Wiener's lemma

Examensarbete

Wiener's lemma

Författare: Henrik Fredriksson Handledare: Joachim Toft Examinator: Börje Nilsson Datum: 2013-06-20 Kurskod: 4MA11E Ämne: Matematik Nivå: Magister

Wiener’s lemma

Henrik Fredriksson

June 20, 2013

Abstract

In this thesis we study Wiener’s lemma. The classical version of the lemma, whose realm is a Banach algebra, asserts that the pointwise inverse of a nonzero function with absolutely convergent Fourier expansion, also possesses an absolutely convergent Fourier expansion. The main purpose of this thesis is to investigate the validity in algebras endowed with a quasi-norm or a p-norm.

As a warmup, we prove the classical version of Wiener’s lemma using elemen- tary analysis. Furthermore, we establish results in Banach algebras concerning spec- tral theory, maximal ideals and multiplicative linear functionals and present a proof Wiener’s lemma using Banach algebra techniques.

Let ν be a submultiplicative weight function satisfying the Gelfand-Raikov-Shilov condition. We show that if a nonzero function f has a ν-weighted absolutely con- vergent Fourier series in a p-normed algebra A. Then 1/f also has a ν-weighted absolutely convergent Fourier series in A.

Key words: Wiener’s lemma, Banach algebra, quasi-norm, p-norm, submulti- plicative weight function.

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Acknowledgments

I would like to express my deepest gratitude to my supervisor, Prof. Joachim Toft at Linnæus University for formulating and introducing me to the subject of this thesis.

His great guidance and fruitful remarks have been tremendously helpful throughout the work of thesis. I feel honored and thankful to Prof. Toft for once again supervising me.

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Contents

Abstract ii

Acknowledgments iii

1 Introduction 2

2 Newman’s proof 3

3 Banach Algebras 5

4 Generalizations and Variations 20

References 26

Chapter 1

Introduction

The classical formulation of Wiener’s lemma is the following; if a non-vanishing func- tion f has an absolutely convergent Fourier series, then 1/f has an absolutely con- vergent Fourier series. A direct approach and the most cumbersome method to verify that 1/f possesses this property is to compute or estimate its Fourier series and ex- amine if the series is absolutely convergent. With Wiener’s lemma one gets an easier condition.

The original proof by Wiener [15], needed for the ”Tauberian theorems”, make use of localization properties and partion-of-unit argument. The proof using Neumann series in Chapter 2 is due to Newman [11].

Using Banach algebra techniques, Gelfand [4] proved the lemma in a elegant way.

The details are given in Chapter 3.

A guaranteed condition for absolute convergence of a Fourier series, is that the coefficients belongs to `₁(Z^d). One way to obtain faster convergence for a partial sum, is to impose on the series with a weight function. Weakening the subadditive property of the norm in a Banach algebra A and instead require that kx+ykA ≤ C(kxkA+kykA) for some constant C ≥ 1 gives rise to a quasi-Banach space if A is complete by the induced metric. In Chapter 4 we investigate the validity of Wiener’s lemma in quasi- Banach spaces with assist of submultiplicative weight functions.

At first glance, one may ask why such a modest statement as Wiener’s lemma can be interesting to study. Wiener’s lemma is the heart of the abstract theory of commutative Banach algebras developed by Gelfand. Therefore, the way of proving Wiener’s lemma is remarkable in the sense that it mostly uses techniques from algebra despite the fact that Fourier series originates in analysis.

The concept of invertibility is of great importance of solving systems of linear equations or operator equations. Therefore one may say from an abstract point of view that Wiener’s lemma is about invertibility in Banach algebras.

Chapter 2

Newman’s proof

This chapter contains a proof of Wiener’s lemma using elementary analysis, given by Newman [11] in .

A periodic function of period one can be identified as a function defined on the torus T := {z ∈ C : |z| = 1}. Let A(T) denote the space of all continuous functions defined on T with absolutely convergent Fourier series. Define a norm on A(T) by

(2.1) kf kA(T) :=X

n∈Z

| bf (n)|, f ∈ A(T).

We leave the details of the space A(T) and its norm (2.1) until Chapter 3 and proceed with the proof.

Let f ∈ A(T), then f ∈ A(T). Since 1/f = f / |f |², it is sufficient to prove that 1/|f |² ∈ A(T). Hence, there is no loss of generality to show this for positive functions in A(T), since we can replace f with |f |². Furthermore, by normalization, we may also assume that 0 < f ≤ 1 on T. Now define g = 1 − f . If f (z) 6= 0 for all z ∈ T, then f is invertible in C(T) (the space of continuous functions on T) due to the convergence of the series

(2.2)

∞

X

n=0

g(z)ⁿ= 1

1 − g(z) = 1

f (z) ∈ C(T).

Our goal is to show that the Neumann series (2.2) also converges in A(T). Consider the Fourier series g =P

n∈Zbg(n)e^2πinz and choose N ∈ N so large such that for any given ε > 0 we haveP

|n|≥N|bg(n)| < ε. Introduce the function p(z) defined by

p = X

|n|≤N

bg(n)e^2πinz,

then for sufficiently large N , kg − pkA(T) < ε holds. Hence we have the approximation g(z) = p(z) + r(z) for some r with krk_A(T) < ε. By the binomial theorem, we get the

estimate

kgⁿkA(T) ≤

n

X

k=0

n k



kpkⁿ_A(T)· krk^n−k_A(T) ≤

n

X

k=0

n k



kpkⁿ_A(T)· ε^n−k

= (kpkA(T)+ ε)ⁿ≤ (kgkA(T)+ 2ε)ⁿ

= (k1 − f kA(T)+ 2ε)ⁿ ≤ (1 − δ + 2ε)ⁿ,

where δ = infz∈T|f (z)| > 0. Since (1−δ +2ε) < 1 the Neumann series (2.2) converges in A(T)-norm and we conclude that 1/f ∈ A(T).

Chapter 3

Banach Algebras

In this chapter we introduce the concept of Banach algebras. We are especially interested in the Banach algebra A(T^d). An element in A(T^d) is a function such that it can be represented by an absolutely convergent Fourier series on the d-dimensional torus T^d.

The section about spectral theory gives conditions about invertibility in a Banach algebra and shows a relationship between Banach algebras and holomorphic functions.

In particular, using Liouville’s theorem, we show that the spectrum of an element in a Banach algebra is a nonempty compact subset of the complex numbers. For this reason, the chapter is devoted to complex Banach algebras.

In Section 3.4 we identify all multiplicative linear functionals in A(T^d) and use this to prove Wiener’s lemma.

3.1 General Theory

A vector space A over the complex scalar field C is said to be a complex algebra if for every x, y, z ∈ A and α, β ∈ C there is a multiplication defined on A satisfying

(i) (xy)z = x(yz),

(ii) x(y + z) = xy + xz and (iii) αβ(xy) = (αx)(βy).

If there exists an element e ∈ A with the property ex = xe = x for every x ∈ A then e is called an identity and A is said to be an algebra with identity. An identity e ∈ A is unique if it exists. To see this, suppose there exists two identities, e and e⁰. Then e = ee⁰ = e⁰.

If xy = yx for all x, y ∈ A, then A is called a commutative algebra.

Let X be a linear subspace of A. If x, y ∈ X such that xy ∈ X, then one says that X is a subalgebra of A.

A normed algebra is an algebra A equipped with a norm, k·kA : A → [0, ∞) such that the norm is submultiplicative, that is, kxykA ≤ kxkA· kykA holds for all x, y ∈ A.

If A is a normed algebra with identity, then it follows that kekA≥ 1. However, there always exists a norm k·k₀, equivalent to k·k_A, satisfying kek₀ = 1 (see Kaniuth [9], Katznelson [10]). We may therefore without loss of generality assume that kekA = 1.

If (A, k·kA) is a Banach space then A is called a Banach algebra.

Let (x_n) and (y_n) be Cauchy sequences in a Banach algebra A with limits x and y respectively. Then the inequality

kx_ny_n− xykA = k(x_n− x)y − x_n(y − y_n)x_nkA

≤ kxnkAkyn− ykA+ kynkAkxn− xkA

shows that multiplication is a continuous map from A × A to A.

Example 3.1.1. Let X be a compact subset of the complex numbers. Let A(X) denote the space of all continuous functions f : X → C which are holomorphic on the interior of X. The space A(X), with pointwise defined operations, i.e. if addition and multiplication are defined by

• (f + g)(x) = f (x) + g(x),

• (f g)(x) = f (x)g(x) and

• (λf )(x) = λf (x)

for all f, g ∈ A, λ ∈ C is an algebra with the constant function e = 1 serving as identity. With the supremum norm

kf kL∞(X) = sup

x∈X

|f (x)| ,

the algebra A(X) becomes a Banach algebra. Indeed, the supremum norm is sub- multiplicative. Let f, g ∈ A(X). Then

kf gkL∞(X) = sup

x∈X

|f (x)g(x)| ≤ sup

x∈X

|f (x)| · sup

x∈X

|g(x)| = kf kL∞(X)· kgkL∞(X). Since convergence in the supremum norm is equivalent to uniform convergence, the limit of a sequence of holomorphic functions is also holomorphic. Thus, A(X) is complete.

Example 3.1.2. Let `1(Z) be the Banach space of all infinite sequences of complex numbers x = (xn) such that

kxk_`1(Z) =X

n∈Z

|x_n| < ∞.

Define multiplication by the convolution of the sequences x, y ∈ `₁(Z) by x · y = (x_k∗ y_k)(n) =X

k∈Z

x_ky_n−k.

Then `₁(Z) is Banach algebra with identity since

kx · yk_`1(Z) =X

n∈Z

X

k∈Z

x_ky_n−k     

≤ X

n,k∈Z

|x_k| |y_n−k| = kxk_{`1(Z)</sub>kyk<sub>`1(Z)}.

with identity given by the sequence (δ_n⁰), where δ⁰_n= 1 if n = 0 and 0 otherwise.

Example 3.1.3. Let A be a Banach space and define multiplication by xy = 0 for all x, y ∈ A, then A is a Banach algebra.

Example 3.1.3 illustrates a Banach algebra without an identity. The existence of identity is crucial for theory of spectra. But as we now will show, an algebra without idenitity can always be embedded in an algebra with identity.

Consider the set of all pairs (x, λ), x ∈ A, λ ∈ C. Let A_e = A × C, i.e all such pairs (x, λ). Define the linear space operations componentwise according to

(x, λ) + (y, µ) = (x + y, λ + µ), µ(x, λ) = (µx, µλ) and define multiplication by

(x, λ)(y, µ) = (xy + λy + µx, λµ).

for all x, y ∈ A, λ, µ ∈ C. Then A_e becomes an algebra with identity e = (0, 1). Note that Ae is commutative if and only if A is commutative. If A is a normed algebra, we define the norm on A_e by

k(x, λ)kAe = kxkA+ |λ| , x ∈ A, λ ∈ C.

By this procedure we say that we adjoin an identity to A and the Banach algebra Ae

is called the unitisation of A.

The space A(T^d). Let f be a Z^d-periodic function defined on R^d, that is f (x) = f (x + k) for all k ∈ Z^d. Such f can be identified as a function defined on the d- dimensional unit cube [0, 1]^d or a function of the d-dimensional torus T^d := R^d/Z^d. The n’th Fourier coefficient of f are given by

(3.1) f (n) =b

Z

[0,1]^d

f (x)e^−2πihn,xidx,

where hn, xi = Pd

i=1n_ix_i is the inner product on R^d. The function f can then be expressed by its Fourier series

(3.2) f = X

n∈Z^d

f (n)eb ^2πihn,xi.

If P

n∈Z^d| bf (n)| < ∞, the Fourier series converges absolutely and f is a uniformly continuous function. Let A(T^d) denote the space of functions on T^d which can be represented with an absolutely convergent Fourier series, i.e. f ∈ A(T^d) if and only if and ( bf (n)) ∈ `₁(Z^d).

If one defines multiplication pointwise and introduce the Wienernorm, defined by

(3.3) kf k_A(T^d₎:= X

n∈Z^d

| bf (n)|,

then A(T^d) is a Banach algebra. To verify this, let f = P

k∈Z^df (k)eb ^2πihk,xi∈ A(T^d) and g =P

m∈Z^dbg(m)e^2πihm,xi ∈ A(T^d). Then f g = X

k∈Z^d

f (k)eb ^2πihk,xi

! X

m∈Z^d

bg(m)e^2πihm,xi

!

= X

k,m∈Z^d

f (k)b bg(m)e^2πihk+m,xi

= X

n∈Z^d

X

k∈Z^d

f (k)b bg(n − k)e^2πihn,xi

= X

n∈Z^d

( bf ∗bg)(n)e^2πihn,xi.

Interchanging the summations is admissible since by assumption the Fourier series of f and g converges absolutely and we see that pointwise multiplication in A(T^d) leads to convolution of the Fourier coefficients.

The Wienernorm is submultiplicative due to

kf gk_A(T^d₎ = X

n∈Z^d

|( bf ∗bg)(n)| = X

n∈Z^d

X

k∈Z^d

f (k)b bg(n − k)     

≤ X

k,n∈Z^d

| bf (n)| |bg(n − k)| = kf k_A(T^d₎kgk_A(T^d₎.

The commutativity of convolution follows from (f ∗ g)(n) = X

k∈Z^d

f (k)b bg(n − k) collecting and summing terms for whichm=n−k



= X

m∈Z^d

f (m − n)b bg(m) = (g ∗ f )(n).

This shows that the binomial theorem is applicable in the proof by Newman as de- scribed in Chapter 2.

The identity on A(T^d) is the constant function one. Hence A(T^d) is a commutative Banach algebra with identity under pointwise multiplication.

Definition 3.1.5. Let A and B be algebras. An homomorphism is a map ϕ : A → B such that for all x, y ∈ A

(i) ϕ(x + y) = ϕ(x) + ϕ(y) and (ii) ϕ(xy) = ϕ(x)ϕ(y).

If ϕ is bijective, ϕ is called an isomorphism. Put differently, a homomorphism (or iso- morphism) preserves the properties of the binary operations of an algebraic structure into another.

Suppose A and B are normed algebras. If ϕ is norm-preserving in the sense that if kf kA = kϕ(f )kB holds. Then we say that ϕ is isometric.

The algebras A(T^d) and `<sub>1</sub>(Z<sup>d</sup>) are isometrically isomorphic. Consider the map ϕ : A(T<sup>d</sup>) → `₁(Z), defined by ϕ(f ) = ( bf (n)). The injectivity follows from the uniqueness of Fourier series.

To show surjectivity, let (c_n) ∈ `₁(Z^d) be arbitrary and define f = X

k∈Z^d

c_ke^2πihk,xi, f ∈ C(T^d).

For each n ∈ Z^d, we have

f (n) =b Z

[0,1]^d

X

k∈Z^d

cke^2πihk,xi

!

e^−2πihn,xidx

= X

k∈Z^d

c_k Z

[0,1]^d

e^{2πihk−n,xi}dx = c_n.

Furthermore,

kf k_A(T^d₎= X

n∈Z^d

| bf (n)| = kϕ(f )k_`1(Z^d₎. Thus ϕ is an isometric isomorphism.

3.2 Spectral Theory

In this section we introduce the spectrum of an element x in a Banach algebra A and establish results concerning invertibility of elements in Banach algebras.

Definition 3.2.1. Let A be a complex algebra with identity. An element x ∈ A is said to be invertible if there exists an element y ∈ A such that xy = yx = e. If such y exists, then it is unique and we denote it by x⁻¹. Let G(A) denote the set of all invertible elements of A. The subgroup G(A) is called the general linear group of A.

Definition 3.2.2. The spectrum of x ∈ A is the set

σ_A(x) = {λ ∈ C : λe − x /∈ G(A)}

and the resolvent of x is the set ρA(x) = C\σA(x).

Definition 3.2.3. The spectral radius of x in a normed algebra A, is the number rA(x) = sup {|λ| : λ ∈ σA(x)} .

Proposition 3.2.4 (Spectral radius formula). Let A be a normed algebra. Then for all x ∈ A,

rA(x) = lim

n→∞kxⁿk^1/n_A . Proof. See e.g. Rudin [14].

Proposition 3.2.5. Let A be a Banach algebra with identity. If x ∈ A such that rA(x) < 1 then (e − x) is invertible and

(e − x)⁻¹ = e +

∞

X

n=1

xⁿ.

Furthermore, if kxkA < 1 we have the estimate k(e − x)⁻¹kA ≤ 1

1 − kxkA

.

Proof. By Proposition 3.2.4 for any fixed ξ such that rA(x) < ξ < 1 there exists N ∈ N such that kxⁿk^1/n_A ≤ ξ when n ≥ N . Hence kxⁿkA ≤ ξⁿ when n ≥ N . Clearly P∞

n=1kxⁿkA converges since ξ < 1. The partial sums y_m = e +Pm

n=1xⁿ converges with respect to the norm in A since it is complete with limit y = e +P∞

n=1xⁿ. This gives

(e − x)y_m = y_m(e − x) = e +

m

X

n=1

xⁿ

!

(e − x)

= e − x +

m

X

n=1

xⁿ−

m

X

n=1

xⁿ⁺¹

= e − x +

m

X

n=1

(xⁿ− xⁿ⁺¹) = e − x^m+1.

Now y_m → y and x^m+1 → 0 as n → ∞. Hence, we conclude that (e−x)y = y(e−x) = e.

The estimate follows from k(e − x)⁻¹k_A ≤ 1 +

∞

X

n=1

kxⁿk_A ≤ 1 +

∞

X

n=0

kxkⁿ_A− kxk⁰_A = 1 1 − kxkA

.

Proposition 3.2.6. Let A be a Banach algebra with identity, then G(A) is open.

Proof. Let x ∈ G(A) be arbitrary and let y ∈ A be such that ky − xkA < kx⁻¹k⁻¹_A . We want to show that the open ball with radius kx⁻¹k⁻¹_A with center in x is contained in G(A). Choose x ∈ A such that ke − xkA < 1. It follows that rA(e − x) < 1 and x = e − (e − x) is invertible. Then

ke − x⁻¹ykA = kx⁻¹(x − y)kA ≤ kx⁻¹kA· kx − ykA< 1

shows that x⁻¹y ∈ G(A) since x⁻¹y = e − (e − x⁻¹y). Hence, y ∈ G(A) since G(A) is a group. Thus, G(A) is open in A.

Theorem 3.2.7. Let A be a Banach algebra with identity. Then σA(x) 6= ∅ for all x ∈ A.

Proof. We prove the theorem by contradiction. Suppose σA(x) = ∅, then ρA(x) = C.

Let x ∈ A be fixed and define f (λ) = (λe − x)⁻¹. Using the identity x⁻¹ − y⁻¹ = x⁻¹(y − x)y⁻¹ we get

f (λ + h) − f (λ)

h = ((λ + h)e − x)⁻¹− (λe − x)⁻¹ h

= ((λ + h)e − x)⁻¹(λe − x − (λ + h)e + x)(λe − x)⁻¹ h

= ((λ + h)e − x)⁻¹(he)(λe − x)⁻¹ h

= ((λ + h)e − x)⁻¹(λe − x)⁻¹ → (λe − x)⁻², h → 0.

Thus, f is holomorphic in ρA(x).

We estimate f for large values of λ. If |λ| > kxk_A, by Proposition 3.2.5, kf (λ)k_A = k(λe − x)⁻¹k_A ≤ 1

|λ| (1 − kxkA/ |λ|) = 1

|λ| − kxkA

→ 0, |λ| → ∞.

The A-valued function f is a bounded entire function on C, tending to zero as |λ| →

∞. By Liouville’s theorem f must be a constant and f (λ) = (0e − x)⁻¹, giving that x⁻¹= 0. This can not be true, because then we would have 0 = x⁻¹x = e.

Since G(A) is open and C\σA(x) is the inverse image of the continuous mapping λ 7→ λe − x, it follows that if |λ| > r_A(x) then r(x/λ) < 1 and λe − x = λ(e − x/λ) is invertible. Hence, σA(x) is contained in the disk {λ ∈ C : |λ| ≤ rA(x)}.

Theorem 3.2.8 (Gelfand-Mazur). Let A be a Banach algebra with identity such that A is also a field. Then A is isomorphic to to the field of complex numbers. If kekA = 1 then A is isometrically isomorphic to C.

Proof. Suppose that x 6= λe for all λ ∈ C, then λe − x 6= 0 is invertible for all λ. This implies that σ(x)A = ∅ which is impossible.

If kekA= 1 then kxkA= kλekA = |λ| kekA = |λ|.

3.3 Ideals and Quotient spaces

In this section we study ideals in rings and quotient rings. We bear in mind that an algebra is also a ring. Thus all results given in this section also holds for algebras.

Definition 3.3.1. A right ideal of a ring R is a set I ⊆ R such that xy ∈ I whenever x ∈ R, y ∈ I. An left ideal is defined in an analogous way. One merely says that I is an ideal if it is both an right and left ideal.

An ideal in a commutative ring is obviously both a left and a right ideal.

Every ring has two trivial ideals, namely I = R and I = {0}. Any other ideals of R are called proper ideals. A maximal ideal of R is a nontrivial ideal I such that there is no proper ideal containing I.

Example 3.3.2. Let X be a subset of the complex numbers. Let Cb(X) denote the space of all bounded continuous functions f : X → C and let C_c(X) denote the space of all continuous functions f : X → C with compact support. Then C_c(X) is an ideal of C_b(X) since for all f ∈ C_b(X), g ∈ C_c(X), the product f g is continuous and clearly has compact support .

Let R[x] be a polynomial ring in indeterminate x with coefficients in a ring R. The polynomial ring Z[x] is a subset of Q[x], but not an ideal since for f ∈ Z[x], g ∈ Q[x]

the product f g not necessarily is a polynomial with integer coefficients.

Proposition 3.3.3. Let R be a ring with identity e and let I be an ideal in R. If I contains an invertible element then I = R.

Proof. Suppose an element x ∈ I is invertible in R. Take x⁻¹∈ R, then e = xx⁻¹ ∈ I. But this implies that ey = y ∈ I for all y ∈ R. Hence, I = R.

Proposition 3.3.4. Every proper ideal I in a ring with identity is contained in a maximal ideal.

Before proving Proposition 3.3.4, we introduce some results from set theory.

A partially ordered set (P, ) is said to be linearly ordered or totally ordered if a  b or b  a for all a, b ∈ P.

Zorn’s lemma. If every linearly ordered subset of a nonempty partially ordered set P has a upper bound contained in P, then P contains at least one maximal element.

Proof of Proposition 3.3.4. Suppose R is a ring with identity e. Furthermore, let J be the set of all proper ideals in R. The family J is a partially ordered set by the inclusion relation and, by Zorn’s lemma, contains a maximal linearly ordered subfamily J₀. Since e /∈ I and J is the union of all such I, it follows that e /∈ J . Hence J is a proper ideal containing I. That J is maximal follows from the maximality of J₀.

Corollary. An element x in a commutative ring R with identity is invertible in R if and only if x is not contained in any maximal ideal.

Proof. Assume x ∈ R is invertible and suppose that x is contained in an ideal I.

Then xx⁻¹ = e ∈ I and by Proposition 3.3.3 the ideal I can not be a maximal since I = R.

Conversely, assume that x do not belong to any maximal ideal. Form the nonempty set

xR = {xy : y ∈ R} .

This is clearly an ideal in R. By Proposition 3.3.4 the set xR must be contained in a maximal ideal in R. This is a contradiction to the assumption that x is not contained in a maximal ideal. Consequently xR = R. Using that xR is commutative and e ∈ xR implies the existence of an element ξ ∈ xA such xξ = ξx = e. Hence x⁻¹= ξ.

Definition 3.3.5. Let I be an ideal in a ring R. Denote the quotient space consti- tuted by the congruence relation x ≡ y (mod I) by R/I. That is x ∼ y if x − y ∈ I.

This equivalence relation decomposes R into cosets of the form x + I = {y : x − y ∈ I} .

The set x + I is called a class modulo I containing x. One says that the element x is a representative of the class x + I and is independent of the representative of the class.

The quotient space R/I becomes a ring if addition and multiplication on these classes are defined by

• (x + I) + (y + I) = (x + y) + I,

• α(x + I) = αx + I and

• (x + I)(y + I) = xy + I.

and R/I is called the quotient ring of R modulo I.

Definition 3.3.6. An ideal I in a ring R is called modular if there exists an element y ∈ R such that

xy ≡ yx ≡ x (mod I).

If such an element exists, then it it called an identity modulo I. The ideal I is said to be a maximal modular ideal if it is both modular and maximal.

Obviously, if R contains an identity e, then every ideal in R is modular since e is an identity modulo I.

If an ideal I contains a modular ideal then I is itself modular. As a consequence I is a maximal modular ideal if and only if it is maximal with respect to all proper modular ideal.

As mentioned in the beginning of this section, all previous results are still valid if one replace statements about rings with algebras. We now return working exclusively with algebras.

Proposition 3.3.7. Let A be a Banach algebra. The closure I of an ideal I in A is an ideal and every maximal ideal is closed.

Proof. By the completeness of A, there exists a Cauchy sequence (x_n) ∈ I with limit x ∈ I . Let y ∈ A be arbitrary. Then xy ∈ I and the inequality

kxy − x_nykA = k(x − x_n)ykA ≤ kx − x_nkAkykA

shows that x_ny → xy ∈ I as x_n → x. Hence I is an ideal.

Suppose that I is maximal. As we just saw I is an ideal. Clearly I ⊆ I, but the assumption that I is maximal implies I ⊆ I and consequently I = I.

Proposition 3.3.8. Let A be a normed algebra and I a proper ideal in A. Then A/I is a normed algebra with the quotient norm

kx + IkA/I := inf

y∈x+IkykA

If A is a Banach algebra, so is A/I.

Proof. Since 0 ∈ 0 + I we get k0 + IkA/I = infy∈0+IkykA = 0.

Conversely, suppose that k0 + IkA/I = 0. Then 0 + I contains a sequence with limit 0 in A. This limit belongs to 0 + I because it is closed by Proposition 3.3.7.

Hence k0 + IkA/I = 0.

Let x + I, y + I ∈ A/I. By definition, for any given ε > 0 there exists ξ ∈ x + I, η ∈ y + I such that

kξkA ≤ kx + IkA/I+ε

2 and kηkA ≤ ky + IkA/I+ ε 2.

By the triangle inequality of k·kA,

(3.4) kξ + ηk_A ≤ kξk_A+ kηk_A ≤ kx + Ik_A/I+ ky + Ik_A/I+ ε.

But ξ + η ∈ (x + y) + I, so

(3.5) k(x + y) + IkA/I ≤ kξ + ηkA

Combing (3.4) and (3.5) yields

k(x + y) + Ik_A/I ≤ kx + Ik_A/I+ ky + Ik_A/I+ ε.

This verifies the triangle inequality since ε > 0 is arbitrary.

Now we show that the quotient norm is submultiplicative if A is a normed algebra.

Indeed, using that k·kA is submultiplicative yields k(x + I)(y + I)k_A/I = kxy + Ik_A/I

= inf

z∈Ikxy + zkA

≤ inf

a,b∈Ik(x + a)(y + b)kA

≤ inf

a,b∈Ikx + akA· ky + bkA

= kx + Ik_A/I · ky + Ik_A/I.

It remains to show completeness of A/I if A is a Banach algebra. Let (x_n+ I) be a Cauchy sequence in A/I. Then there exists a subsequence (x_nk+ I) such that

kx_nk+ I − x_nk+1+ Ik_A/I = kx_nk − x_nk+1 + Ik_A/I < 2^−k.

for all k ∈ N. Furthermore, there exists elements of a sequence (y_k) such that y_k− x_nk ∈ I and ky_k− x_nkkA < 2^−k for all k ∈ N. If (y_k) is a Cauchy sequence in A with limit y ∈ A, then it follows that lim_k→∞x_nk + I = y + I ∈ A/I. This implies that the full Cauchy sequence also has limit y + I. Hence A/I is complete.

Proposition 3.3.9. Let I be a maximal ideal in an algebra A such that kekA = 1.

The element e + I satisfies ke + IkA/I = 1

Proof. Choose an element x ∈ A such that r_A(x) < 1. Then (e − x) is invertible by Proposition 3.2.5. But a maximal ideal can not contain any invertible elements, so we must have that ke + IkA/I ≥ 1.

But on the other hand e ∈ e + I. By definition of the quotient norm we have ke + IkA/I ≤ kekA= 1.

This gives that ke + IkA/I = 1.

The next aim in our journey to prove Wiener’s lemma is to determine under which conditions the Gelfand-Mazur theorem is applicable to the quotient algebra A/I. Our goal is to determine under which conditions A/I is a field.

The two following propositions with tell us the requirements for this.

To prove Proposition 3.3.11 we apply the Correspondence theorem for rings, as- serting that there is a one-to-to correspondence between the sets of all ideals in A that contains I and the set of all ideals in A/I (see Fraleigh [3]).

Proposition 3.3.10. Let A be a Banach algebra with identity e. Then A is a field if and only if there exists no nontrivial proper ideals in A.

Proof. Suppose A is a field. Consequently, an ideal I 6= {0} must contain an invertible element. By Proposition 3.3.3 we must have I = A. Hence A only has trivial ideals.

Conversely, suppose that A only has trivial ideals, then A itself is an ideal. By definition, every element in an ideal is on the form ra for all r, a ∈ A. Hence, we can find nonzero elements x, y ∈ A such that xy = e, but this implies that x invertible with inverse y. Thus, y = x⁻¹ and every nonzero element is invertible. This shows that A is a field.

Proposition 3.3.11. Let I be a maximal modular ideal in Banach algebra A. Then A/I is a field.

Proof. Take a noninvertible element x + I ∈ A/I. Then the set J = {xy + I : y + I ∈ A/I}

is an ideal in A/I. By construction J is nontrivial and nonempty since it does not contain e + I and J 6= {0} since x + I ∈ J .

Since the the inverse image of an ideal by a homomorphism is an ideal, there exists a nontrivial ideal containing I as a proper subset. Since we supposed that I was maximal we have a contradiction. Hence A/I is a field.

Proposition 3.3.12. Let A be Banach algebra with identity and let I be a maximal ideal of A. Then (A/I, k·kA/I) is isometrically isomorphic to the complex numbers, i.e the exists a bijection ϕ : A/I → C such that x = λ(e + I) where λ ∈ C.

Proof. Under the given conditions, Proposition 3.3.9, Proposition 3.3.10 and Propo- sition 3.3.11 gives that A/I is a field and ke + IkA/I = 1. The proof now follows from Gelfand-Mazur’s theorem (Theorem 3.2.8).

3.4 Multiplicative linear functionals

A linear functional ϕ : A → C defined on a Banach algebra A is said to be multi- plicative if ϕ(xy) = ϕ(x)ϕ(y) for all x, y ∈ A.

Let ∆(A) denote the set of all all nonzero linear multiplicative functionals defined on A. The set ∆(A) is also called the structure space or the maximal ideal space of A. The expression maximal ideal space is justified by Proposition 3.4.3.

Proposition 3.4.1 (Gleason-Kahane-Zelasko, [9]). Let ϕ be a linear functional de- fined on a Banach algebra A with identity. The following statements are equivalent:

(i) ϕ is nonzero and multiplicative.

(ii) ϕ(e) = 1 and ϕ(x) 6= 0 for all invertible x ∈ A.

(iii) ϕ(x) ∈ σA(x) for every x ∈ A.

Corollary. Let A be a Banach algebra with identity. If x ∈ G(A) and ϕ ∈ ∆(A).

Then ϕ(x⁻¹) = ϕ(x)⁻¹.

Proof. Since ϕ is multiplicative and ϕ(e) = 1 one has 1 = ϕ(e) = ϕ(xx⁻¹) = ϕ(x)ϕ(x⁻¹).

On the other hand

ϕ(x)ϕ(x)⁻¹ = 1.

Whence, ϕ(x⁻¹) = ϕ(x)⁻¹.

Definition 3.4.2. By the kernel of ϕ ∈ ∆(A) in Banach algebra A, we mean the set ker (ϕ) = {x ∈ A : ϕ(x) = 0} .

The following proposition gives the relation between the kernels of multiplicative linear functionals and ideals in a Banach algebra.

Proposition 3.4.3. The kernel of linear multiplicative functional ϕ ∈ ∆(A) is an ideal. Conversely, every maximal ideal is the kernel for some ψ ∈ ∆(A).

Proof. That ker (ϕ) is an ideal is obvious since for x ∈ ker (ϕ), y ∈ A one has ϕ(xy) = ϕ(x)ϕ(y) = 0y = 0,

giving that xy ∈ ker (ϕ).

Let I be maximal ideal, then A/I is a field and consequently a Banach algebra.

By Gelfand-Mazur’s theorem, A/I is isomorphic to the complex numbers. Let ψ denote this isomorphism and let ϕ denote the canonical homomorphism from A to A/I. Define φ = ψ ◦ ϕ. Then φ ∈ ∆(A) with kernel I since it is the kernel of ϕ.

Corollary. A complex number λ belongs to σ^A(x) if and only if ϕ(x) = λ for some ϕ ∈ ∆(A).

Proof. Suppose λ ∈ σA(x), then λe − x is not invertible in A. Thus, for arbitrary y ∈ A, the set

I = {(λe − x)y : x, y ∈ A, λ ∈ σA(x)}

is a proper ideal in A, contained in a maximal ideal. By Proposition 3.4.3 there exists ϕ ∈ ∆(A) such that ϕ(λe − x) = 0. Whence, ϕ(x) = λ since ϕ(e) = 1.

Conversely, if λ ∈ ρA(x) then (λe − x) is invertible and (λe − x)y = e for some y ∈ A. For all ϕ ∈ ∆(A) one has

ϕ(λe − x)ϕ(y) = 1, giving that ϕ(λe − x) 6= 0. Thus, ϕ(x) 6= λ.

Proposition 3.4.4. Let A be a Banach algebra. Every ϕ ∈ ∆(A) is bounded on A and |ϕ(x)| ≤ rA(x) for all x ∈ A. In particular kϕk ≤ 1 and kϕk = 1 if A possesses an identity.

Proof. Let x ∈ A and λ ∈ C such that |λ| > rA(x), then rA(x/λ) < 1 and λe − x = λ(e − x/λ) is invertible. By the Gleason-Kahane-Zelasko theorem, ϕ(x) ∈ σA(x), so we must have ϕ(x) 6= λ for all |λ| > rA(x). This gives that |ϕ(x)| ≤ rA(x) and

kϕk = sup

kxkA≤1

|ϕ(x)| ≤ sup

kxkA≤1

rA(x) ≤ 1,

since rA(x) ≤ kxk.

The assertion kϕk = 1 follows from ϕ(e) = 1.

The maximal ideal space ∆(A(T^d)). We identify the maximal ideal space ∆(A(T^d)).

Let ϕ be a multiplicative linear functional defined on A(T^d). For every f ∈ A(T^d) we have

ϕ(f ) = ϕ X

n∈Z^d

f (n)eb ^2πihn,xi

!

= X

n∈Z^d

f (n)ϕ eb ^2πihn,xi
.

For each n ∈ Z^d, consider the function x 7→ e^2πihn,xi ∈ A(T^d). By Proposition 3.4.4 and using that ke^2πihn,xik_A(Td) = 1 for all x ∈ T^d, we get

|ϕ(e^2πihn,xi)| ≤ kϕk · ke^2πihn,xikA(T^d) = 1, and

|ϕ(e^−2πihn,xi)| ≤ kϕk · ke^−2πihn,xik_A(T^d₎ = 1.

Since ϕ is multiplicative and ϕ(e) = 1 we obtain

1 = |ϕ(e^2πihn,xie^−2πihn,xi)| = |ϕ(e^2πihn,xi)| · |ϕ(e^−2πihn,xi)| ≤ 1, and this implies that |ϕ(e^2πihn,xi)| = 1. Hence, there exists a λ ∈ T^d such that (3.6) ϕ(e^2πihn,xi) = e^2πihn,λi.

This gives

ϕ(f ) = X

n∈Z^d

f (n)eb ^2πihn,λi = f (λ).

We see that each λ ∈ T^d gives rise to a functional of the form (3.7) ϕ_λ(f ) = f (λ), f ∈ A(T^d).

Consequently, every functional ϕ ∈ ∆(A(T^d)) is on the form (3.7). Since the kernel for ϕ is a maximal ideal (Proposition 3.4.3) we see that every maximal ideal of A(T^d) coincides with the zeros of f on T^d.

By using these results, we can now prove Wiener’s lemma.

Theorem 3.4.6 (Wiener’s lemma). If f ∈ A(T^d) and f (x) 6= 0 for all x ∈ T^d, then 1/f ∈ A(T^d).

Proof. The assumption f (x) 6= 0 for all x ∈ T^d, gives that f do not belong to any maximal ideal and is therefore invertible in A(T^d); that is, 1/f has an absolutely convergent Fourier series.

Chapter 4

Generalizations and Variations

As the title foretells, in this final chapter we give generalizations and variations of Wiener’s lemma. In particular, we show

(i) a weighted versions of Wiener’s lemma (Theorem 4.1.6), (ii) validity in p-normed algebras (Theorem 4.2.5) and

(iii) a weighted version in p-normed algebras (Theorem 4.2.6).

We mention in advance that Theorem 4.2.6 is a generalization of the other two the- orems and therefore omit the proofs using algebra techniques. However, we give a brief description of how to prove Theorem 4.1.6 using Neumann series.

4.1 Submultiplicative weight functions

To obtain faster convergence of Fourier series, one may manipulate the decay of the Fourier coefficients by a weight function. Following the papers by Gr¨ochenig [6, 7], we introduce submultiplicative weight functions and study a weighted version of Wiener’s lemma.

Definition 4.1.1. A submultiplicative weight function ν defined on R^d is a nonneg- ative function such that

(4.1) ν(x + y) ≤ ν(x)ν(y) for all x, y ∈ R^d. We assume that ν is continuous.

If ν(−x) = ν(x) for all x ∈ R^d then ν is said to be symmetric.

For a symmetric submultiplicative weight function one has ν(x) ≥ 1 for all x ∈ R^d. This assertion follows from

ν(x) = ν(x + y − y) ≤ ν(x)ν(y)².

A standard class of symmetric submultiplicative weight functions is (4.2) m_a,b,s,t(x) = e^a|x|b(1 + |x|)^s(log(e + |x|))^t.

where a, s, t ≥ 0 and 0 ≤ b < 1. This class contains

(i) the polynomial weight functions m_s(x) = (1 + |x|)^s, s ∈ R, (ii) the exponential weight functions m(x) = e^a|x|, a ∈ R,

(iii) and the subexponential weight functions m(x) = e^a|x|b for 0 ≤ b < 1 and a ∈ R.

Lemma 4.1.2 ([6]). Let ν be a symmetric submultiplicative weight function, then there exists a constant a ≥ 0 such that

ν(x) ≤ e^a|x|, x ∈ R^d.

Every submultiplicative weight function ν grows at most exponentially.

For each weight function ν on Z^dwe can associate a weighted `<sub>1</sub>(Z<sup>d</sup>)-space, denoted by `^ν₁(Z^d) with the norm

kxk_`^ν

1(Z)= X

n∈Z^d

|x_n| ν(n).

The space A^ν(T^d). Let ν be a submultiplicative weight function. We say that a function f , equal its Fourier series (3.2), has a weighted absolutely convergent Fourier series if

(4.3) kf k_A^ν_(T^d₎ := X

n∈Z^d

| bf (n)|ν(n) < ∞.

Let A^ν(T^d) denote the Banach algebra of all such functions with finite norm as defined in (4.3).

Proposition 4.1.4. If ν is a submultiplicative weight function, then A^ν(T^d) is a Banach algebra with respect to pointwise multiplication.

Proof. Using that ν(n) = ν(k + (n − k)) ≤ ν(k)ν(n − k), the proof is analogous to what we showed in the Banach algebra A(T^d). We give the details for the submulti- plicativity.

Let f, g ∈ A^ν(T^d) then

kf gk_A^ν_(T^d₎ = X

n∈Z^d

X

k∈Z^d

f (k)b bg(n − k)     

ν(n)

≤ X

n,k∈Z^d

| bf (n)||bg(n − k)|ν(k)ν(n − k)

= kf k_Aν(T^d)· kgk_Aν(T^d). Hence k·k_A^ν_(T^d₎ is submultiplicative.

Suppose f is a nonzero function in A^ν(T^d), are we then guaranteed that 1/f ∈ A^ν(T^d)? In other words, is Wiener’s lemma valid in A^ν(T^d) for all submultiplicative weight functions ν? Gelfand, Raikov, and Shilov [4] showed preserved validity of the lemma if ν satisfies

(4.4) lim

n→∞ν(nx)^1/n = 1 for all x ∈ R^d.

Definition 4.1.5. A submultiplicative weight function ν defined on R^d is said to satisfy the GRS (Gelfand-Raikov-Shilov) condition if (4.4) holds for all x ∈ R^d.

It is easy to see that a subexponential weight function e^a|x|b for a > 0 and 0 ≤ b < 1 satisfies the GRS-condition, but a exponential weight e^a|x|, a > 0 violates the same condition.

Theorem 4.1.6 (Weighted version of Wiener’s lemma). Let ν be a submultiplicative weight function satisfying the GRS-condition. If f ∈ A^ν(T^d) and f (x) 6= 0 for all x ∈ T^d, then 1/f ∈ A^ν(T^d).

We give the outlines of the proof, using the same arguments as in Newman’s proof in Chapter 2. We refer to [7] for the details).

To prove the theorem it is useful to set

˜

ν(n) = max

kkk∞≤nν(k)

where kkk∞= max_j|k_j|, k ∈ R^d is the maximum norm on R^d.

Lemma 4.1.7. The weight ˜ν is submultiplicative and increasing on N. If ν satisfies the GRS-condition, then ˜ν also satisfies the GRS-condition.

Proof of Theorem 4.1.6. The procedure is in the same spirit as Chapter 2, except we have the estimate

(4.5)

∞

X

n=0

kgⁿk_Aν(T^d)≤

∞

X

n=0

˜

ν(n)(1 − δ + 2ε)ⁿ.

Recall that g = 1 − f where f ∈ A^ν(T^d) is such that 0 < f ≤ 1 and δ = inf_x∈T^d|f (x)| > 0. The seriesP∞

n=0gⁿ converges to 1/f if lim sup

n→∞

(˜ν(n)(1 − δ + 2ε)ⁿ)^1/n = lim sup

n→∞

˜

ν(n)^1/n(1 − δ + 2ε)
< 1, which is the case by Lemma 4.1.7.

4.2 Quasi-norms and p-norms

Definition 4.2.1. A quasi-norm on a vector space A is a map k·kA : A → [0, ∞) such that for x, y ∈ A, α ∈ C one has

(i’) kxkA = 0 if and only if x = 0, (ii’) kαxk_A = |α| kxk_A and

(iii’) kx + ykA ≤ C(kxkA+ kyk_A) for some constant C ≥ 1.

Aoki [1] and Rolewicz [12]) showed that, if C = 2^1/p−1 where 0 < p ≤ 1, then there exists a map |||·||| such that

(i) |||x||| > 0 if x 6= 0, (ii) |||αx||| = |α|^p|||x||| and

(iii) |||x + y||| ≤ (|||x|||^p+ |||y|||^p)^1/p. for all x, y ∈ A, α ∈ C.

A mapping fulfilling the conditions (i),(ii) and (iii) above, is called a p-norm or a p-homogeneous norm and (A, |||·|||) is said to be p-normable algebra. If A is complete under the induced metric

d(x, y) = |||x − y|||^p, then A is called a quasi-Banach space.

Example 4.2.2. That kxk`p(N) = (P∞

n=1|x_n|^p)^1/pdefines a norm on `_p(N) and kf kL_p(0,1)= (R1

0 |f (x)|^p dx)^1/p defines a norm on L_p(0, 1) for 1 ≤ p < ∞ follows from Minkowski’s inequality. On the other hand the spaces `_p(N) and L_p(0, 1) are quasi-Banach spaces for 0 < p ≤ 1.

Proposition 4.2.3. Let x = (xn) such that P∞

n=1|x_n|^p < ∞ for 0 < p ≤ 1. Then kxk = (P∞

n=1|x_n|^p)^1/p is a quasi-norm on `_p(N) for 0 < p ≤ 1.

Proof. The conditions (i) and (ii) in Definition 4.2.1 are evident. To verify (iii), define f (t) = (1 + t)^p− 1 − t^p for t ≥ 0.

It follows that

f⁰(t) = p(1 + t)^p−1− pt^p−1< 0 for all t > 0.

Furthermore is f (0) = 0, and hence f (t) < 0 for all t > 0. Put t = a/b such that both a and b are nonzero, then

fa b



= 1 + a

b

p

− 1 − a^p

b^p = a + b b

p

− b^p b^p − a^p

b^p < 0

which can rewritten as (a + b)^p < a^p+ b^p, where non strict inequality occurs if a = 0 or b = 0. Replacing a with x = (x_n) ∈ `<sub>p</sub>(N), b with y = (y<sub>n</sub>) ∈ `_p(N) and summing over all n ∈ N yields

kx + yk =

∞

X

n=1

|x_n+ y_n|^p

!1/p

≤

∞

X

n=1

|x_n|^p+

∞

X

n=1

|y_n|^p

!1/p

≤ 2^1/p−1







∞

X

n=1

|x_n|^p

!1/p

+

∞

X

n=1

|y_n|^p

!1/p





= 2^1/p−1(kxk + kyk).

The completeness is proved in the same way as in the case 1 ≤ p < ∞.

Proposition 4.2.3 is also true if x, y ∈ L_p(0, 1) for 0 < p ≤ 1.

Results of Banach space theory relying on completeness is still valid for quasi- Banach spaces, e.g. the boundedness principle, open mapping theorem and closed graph theorem. On the other hand, Hahn-Banach’s theorem, depending on convexity is not valid for quasi-Banach spaces.

As Proposition 4.2.3 showed, `<sub>p</sub>(N) and L<sub>p</sub>(0, 1) are both quasi-Banach algebras but only `_p(N) has non-trivial dual space.

Zelazko [18] extended the results of spectral theory and ideals in Banach algebras to p-normable Banach algebras. We may therefore use the same arguments as in Chapter 3 when proving Wiener’s lemma in p-normable algebras.

The space A^ν_p(T^d). Let 0 < p ≤ 1. A function f equals its Fourier series (3.2) belongs to the p-normed algebra A^ν_p(T^d) if

kf k_A^ν_p(T^d₎:= X

n∈Z^d

| bf (n)ν(n)|^p

!1/p

< ∞.

We set A_p(T^d) = A^ν_p(T^d) when ν(n) := 1 for all n ∈ R^d.

Theorem 4.2.5 (Wiener’s lemma). If f ∈ Ap(T^d) and f (x) 6= 0 for all x ∈ T^d, then 1/f ∈ A_p(T^d).

We now combine the previous results in this chapter and present the main result of this thesis; a weighted version of Wiener’s lemma in p-normable algebras.

Theorem 4.2.6 (Wiener’s lemma). Let ν be a submultiplicative weight function sat- isfying the GRS-condition. Let f ∈ A^ν_p(T^d) for 0 < p ≤ 1. If f (x) 6= 0 for all x ∈ T^d, then 1/f ∈ A^ν_p(T^d).

Proof. Let ϕ ∈ ∆(A^ν_p(T^d)). For every f ∈ A^ν_p(T^d) we have

ϕ(f ) = ϕ X

n∈Z^d

f (n)eb ^2πihn,xi

!

= X

n∈Z^d

f (n)ϕ eb ^2πihn,xi
.

For each n ∈ Z^d, consider the function x 7→ e^2πihn,xi∈ A^ν_p(T^d). By using that kϕk = 1 and ke^2πihn,xik_A^ν

p(T^d) = ν(n) for all x ∈ T^d, we get (4.6) |ϕ(e^2πihn,xi)| ≤ kϕk · ke^2πihn,xik_Aν

p(T^d)= ν(n) and

(4.7) |ϕ(e^−2πihn,xi)| ≤ kϕk · ke^−2πihn,xik_A^ν_p(T^d₎ = ν(−n).

Since ϕ is multiplicative and ϕ(e) = 1 we obtain

(4.8) 1 = |ϕ(e^2πihn,xie^−2πihn,xi)| = |ϕ(e^2πihn,xi)| · |ϕ(e^−2πihn,xi)|.

From (4.6), (4.7) and (4.8) it follows that

ν(n) ≥ |ϕ(e^2πihn,xi)| = 1

|ϕ(e^−2πihn,xi)| ≥ 1 ν(−n). Write n = km, k > 0 ∈ Z, m ∈ Z^d. Then

1

ν(−km)^1/k ≤ |ϕ(e^2πikhm,xi)|^1/k ≤ ν(km)^1/k.

The weight ν fulfills the GRS-condition. By letting k tend to infinity, the latter estimate become

1 ≤ |ϕ(e^2πihm,xi)| ≤ 1.

It follows that |ϕ(e^2πihm,xi)| = 1 and there exists λ ∈ T^d such that ϕ(e^2πihm,xi) = e^2πihm,λi.

This gives

ϕ(f ) = X

n∈Z^d

f (n)eb ^2πihn,λi = f (λ).

We see that each λ ∈ T^d gives rise to a functional of the form (4.9) ϕλ(f ) = f (λ), f ∈ A^ν_p(T^d).

By the same arguments as in the proof of Theorem 3.4.6; the assumption f (x) 6= 0 for all x ∈ T^d gives that f do not belong to any maximal ideal and is therefore invertible in A^ν_p(T^d). Hence, 1/f has a ν-weighted absolutely convergent Fourier series in the p-normed algebra A^ν_p(T^d).

References

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Tokyo, vol. 18, 588–594, 1942.

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