SERGEI CHOBANYAN, SHLOMO LEVENTAL, AND HABIB SALEHI Received 26 March 2005; Accepted 7 September 2005 The goal of the paper is twofold: 1 to show that the exact valueD2in the Me Rademach
Trang 1SERGEI CHOBANYAN, SHLOMO LEVENTAL, AND HABIB SALEHI
Received 26 March 2005; Accepted 7 September 2005
The goal of the paper is twofold: (1) to show that the exact valueD2in the Me Rademacher inequality equals 4/3, and (2) to give a new proof of the Me ´nshov-Rademacher inequality by use of a recurrence relation The latter gives the asymptotic estimate lim supn D n / log22n ≤1/4.
Copyright © 2006 Sergei Chobanyan et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
The Me ´nshov-Rademacher inequality deals with the estimation of
D n =sup E max
1≤ k ≤ n
k
l =1
α l ϕ l
2
where sup is taken over all probability spaces (Ω,Ᏺ,P), all real orthonormal systems (ϕ1, , ϕ n) on them, and all real coefficient collections (α1, , α n) withn
1α2
i =1 Rademacher [9] and Me ´nshov [7] independently proved that there exists an absolute constantC > 0 such that for each n ≥2,
A traditional proof using a bisection method (see, e.g., Doob [2] and Lo`eve [6]) leads to the inequality
D n ≤log2n + 2 2
Kounias [4] used a trisection method to get a finer inequality:
D n ≤
log
2n
log23+ 2
2
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2006, Article ID 68969, Pages 1 7
DOI 10.1155/JIA/2006/68969
Trang 2The aim of this paper is twofold: to show that the exact starting valueD2=4/3 and to
establish a recurrence relation which leads to a refinement of (1.4) and an asymptotic constant≤1/4 Note that there are several other proofs of the Me ´nshov-Rademacher
in-equality and its generalizations, see, for example, Somogyi [10] and M ´oricz and Tandori [8]
Section 2 deals with the proof ofD2=4/3, whileSection 3 is devoted to the proof
of the Me ´nshov-Rademacher inequality with the asymptotic constant≤1/4.Section 4 contains alternative proofs to those results using the concept of main triangle projec-tion, a subject which was studied in depth in Gohberg and Kre˘ın [3] and Kwapie ´n and
“Pełczy ´nski” [5]
2 The value ofD2
Theorem 2.1 D2=4/3.
The proof of the theorem is based on the following lemma which may be of indepen-dent interest
Lemma 2.2 Let c > 0 , p c ≡ c2/(1 + c2), and define
f (p, c) = sup
X ∈ Ꮽ(p,c)E
where
Ꮽ(p,c) = {X ∈ L0(Ω,Ᏺ,P) : E(X)=0, E(X2)=1,P(X > −c) = p}. (2.2)
Then
Proof of Lemma 2.2 To show that the left-hand side is greater than or equal to right-hand
side, we observe that E(X p1X p > − c)= p(1 − p), where the distribution of X p ∈ Ꮽ(p,c) is
given by
p = P
X p =
(1− p) p
=1− P
X p = −
p
(1− p)
To see that the left-hand side is less than or equal to right-hand side, we define
h p(x) = x ·1x> − c − p · x −
p(1 − p)
The maximum ofh p(x) is achieved at x = (1− p)/ p and at − p/(1 − p) for the regions
x > −c and x ≤ −c, respectively We conclude that for any X ∈ Ꮽ(p,c),
0≤E
h p
X p
−E
h p(X)
=E
X p ·1Xp > − c
−E
X ·1X>− c
Trang 3Let us note also thatᏭ(p,c) is empty for p < p c Indeed, by the Chebyshev inequality,
E(X) =0 and E(X2)=1 implyP(X ≤ −c) ≤1/(1 + c2)=1− p c
Proof of Theorem 2.1 The result follows by standard calculations from the representation
a2 +b2=1,b2/(1+3a2 )<p<1
a2+b2p + 2ab · p(1 − p) (2.7)
To prove (2.7) convert an orthonormal pair (ϕ1,ϕ2) defined on (Ω,Ᏺ,P) into (X ≡ ϕ1/
ϕ2, 1) The new pair is orthonormal with respect to the measuredP = ϕ2dP Also
EPmax
aϕ1 2 ,
aϕ1+bϕ2 2
=EPmax
(aX)2, (aX + b)2
= a2+b2P (X > −b/2a) + 2ab ·EP
X ·1X>− b/2a
≤ a2+b2p + 2ab · f
p, b
2a
,
(2.8)
wherep = P (X > − b/2a) Now (2.7) follows fromLemma 2.2withc = b/2a.
3 An induction proof of the Me ´nshov-Rademacher inequality
Theorem 3.1 (i)
D m ≤1
4
3 + log2m 2
In particular, (ii)
lim sup
m
D m
log22m ≤1
Lemma 3.2 The following recurrence relation holds true for any n ∈ N:
Proof of Lemma 3.2 We have for any n ∈ N,
max
k ≤2n
k
1
α i ϕ i
2
≤max
max
k ≤ n
k
1
α i ϕ i
2 ,
n
1
α i ϕ i
+ maxn<k ≤2n
k
n+1
α i ϕ i
2
≤max
k ≤ n
k
1
α i ϕ i
2 + 2
n
1
α i ϕ i
n<kmax≤2n
k
n+1
α i ϕ i
+ maxn<k ≤2n
k
n+1
α i ϕ i
2
.
(3.4) Taking expectations in (3.4) and using the Cauchy-Schwartz inequality, we come to the
Trang 4desired recurrence relation:
D2n ≤ pD n+ 2 p(1 − p)D n+ (1− p)D n = D n+ D n, (3.5) wherep =n
1α2i
Proof of Theorem 3.1 Lemma 3.2implies that for anyn ∈ N,
D12/2 n ≤ D1/2
SinceD1=1, this implies that for eachn ∈ N,
D12/2 n ≤1 +n
Let us take now 2n ≤ m < 2 n+1 Then
D m ≤ D2n+1 ≤
1 +n + 1 2
2
≤
1 +log2m + 1 2
2
Remark 3.3 (1) The proof ofTheorem 3.1is a refinement of that appeared in Chobanyan [1]
(2) Kounias’s result mentioned in the introduction leads to lim sup(D n / log22n) ≤
(log 2/ log 3)2which is larger than 1/4 ofTheorem 3.1
4 An alternative approach: the main triangle projection
Consider the space L(Rn) of all linear operators (matrices) acting inRn The correspon-dence between the operators and matrices is given bya i j =(Ae j,e i), i, j =1, , n The main triangle projection T n: L(Rn)→L(Rn) is a linear operator introduced as follows For
anA ∈L(Rn), the matrix of the operatorB = T n A has the form b i j = a i j ifi + j ≤ n + 1
andb i j =0 otherwise
We assume thatRnis endowed with the Euclidean norm, and the norm in L(Rn) is the usual operator norm
Theorem 4.1 D n = T n 2, n ∈ N.
Proof Let us prove first that T n 2≡sup A ≤1T n A2≤ D n Since the orthogonal
op-erators (and only them) are the extreme points of the unit ball of L(Rn), it suffices to show that for any orthogonal operatoru ∈L(Rn), T n u2≤ D n Let us relate with u the
orthonormal systemϕ1, , ϕ ndefined on (Ω,P), where Ω= {1, , n}, P( j) =1/n, j =
1, , n, as follows:
ϕ k(j) = √ n
ue k,e j , k, j =1, , n. (4.1)
Trang 5We have for any vectorα =(α1, , α n)∈ R nwith|α| =1,
D n ≥E max
k ≤ n
k
i =1
α i ϕ i
2
=
n
j =1
max
k ≤ n
k
i =1
α i
ue i,e j
2
≥
n
j =1
n −j+1
i =1
α i
ue i,e j
2
=T n u
α 2
.
(4.2)
Taking supremum over all orthogonalu’s and α’s from the unit ball ofRn, we getD n ≥
T n 2 To prove the inverse inequality, consider an orthonormal system (ϕ1, , ϕ n)⊂
L2(Ω,Ᏺ,P) and any vector α =(α1, , α n)∈ R nwith|α| =1
I(α, ϕ) ≡E max
k ≤ n
k
i =1
α i ϕ i
2
=
n
k =1
E1S k
k
i =1
α i ϕ i
2
where S k = {ω ∈ Ω : the minimum of l s at which|l
i =1α i ϕ i(ω)|attains its maximum equalsk} Then we have
I(α, ϕ) =sup
g
n
k =1
Eg k1S k
k
i =1
α i ϕ i
2
where supremum is taken over all collectionsg =( 1, , g n) such thatg k’s vanish outside
ofS kandg k 2=1,k =1, , n We have further
I(α, ϕ) =sup
g
n
k =1
k
i, j =1
α i α jE g k ϕ i ϕ j
=sup
g
n
i, j =1
n
k =max(i, j)
α i α jE g k ϕ i ϕ j =sup
g
T n Aα 2
,
(4.5)
where (Ae j,e i)=Eg n − j+1 · ϕ i,i, j =1, , n We have
A =sup
| α |=1
n
i =1
n
j =1
Eα j g n − j+1 ϕ i
2
=sup
| α |=1
n
i =1
Ef ϕ i 2
=sup
| α |=1
Ef2=1, (4.6)
where f = α j g j, ifω ∈ S j, j =1, , n Therefore, (4.5) impliesD n ≤ T n 2 The theorem
The following corollary is ourTheorem 2.1
Corollary 4.2 D2=4/3.
Proof We have according toTheorem 4.1,
D2=T2 2
=sup
u
T2u 2
=sup
a b
b 0
2 :a2+b2=1
=4
Trang 6
Remark 4.3 It follows from the proof ofTheorem 4.1thatD n =sup E[maxj(j
l =1a l ϕ l)2], where the supremum is over all real orthonormal systemsϕ1, , ϕ n, where eachϕ j, j =
1, , n takes at most n values, and all reals α1, , α nwith|α| =1
The following lemma establishes a finer recurrence relation thanLemma 3.2 However, the two lemmas are asymptotically equivalent
Lemma 4.4
D2n ≤4
3D n if D n ≤3, D2n ≤ D n −1
2+
D n −3
4 if D n ≥3. (4.8)
Proof We have for any n ∈ N:
T2n =sup
A T n B
T n C 0
where the supremum runs over all matricesA, B, C, and D in L(Rn) such that(A B C D) ≤
1 For such matrices A, B, C, and D we check that |uA|2+|uT n B|2≤ T n 2|u|2 and
|Ax|2+|T n Cx|2≤ T n 2|x|2for allu, x ∈ R n Therefore,T2n ≤sup{(u, Ax) + (u, F y) +
(v, Gy) : u, v, x, y ∈ R n,|u|2+|v|2≤1,|x|2+|y|2≤1,A, F, G ∈L(Rn),A ≤1,|wA|2+
|wF|2≤ D n |w|2,|Az|2+|Gz|2≤ D n |z|2for allw, z ∈ R n
The last supremum can easily
be computed and its square equals supa ∈[0,1](D n − a/2 +
D n a−3a2/4) Hence, D2n ≤
4/3D nifD n ≤3 andD2n ≤ D n −1/2 +
D n −3/4 if D n ≥3 This completes the proof of
Finally, it is known that for the Hilbert matrix (H n(i, j)=1/(i− j), if i = j and H n(i, i)=
0,i, j =1, , n, n ≥2),
T n H n
This along withTheorem 3.1implies the following bilateral estimate:
1
π2log22e ≤lim inf D n
log22n ≤lim sup D n
log22n ≤1
Acknowledgments
This work was supported in part by the US Civilian Research and Development Foun-dation Award GEMI-3328-TB-03 We want to express our gratitude to the anonymous referee for bringing to our attention the relationship betweenD n and the norm of the main triangle projection Furthemore, the results/proofs inSection 4are based on ideas, suggestions, and comments made by the referee
References
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Sergei Chobanyan: Muskhelishvili Institute of Computational Mathematics, Georgian Academy of Sciences, 8 Akuri Street, Tbilisi 0193, Georgia
E-mail address:chobanya@stt.msu.edu
Shlomo Levental: Department of Statistics & Probability, Michigan State University, East Lansing,
MI 48824, USA
E-mail address:levental@stt.msu.edu
Habib Salehi: Department of Statistics & Probability, Michigan State University, East Lansing,
MI 48824, USA
E-mail address:salehi@stt.msu.edu
... betweenD n and the norm of the main triangle projection Furthemore, the results/proofs inSection 4are based on ideas, suggestions, and comments made by the refereeReferences... supported in part by the US Civilian Research and Development Foun-dation Award GEMI-3328-TB-03 We want to express our gratitude to the anonymous referee for bringing to our attention the relationship... n Since the orthogonal
op-erators (and only them) are the extreme points of the unit ball of L(Rn), it suffices to show that for any orthogonal