We shall see that the Sidon constant of the Rademacher system equals π/2

RADEMACHER VARIABLES IN CONNECTION WITH COMPLEX SCALARS

J. A. SEIGNER

Abstract. We shall see that the Sidon constant of the Rademacher system equals π/2. This is also the best constant for the contraction principle if complex scalars are involved.

1. The Rademacher System and Its Sidon Constant

Rademacher variables are generally understood as an i.i.d sequence of random variables taking the values−1 and +1 each with probability 1/2. We model them as follows. LetEdenote the multiplicative group of the two elements−1 and +1 inC. Let us consider the cartesian powerE^∞=Q

j∈NEand the natural maps rj:E^∞→T, (j∈N),

which assign to any sequenceε= (εj)^∞_j=1 theirjth coordinateεj. HereTdenotes the group of complex numbers of modulus 1.

If we equip E^∞ with the coarsest topology that will make all rj continuous we find by Tychonoff’s theorem that E^∞ is compact. Moreover, if we define multiplication inE^∞ coordinate wise therj become homorphisms.

As we are usually concerned with only finitely many Rademacher variables at a time, and since we are interested in their distributional properties, barely, we may equally think ofr1, . . . , rn (nfixed) as to be defined onEⁿ rather thanE^∞. This should cause no troubles.

In either case, it turns out that we move in a convenient setting.

Given a compact abelian group Ga continuous homeomorphismχ:G→Tis calledcharacter. We say a sequence of charactersX = (χ1, χ2, . . .) is a Sidon set, provided we can find a constantC such that however we choose a naturaln and complex numbersa1, . . . , an we have

(1)

Xn j=1

|aj| ≤C Xn j=1

ajχj_∞.

Received March 14, 1997.

1980Mathematics Subject Classification(1991Revision). Primary 46B20, 46E40; Secondary 43A46.

The author is supported by the DFG.

Of course, k · k_∞ is shorthand for the norm in C(G), the space of continuous functions onG. IfXis a Sidon set, we label the smallest of all admissible constants C as

S(X).

This is theSidon constantofX (cf. [4]).

The system of Rademacher variablesR= (r1, r2, . . .) is an obvious example for a Sidon set, for if we split the term Pn

j=1ajrj into its real and imaginary parts we certainly get away with C = 2 in (1). We will see, that we can do slightly better, albeit the precise value ofS(R) is rather of aesthetic interest. The proof rests upon the following fact, which is almost a blueprint of [1, Lemma 3.6, p. 21].

Lemma 1. Forj= 1, . . . , n letKj be compact topological spaces and let fj be continuous complex functions on Kj. Suppose there exist points aj, bj ∈Kj such that

kfjk_∞=|fj(aj)|=|fj(bj)| and fj(aj) =−fj(bj).

If we define f∈CQn j=1Kj

by

f(t1, . . . , tn) = Xn j=1

fj(tj), (t1, . . . , tn)∈ Yn j=1

Kj,

then _n

X

j=1

kfjk_∞≤π 2kfk_∞. Moreover, the constantπ/2is best possible.

Proof. Let us fix someϑ∈[0,2π) for an instant. Define tj =

aj, if Re (e^iϑfj(aj))>0

bj, if Re (e^iϑfj(bj))≥0 j= 1, . . . , n and chooseσj ∈[0,2π) such that

kfjk_∞=e^−iσjfj(bj) =e^i(π−σj)fj(aj).

Then we get

Re (e^iϑfj(tj)) = maxn

Re (e^i(ϑ+σj))kfjk_∞, Re (e^i(ϑ+σj+π))kfjk_∞o

=kfjk_∞|cos(ϑ+σj)|. Hence,

kfk_∞≥e^iϑ Xn j=1

fj(tj)≥ Xn j=1

Re

e^iϑfj(tj)

= Xn j=1

kfjk_∞|cos(ϑ+σj)|.

Integration with respect toϑwill settle our issue, since 2πkfk_∞≥

Xn j=1

kfjk_∞ Z 2π

0

|cos(ϑ+σj)|dϑ= 4 Xn j=1

kfjk_∞.

The fact that π/2 is best possible will be clear by the example included in the

proof of the following theorem.

Theorem 2. The Sidon constant of the Rademacher system equalsπ/2.

Proof. Given a1, . . . , an∈Cwe definefj:E→Cbyfj(−1) =−aj,fj(+1) = aj. Letf:Eⁿ →Cbe given byf(ε1, . . . , εn) =Pn

j=1fj(εj), then we may just as well write

f = Xn j=1

ajrj,

wherer1, . . . , rn are to be understood as defined onEⁿrather thanE^∞. Now, the preceeding lemma applies and we get

Xn j=1

|aj| ≤ π 2

Xn j=1

ajrj_∞.

As for the optimality ofπ/2 fixn∈Nfor an instant. Letβ =e^i2π/n be ann-the root of unity. We are going to consider the functiongn=Pn

j=1β^j−1rj onEⁿ. If sign (a)∈Tis defined by|a|/a (a6= 0) and sign (0) = 1 then

|gn(ε)|= sign gn(ε)

gn(ε) = Xn j=1

Re

β^j−1sign (gn(ε))

εj (ε= (ε1, . . . , εn)).

Since, obviouslyPn

j=1Re (β^j−1e^iϑ)εj≤ |gn(ε)|it follows that kgnk_∞= max

εj=±1 max

0≤ϑ<2π

Xn j=1

Re (β^j−1e^iϑ)εj. Note thatϑ7→maxε_j=±1Pn

j=1Re (β^j−1e^iϑ) is 2π/n-periodic and determineϑn ∈ [0,^2π_n) andε^∗₁, . . . , ε^∗_n such that

kgnk_∞= Xn j=1

Re (β^j−1e^iϑn)ε^∗_j = Xn j=1

cos

ϑn+2π(j−1) n

ε^∗_j.

By maximality every summand cos

ϑn+^2π(j_n⁻¹⁾

ε^∗_j is bound to be non-negative.

Thus, in actual fact we have kgnk_∞=

Xn j=1

cos

ϑn+2π(j−1)

n .

SincePn

j=1|β^j−1|=nwe may concluden≤S(R_n)kgnk_∞ or, equivalently, S(R_n)⁻¹≤kgnk_∞

n = 1

n Xn j=1

cos

ϑn+2π(j−1)

n .

By continuity of the cosine we find that the right hand side tends toR1

0 |cos(2πt)|dt

= ²_π asn→ ∞. We concludeS(R) = sup_nS(R_n)≥ ^π

2.

2. The Contraction Principle Using Complex Scalars

The result on the Sidon constant of the Radmacher system can be applied to the complex version of the contraction principle. It is well known, and easily seen, that given realsa1, . . . , an and vectorsx1, . . . , xn in some (real or complex) Banach spaceX we always have

Xn j=1

ajxjrj_LX

p(En)

≤ max

j=1,...,n|aj| Xn j=1

xjrj_LX

p(En)

for 1≤p≤ ∞(cf. [3, p. 91]).

If we want to extend this result to complex scalars and complex Banach spaces the basic tool is Pe lczy´nski’s celebrated result on commensurate sequences [5]

which we state in a disguised form.

Lemma 3 (Pe lczy´nski [5], Pisier [6]). Letχ1, . . . , χn and ψ1, . . . , ψn be char- acters on compact abelian groupsGandH, respectively. IfC≥1is such that

Xn j=1

ajψj_∞≤C Xn j=1

ajχj_∞ for a1, . . . , an∈C

then we find for alls∈H and1≤p≤ ∞

Xn j=1

xjψj(s)χj_LX

p(G)≤C Xn j=1

xjχj_LX

p(G),

regardless of the choice of the Banach spaceX and vectorsx1, . . . , xn inX. As for the proof there is no point in going into details. Everything can be found in Pe lczy´nski’s paper ([5, Theorem 1], compare [6, Th´eor`eme 2.1]). Three notes may be helpful.

• Here,L^X_p (G) is the space of Bochner-p-integrableX-valued functions onG (with respect to the Haar measure). Of course, Pn

j=1xjϕj:G −→ X is continuous, so we need not bother about integrability.

• The claim remains true also for Orlicz spacesL^X_φ(G) with literally the same proof as in [5], since all that is employed is Young’s inequality.

• We suggest to call the best constant C in the inequality above relative Sidon constant of Ψn = (ψ1, . . . , ψn) vs. X_n = (χ1, . . . , χn) for the fol- lowing reason. If ψ1, . . . , ψn are Steinhaus variables, i.e.ψj:Tⁿ →T is the projection on thejth coordinate, then

Xn j=1

ajψj_∞= Xn j=1

|aj| and we find that the best constantCequals S(X_n).

Corollary 4. The best constant in the principle of contraction for complex scalars isπ/2.

Proof. We have to proof the inequality

Xn j=1

ajxjrj_LX

p(En)

≤π 2

Xn j=1

xjrj_LX

p(En)

whenevera1, . . . , an are complex scalars of modulus≤1. Just as in the real case (cf. [3, pp. 95]), we may argue by convexity to see that the function

(a1, . . . , an)7→

Xn j=1

ajxjrj_LX

p(En)

takes its maximum on {a : kak_∞ ≤ 1} ⊂ `ⁿ∞ in an extreme point, say in s = (s1, . . . , sn) where|s1|=· · ·=|sn|= 1. Ifψ1, . . . , ψn are Steinhaus variables the lemma implies

Xn j=1

xjψj(s)rj_LX

p(En)≤C Xn j=1

xjrj_LX

p(En)

withC=S(R_n)%π/2 (n→ ∞).

As for the question of optimality it is useful to note that if we restrict our attention to scalarsaj of modulus 1 the inequalities

Xn j=1

ajrjxj_LX

p(En)

≤Cn Xn j=1

rjxj_LX

p(En) for |a1|=· · ·=|an|= 1 (2)

and

Xn j=1

rjxj_LX

p(En)

≤Cn Xn j=1

ajrjxj_LX

p(En) for |a1|=· · ·=|an|= 1 (3)

are equivalent, leading to the same constantCn.

Let us consider a second set ofnRademacher variables which we would like to interpret as vectorsxj inX =C(Eⁿ) given byxj(eε1, . . . ,eεn) =εej.

An inspection on the proof of Theorem 2 reveals that 1

n Xn j=1

e^2πij/nxj_∞

tends to 2/π as n→ ∞. The key observation is that for arbitrary complex num- bersaj

Xn j=1

ajxj_∞= max

e εj=±1

Xn j=1

ajεej= max

e εj=±1

Xn j=1

ajεjeεj however we chooseε1, . . . , εn∈ {−1,+1}. Accordingly,

Xn j=1

e^2πij/nrjxj_LX

p(En)= Xn j=1

e^2πij/nxj_∞.

On the other hand 1 n

Xn j=1

rjxj_LX

p(En)= 1 n

Xn j=1

xj_∞= 1.

Combining these, the best constantsCn in

Xn j=1

rjxj_LX

p(En)

≤Cn Xn j=1

ajrjxj_LX

p(En)

for |a1| =· · · =|an|= 1 satisfy lim inf

n→∞ Cn ≥π/2. By the equivalence of (2) and

(3) our proof is complete.

3. Concluding Remark

Rademacher variables also are involved when`<sup>n</sup><sub>1</sub> is to be embedded into`∞or a suitable`^N∞ (N ≥n).

Let us recall the real situation. If we take N = 2ⁿ we may identify `^N∞ with

`<sup>∞</sup>(E<sup>n</sup>). If the unit vectors in `^∞(Eⁿ) are labelled eε (ε ∈ Eⁿ), we can define vectorsuj in`^∞(Eⁿ) via

uj = X

ε∈En

rj(ε)eε. Then, for realsa1, . . . , an we certainly have

Xn j=1

|aj|= max

εj=±1

Xn j=1

ajεj= Xn j=1

ajuj_∞,

which amounts to saying that

`<sup>n</sup><sub>1</sub> →`^∞(Eⁿ), ej7→uj (j= 1, . . . , n) is an isometric embedding.

If we look at our previous discussion, the same mapping reinterpreted as acting between the corresponding complex spaces will have operator norm arbitrarily close toπ/2 asn→ ∞— and not any better.

Nevertheless, the universal character of`<sup>∞</sup>still allows us to embed`ⁿ₁ −→ı Wn,→

`^∞ withdim(Wn) =nandkık kı⁻¹k ≤1 +δfor arbitrarily smallδ >0.

Again, isometry is possible if we invokeKronecker’s theoremon diophantine approximation (for a proof and related discussions consult e.g. [2, Chapt. XXIII, pp. 371–391]).

Theorem 5(Kronecker 1884). Letβ1, . . . , βn be in Rsuch that the set 1, β1, . . . , βn

is linearly independent over the fieldQ. Given arbitraryξ1, . . . , ξn inRandδ >0 there exist a natural numbermand integers k1, . . . , kn such that

ξj−mβj−kj< δ (j= 1, . . . , n).

We employ this result for our purposes.

Since

e^ia−e^ib≤ |a−b| (a, b∈R) we see that

exp (2πiαj)−exp (2πimβj)=exp 2πiαj−exp 2πi(mβj+kj)≤2πδ.

We conclude that the sequence of vectors{xm}^∞_m=1inCⁿ defined by xm=

Xn j=1

exp(2πimβj)ej∈Cⁿ (m∈N) is dense inTⁿ.

Put

wj =

e^2πimβj∞

m=1

∈`^∞. (j= 1, . . . , n) Given complex numbersa1, . . . , an, by density we get

Xn j=1

ajwj_∞= sup

m∈N

Xn j=1

aje^2πimβj= sup

k(ξj)ⁿ₁k∞=1

Xn j=1

ajξj= Xn j=1

|aj|. Finally, ifWn =span{w1, . . . , wm}we get the desired isometryı

`<sup>n</sup><sub>1</sub> →ı Wn ⊂`∞

ej7→wj (j= 1, . . . , n).

Acknowledgement. I would like to thank my teacher Prof. Dr. Albrecht Pietsch for his support and advice.

References

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J. A. Seigner, Friedrich-Schiller-Universit¨at Jena, Fakult¨at f¨ur Mathematik und Informatik, D-07743 Jena, Germany