =δjk, ∀j, k∈J.

The inequality in Theorem2.1 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 usingL²inequalities to produceL¹estimates on maximal functions. Such estimates are related to the stronger results of Olevski˘ı [11], Kashin and Szarek [4], and Bochkarev [1]. We conclude with an observation about the statement of Theo- rem2.1in 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 Rademacher- Menshov Theorem.

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2. The Kwapie ´n-Pełczy ´nski Inequality

Theorem 2.1. There is a positive constant c with the following property. For ev- eryn ≥ 1, every Hilbert space H, and every bi-orthogonal pair {v₁, . . . , v_n} and {w₁, . . . , w_n}inH,

(2.1) logn≤c max

1≤m≤nkwmk_H max

1≤k≤n

k

X

j=1

vj

H

.

Proof. Equip [0,1] with a Lebesgue measure λ and let V = L²([0,1], H) be the space ofH-valued square integrable functions on[0,1], with inner product

hF, Gi_V = Z 1

0

hF(x), G(x)i_Hdx and norm

kFk_V = Z 1

0

kF(x)k²_Hdx

.

Suppose that {F₁, . . . , F_n} is an orthonormal set in L²([0,1]) and define vectors p₁, . . . , p_ninV by

p_k(x) =F_k(x)w_k, 1≤k ≤n, x ∈[0,1].

Then

hpk(x), pj(x)i_H =Fk(x)Fj(x) hwk, wji_H , 1≤j, k ≤n,

and so{p₁, . . . , p_n}is an orthogonal set inV. For everyP ∈V, Bessel’s inequality states that

(2.2)

n

X

k=1

|hP, p_ki_V|²

kwkk²_H ≤ kPk²_V .

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Note that here

hP, p_ki_V = Z 1

0

hP(x), w_ki_HF_k(x)dx, 1≤k ≤n.

Now consider a decreasing sequence f₁ ≥ f₂ ≥ · · · ≥ f_n ≥ f_n+1 = 0 of characteristic functions of measurable subsets of[0,1]. For each scalar-valuedG ∈ L²([0,1])define an element ofV by setting

P_G(x) = G(x)

n

X

j=1

f_j(x)v_j.

The Abel transformation shows that P_G(x) = G(x)

n

X

k=1

∆f_k(x)σ_k, where ∆f_k = f_k − f_k+1 and σ_k = Pk

j=1v_j, for 1 ≤ k ≤ n. The functions

∆f1, . . . ,∆fn are characteristic functions of mutually disjoint subsets of[0,1]and for each0≤x≤1at most one of the values∆fk(x)is non-zero. Notice that

kP_G(x)k²_H =|G(x)|²

n

X

k=1

∆f_k(x)kσ_kk²_H. Integrating over[0,1]gives

kP_Gk²_V ≤ kGk²₂ max

1≤k≤nkσ_kk²_H. Note that

hP_G(x), p_k(x)i_H =G(x)f_k(x)F_k(x) hv_k, w_ki_H, 1≤k ≤n, and

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hP_G, p_ki_V = Z 1

0

G(x)f_k(x)F_k(x)dx hv_k, w_ki_H, 1≤k≤n.

Combining this with Bessel’s inequality (2.2), we arrive at the inequality (2.3)

n

X

k=1

Z

[0,1]

Gf_kF_kdλ

2 1

kw_kk²_H ≤ kGk²₂ max

1≤k≤nkσ_kk²_H.

This implies that (2.4)

n

X

k=1

Z

[0,1]

GfkFkdλ

2!

≤

1≤j≤nmax kwkk²_H

kGk²₂

1≤k≤nmax kσkk²_H

. We now concentrate on the case where the functionsF₁, . . . , F_n are given by Men- shov’s result (Lemma 1 on page 255 of Kashin and Saakyan [3]). There is a constant c0 >0, independent ofn, so that

(2.5) λ

(

x∈[0,1] : max

1≤j≤n

j

X

k=1

F_k(x)

> c₀log(n)√ n

)!

≥ 1 4.

Let us useM(x)to denote the maximal function M(x) = max

1≤j≤n

j

X

k=1

F_k(x)

, 0≤x≤1.

Define an integer-valued functionm(x)on[0,1]by m(x) = min

( m :

m

X

k=1

F_k(x)

=M(x) )

. Furthermore, letf_k be the characteristic function of the subset

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{x∈[0,1] : m(x)≥k}.

Then n

X

k=1

f_k(x)F_k(x) =S_m(x)(x) =

m(x)

X

k=1

F_k(x), ∀0≤x≤1.

For an arbitraryG∈L²([0,1])we have Z 1

0

G(x)Sm(x)(x)dx=

n

X

k=1

Z 1 0

G(x)fk(x)Fk(x)dx.

Using the Cauchy-Schwarz inequality on the right hand side, we have (2.6)

Z 1 0

G(x)S_m(x)(x)dx

≤√ n

n

X

k=1

Z 1 0

Gf_kF_kdλ

2!1/2

,

for allG∈L²([0,1]). We will use the functionGwhich has|G(x)|= 1everywhere on[0,1],with

G(x)S_m(x)(x) =M(x), ∀0≤x≤1.

In this case, the left hand side of (2.6) is kMk₁ ≥ c₀

4 log(n)√ n, because of (2.5). Combining this with (2.6) we have

c₀

4 log(n)√ n≤√

n

X

k=1

Z 1 0

GfkFkdλ

2!1/2

.

This can be put back into (2.4) to obtain (2.1). Notice thatkGk2 = 1 on the right hand side of (2.3).

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3. Applications

3.1. L¹estimates

In this section we useH =L²(X, µ), for a positive measure space(X, µ). Suppose we are given an orthonormal sequence of functions (h_n)^∞_n=1 in L²(X, µ), and suppose 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≤nkh_jk_∞and S_n^∗(x) = max

1≤k≤n

k

X

j=1

a_jh_j(x)

, forx∈X, n ≥1.

Lemma 3.1. For a set of functions{h₁, . . . , h_n} ⊂L²(X, µ)∩L^∞(X, µ)and max- imal function

S_n^∗(x) = max

1≤k≤n

k

X

j=1

ajhj(x) , we have

|a_jh_j(x)| ≤2S_n^∗(x), ∀x∈X,1≤j ≤n,

and

Pk

j=1a_jh_j(x)

S_n^∗(x) ≤1, ∀1≤k ≤nandxwhereS_n^∗(x)6= 0.

Proof. The first inequality follows from the triangle inequality and the fact that a_jh_j(x) =

j

X

k=1

a_kh_k(x)−

j−1

X

k=1

a_kh_k(x)

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for2≤j ≤n. The second inequality is a consequence of the definition ofS_n^∗. Fixn≥1and let

v_j(x) =a_jh_j(x) (S_n^∗(x))^−1/2 andw_j(x) = a⁻¹_j h_j(x) (S_n^∗(x))^1/2 for allx∈XwhereS_n^∗(x)6= 0and1≤j ≤n. From their definition,

{v1, . . . , vn} and {w1, . . . , wn}

are a bi-orthogonal pair inL²(X, µ). The conditions we have placed on the functions hj give:

kw_jk²₂ =|a_j|⁻² Z

X

|h_j|²(S_n^∗)dµ≤ M_n²

min1≤k≤n|a_k|²kS_n^∗k₁ and

k

X

j=1

vj

2

= Z

X

1 (S_n^∗)

k

X

j=1

ajhj

2

dµ≤

k

X

j=1

ajhj

1

.

We can put these estimates into (2.1) and find that logn≤c M_n

min_1≤k≤n|a_k|kS_n^∗k^1/2₁ max

1≤k≤n

k

X

j=1

a_jh_j

1/2

1

.

We could also say that

1≤k≤nmax

k

X

j=1

ajhj

1

≤ kS_n^∗k₁ and so

log(n)≤c M_n

min1≤k≤n|a_k|kS_n^∗k₁.

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Corollary 3.2. Suppose that(h_n)^∞_n=1 is an orthonormal sequence inL²(X, µ)con- sisting of essentially bounded functions. For each sequence(a_n)^∞_n=1of complex num- bers and eachn ≥1,

1≤k≤nmin |a_k| logn 2

≤c

1≤k≤nmax kh_kk_∞ 2

1≤k≤nmax

k

X

j=1

a_jh_j

1

1≤k≤nmax

k

X

j=1

a_jh_j 1

and

1≤k≤nmin |a_k| logn≤c

1≤k≤nmax kh_kk_∞

1≤k≤nmax

k

X

j=1

a_jh_j

1

.

The constantcis independent ofn, 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 inL²(X, µ)con- sisting of essentially bounded functions with kh_nk_∞ ≤ M for alln ≥ 1. For each decreasing sequence(a_n)^∞_n=1of positive numbers and eachn ≥1,

(a_n logn)² ≤cM²

1≤k≤nmax

k

X

j=1

a_jh_j

1

1≤k≤nmax

k

X

j=1

a_jh_j 1

and

a_n logn ≤cM

1≤k≤nmax

k

X

j=1

a_jh_j

1

.

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In particular, if(a_nlogn)^∞_n=1is unbounded then





1≤k≤nmax

k

X

j=1

ajhj

1





∞

n=1

is unbounded.

The constantcis independent ofn, and the sequences involved here.

3.2. Salem’s Approach to the Littlewood Conjecture

We concentrate on the case whereH = L²(T) and the orthonormal sequence is a subset of{e^inx : n ∈N}. Let

m₁ < m₂ < m₃ <· · · be an increasing sequence of natural numbers and let

h_k(x) = e^im^k^x for allk≥1andx∈T.In addition, let

D_m(x) =

m

X

k=−m

e^ikx

be them^th Dirichlet kernel. For allN ≥m≥1, there is the partial sum X

m_k≤m

a_kh_k(x) =D_m∗ X

mk≤N

a_kh_k

! (x).

It is a fact thatD_m is an even function which satisfies the inequalities:

(3.1) |D_m(x)| ≤

(2m+ 1 for allx,

1/|x| for _2m+1¹ < x <2π− _2m+1¹ .

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Lemma 3.4. Ifpis a trigonometric polynomial of degreeN,then the maximal func- tion of its Fourier partial sums

S^∗p(x) = sup

m≥1

|D_m∗p(x)|

satisfies

kS^∗pk₁ ≤clog (2N+ 1)kpk₁.

Proof. For such a trigonometric polynomialp, the partial sums are all partial sums of p∗D_N, and all the Dirichlet kernels D_m for1 ≤ m ≤ N are dominated by a function whoseL¹ norm is of the order oflog(2N+ 1).

We can combine this with the inequalities in Corollary3.2, since

1≤k≤nmax

k

X

j=1

ajhj

1

≤clog (2mn+ 1)

m

X

j=1

ajhj

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

k=1

ake^im^k^x

satisfy

1≤k≤nmin |a_k| logn

plog(2m_n+ 1) ≤c max

1≤k≤n

k

X

j=1

a_je^im^j^(·) 1

.

This was Salem’s attempt at Littlewood’s conjecture, which was subsequently settled in [5] and [8].

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3.3. Linearly Independent Sequences

Notice that if{v₁, . . . , v_n}is an arbitrary linearly independent subset ofHthen there is a unique subset

w_jⁿ : 1≤j ≤n ⊆span({v₁, . . . , v_n})

so that{v1, . . . , vn}and{w₁ⁿ, . . . , w_nⁿ}are a bi-orthogonal pair. See Theorem 15 in Chapter 3 of [2]. We can apply Theorem2.1to the pair in either order.

Corollary 3.6. For eachn ≥ 2and linearly independent subset {v1, . . . , vn}in an inner-product spaceH, with dual basis{wⁿ₁, . . . , w_nⁿ},

logn ≤c max

1≤k≤nkw_kⁿk_H max

1≤k≤n

k

X

j=1

v_j H

and

logn≤c max

1≤k≤nkv_kk_H max

1≤k≤n

k

X

j=1

wⁿ_j H

.

The constantc > 0is independent ofn,H, and the sets of vectors.

3.4. Matrices

Suppose thatAis an invertiblen×nmatrix with complex entries and columns a1, . . . , an ∈Cⁿ.

Letb₁, . . . , b_nbe the rows ofA⁻¹. From their definition

n

X

j=1

b_ija_jk =δ_ik

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and so the two sets of vectors n

b^T₁, . . . , b^T_no

and {a₁, . . . , a_n} are a bi-orthogonal pair inCⁿ. Theorem2.1then says that

log(n)≤c max

1≤k≤nkbkk max

1≤k≤n

k

X

j=1

aj

.

The norm here is the finite dimensional`² 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.

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References

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[2] K. HOFFMANANDR.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, Provi- dence, RI, 1989.

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[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.

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[14] R. SALEM, On a problem of Littlewood, Amer. J. Math., 77 (1955), 535–540.