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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:[email protected]

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.

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1. Introduction

Suppose thatHis 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, wkiH =δjk, ∀j, k∈J.

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

(2.1) logn_≤c max

1≤m≤nkwmkH 1max≤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, G_iV = Z 1

0 h

F(x), G(x)_iHdx

and norm

kF_kV =

Z 1

0 k

F(x)_k2_Hdx

.

Suppose that _{F1, . . . , Fn} is an orthonormal set in L2([0,1]) and define vectors

p1, . . . , pninV by

pk(x) =Fk(x)wk, 1≤k ≤n, x ∈[0,1].

Then

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

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

(2.2)

n

X

k=1

|hP, pkiV|2 kwkk2H

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Note that here

hP, pkiV =

Z 1

0 h

P(x), wkiHFk(x)dx, 1≤k ≤n.

Now consider a decreasing sequence f1 ≥ f2 ≥ · · · ≥ fn ≥ fn+1 = 0 of characteristic functions of measurable subsets of[0,1]. For each scalar-valuedG _∈ L2_([0_,_1])_{define an element of}_V _{by setting}

PG(x) = G(x) n

X

j=1

fj(x)vj.

The Abel transformation shows that

PG(x) = G(x) n

X

k=1

∆fk(x)σk,

where ∆fk = fk − fk+1 and σk = Pk_j=1vj, 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

kPG(x)k_H2 =|G(x)|2 n

X

k=1

∆fk(x)kσkk2_H.

Integrating over[0,1]gives

kPGk2_V ≤ kGk2₂ max 1≤k≤nkσkk

2 H. Note that

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Bi-orthogonal Pairs

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

(2.3)

This implies that

(2.4) shov’s result (Lemma 1 on page 255 of Kashin and Saakyan [3]). There is a constant

c0 >0, independent ofn, so that

Let us use_M(x)to denote the maximal function

M(x) = max

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

(2.6) 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

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3. Applications

3.1. L1_estimates

In this section we useH =L2₍_{X, µ}₎_{, for a positive measure space}₍_{X, µ}₎_{. Suppose} we are given an orthonormal sequence of functions (hn)∞_n=1 in L2(X, µ), and sup-pose 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

Mn = max

1≤j≤nkhjk∞and S ∗

n(x) = max 1≤k≤n

k

X

j=1

ajhj(x)

, forx_∈X, n _≥1.

Lemma 3.1. For a set of functions_{h1, . . . , hn} ⊂L2(X, µ)∩L∞(X, µ)and

max-imal function

S∗

n(x) = max₁_≤_k_≤_n

k

X

j=1

ajhj(x)

,

we have

|ajhj(x)| ≤2Sn∗(x), ∀x∈X,1≤j ≤n,

and

Pk

j=1ajhj(x)

S∗

n(x) ≤

1, _∀1_≤k _≤nandxwhere_S_n∗(x)₆= 0.

Proof. The first inequality follows from the triangle inequality and the fact that

ajhj(x) = j

X

k=1

akhk(x)− j−1

X

k=1

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are a bi-orthogonal pair inL2₍_{X, µ}₎_{. The conditions we have placed on the functions}

hj give:

We can put these estimates into (2.1) and find that

logn_≤c Mn

We could also say that

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Bi-orthogonal Pairs Corollary 3.2. Suppose that(hn)

∞

n=1 is an orthonormal sequence inL2(X, µ)

con-sisting of essentially bounded functions. For each sequence(an)∞_n=1of complex

num-bers and eachn _≥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(hn)∞_n=1 is an orthonormal sequence inL2(X, µ)

con-sisting of essentially bounded functions with _khnk∞ ≤ M for alln ≥ 1. For each

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In particular, if(anlogn) ∞

n=1is unbounded then





max

1≤k≤n

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_{einx _: _n _∈_N_}_{. Let}

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

hk(x) = eimkx for allk_≥1andx_∈T_._{In addition, let}

Dm(x) = m

X

k=−m

eikx

be themth _{Dirichlet kernel. For all}_N _≥_m_≥₁_{, there is the partial sum}

X

mk≤m

akhk(x) =Dm∗

X

mk≤N

akhk

!

(x).

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

(3.1) _|Dm(x)| ≤

(

2m+ 1 for allx,

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

S∗

Proof. For such a trigonometric polynomialp, the partial sums are all partial sums of p_∗DN, and all the Dirichlet kernels Dm for1 ≤ m ≤ N are dominated by a function whoseL1 _{norm is of the order of}_log(2_N_{+ 1)}_.

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

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

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Close 3.3. Linearly Independent Sequences

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

wn

j : 1≤j ≤n ⊆span({v1, . . . , vn})

so that_{v1, . . . , vn}and{w1n, . . . , wn}n 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_{wn

1, . . . , wn}n ,

logn _≤c max

1≤k≤nkw n

kkH max 1≤k≤n

k

X

j=1

vj

H and

logn_≤c max

1≤k≤nkvkkH 1max≤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, . . . , bnbe the rows ofA−1. From their definition n

X

j=1

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and so the two sets of vectors

n

bT

1, . . . , bTn

o

and _{a1, . . . , an}

are a bi-orthogonal pair inCn_{. Theorem}_2.1_{then says that}

log(n)_≤c max

1≤k≤nkbkk 1max≤k≤n

k

X

j=1

aj

.

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References

[1] S.V. BOCHKAREV, A generalization of Kolmogorov’s theorem to biorthogo-nal systems, Proceedings of the Steklov Institute of Mathematics, 260 (2008), 37–49.

[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.

[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. KWAPIE ´N ANDA. PEŁCZY ´NSKI, The main triangle projection in matrix spaces and its applications, Studia Math., 34 (1970), 43–68. MR 0270118 (42 #5011)

[7] S. KWAPIE ´N, A. PEŁCZY ´NSKI AND S.J. SZAREK, An estimation of the Lebesgue functions of biorthogonal systems with an application to the nonex-istence of some bases inCandL1_{, Studia Math., 66(2) (1979), 185–200.}

[8] O.C. McGEHEE, L. PIGNO AND B. SMITH, Hardy’s inequality and the L1

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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.

[10] D. MENCHOFF, Sur les séries de fonctions orthogonales, (Premiére Partie. La convergence.), Fundamenta Math., 4 (1923), 82–105.

[11] A.M. OLEVSKI˘I, Fourier series with respect to general orthogonal systems.

Translated from the Russian by B. P. Marshall and H. J. Christoffers.,

Ergeb-nisse 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.