Recursos

This appendix contains material about a version of the Wiener lemma that is required in Section 13. We have chosen the notation here to conform with standard conventions in Fourier analysis. (In the application of this material, the roles off :R→Cand its Fourier

transform ˆf is reversed, withb andt playing the role ofx andξ respectively.)

Let A be the Banach algebra of 2π-periodic continuous functions f : R → C such

that their Fourier coefficients ˆfn are absolutely summable, with norm kfkA = P

n∈Z|fˆn|.

Similarly, letR be the Banach algebra of continuous functions f :R→ C such that their

Fourier transform ˆf :R→C lies inL1(R), with norm kfkR=R_−∞∞ |fˆ(ξ)|dξ.

Given β > 1, we define the Banach algebra Aβ = {f ∈ A : supn∈Z|n|

β_|fˆ_{n| < ∞}} with norm kfk_Aβ = P

n∈Z|fˆn|+ supn∈Z|n|

β_|fˆ_{n|. Similarly, we define the Banach algebra}

R_β ={f ∈ R: sup_ξ∈R|ξ|β_{|fˆ(ξ)|<∞}with normkfk}

Rβ = R∞

−∞|fˆ(ξ)|dξ+supξ∈R|ξ|

β_|fˆ(ξ)|. The following Wiener lemmas are standard.

Lemma A.1 Let β >1 and let f, f1 ∈ Aβ. Suppose thatf is bounded away from zero on

the support of f1.

Then there exists g∈ Aβ such that f1=f g.

Lemma A.2 Let β >1 and let f, f1 ∈ Rβ. Suppose f1 is compactly supported and that f

is bounded away from zero on the support of f1.

Then there exists g∈ Rβ such that f1=f g.

A statement and proof of Lemma A.1 can be found in [13, Theorem 1.2.12]. In this paper, we require Lemma A.2, but we could not find it stated in the literature. Hence we provide here a proof of Lemma A.2, using a standard argument to reduce to Lemma A.1.

Lemma A.3 Let > 0. Suppose that f :R → C is a continuous function with suppf ⊂

[−π +, π −]. Let h : R → C denote the 2π-periodic continuous function such that

h|_[−π,π]=f|_[−π,π]. Then f ∈ R_β if and only if h∈_Aβ.

Proof (cf. [21, Theorem 6.2, Ch. VIII, p. 242]) Fix aC∞functionψ:R→Rsupported in

[−π+/2, π−/2] and such thatψ≡1 on [−π+, π−]. Forα∈[−1,1] letψα(x) =eiαxψ(x). Then there is a constantK0 >0 such that

|(ψb_α)_n| ≤K₀n−β, for all α∈[−1,1], n∈Z.

In particular,ψα∈Aβ for allα and sup|α|≤1kψαkAβ <∞.

Definehα(x) =eiαxh(x). Ifh∈Aβ, then hα =hψα∈Aβ and there is a constant K >0 such thatkhαkAβ ≤KkhkAβ for allα∈[−1,1].

Now, (hcα)n= 1 2π Z π −π eiαxh(x)e−inxdx= 1 2π Z ∞ −∞ f(x)e−i(n−α)xdx= 1 2π ˆ f(n−α). HenceRn n−1|fˆ(ξ)|dξ= R1 0 |fˆ(n−α)|dα= 2π R1 0 |(hc_α)_n|dα. It follows that kfkR = 2π ∞ X n=−∞ Z 1 0 |(hc_α)_n|dα= 2π Z 1 0 khαkAdα≤2πKkhkAβ. (A.1)

Next, we observe that any ξ ∈ _R can be expressed as ξ = (n−α) sgnξ where n ≥ 1,

α∈[0,1]. Hence sup ξ∈_R |ξ|β|fˆ(ξ)|= sup n≥1, α∈[0,1] (n−α)β|fˆ((n−α) sgnξ)| ≤ sup n≥1, α∈[0,1] nβ|fˆ((n−α) sgnξ)| ≤ sup n∈Z, α∈[−1,1] |n|β|fˆ(n−α)|= 2π sup n∈Z, α∈[−1,1] |n|β|(hcα)n| ≤2π sup α∈[−1,1] khαkAβ ≤2πKkhkAβ. (A.2)

Combining (A.1) and (A.2), we obtain thatkfkRβ ≤4πKkhkAβ. Hence we have shown

thath∈_Aβ implies thatf ∈ R_β.

Conversely, suppose f ∈ R_β. Then P

n∈_Z R1

0 |fˆ(n−α)|dα =

R∞

−∞|fˆ(ξ)|dξ <∞ and it

follows from Fubini that P

n∈_Z|fˆ(n−α)|< ∞ for almost every α. Fix such an α. Then P n∈Z|(hcα)n| = (1/2π) P n∈Z|fˆ(n−α)| < ∞ so that hα ∈ A. Hence h = (hα)−α ∈ A. Moreover, sup n∈_Z |n|β|ˆhn|= (1/2π) sup n∈_Z |n|β|fˆ(n)| ≤(1/2π) sup ξ∈_R |ξ|β|fˆ(ξ)|<∞, so thath∈_Aβ.

Proof of Lemma A.2 (cf. [21, Lemma 6.3, Ch. VIII, p. 242]) We make the standard abuse of notation that functions on R supported on a closed subset of (−π, π) can be identified

with 2π-periodic functions on R. In particular, the conclusion of Lemma A.3 becomes

f ∈ R_β if and only if f ∈_Aβ.

Without loss, we can suppose that suppf1 ⊂[−2,2]. By Lemma A.3,f1 ∈Aβ.

Choose aC∞functionχ:R→Rsuch that suppχ⊂[−3,3] andχ≡1 on [−2,2]. Then

χ∈ A_β and χ∈ R_β. In particular χf ∈ R_β, and by Lemma A.3χf ∈_Aβ.

Moreover χf = f on suppf1 and hence is bounded away from zero on suppf1. By

Lemma A.1, there existsg0 ∈Aβ such thatf1 =g0(χf) = (g0χ)f.

Sinceg0, χ∈Aβ, we deduce thatg=g0χ∈Aβ. By Lemma A.3,g∈ Rβ. Hencef1=gf

withg∈ R_β as required.

Acknowledgements The research of IM was supported in part by EPSRC Grant EP/F031807/1 (held at the University of Surrey) and by the European Advanced Grant StochExtHomog (ERC AdG 320977). The research of DT was supported in part by the European Advanced Grant MALADY (ERC AdG 246953). IM and DT are grateful to the

Centre International de Rencontres Math´ematiques for funding the Research in Pairs topic “Infinite Ergodic Theory”, Luminy, August 2012, where part of this research was carried out.