Wiener algebra of absolutely convergent Fourier series

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Problem 13.26 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 13 "Fourier Series"

Prove the following facts about the Fourier algebra $A(\mathbf{T})$ .

(a)

$A(\mathbf{T})$ is a Banach space with respect to the norm $\left\| f\right\|_{A} = \left\| \widehat{f}\right\|_{\ell^{1}}$ , and $\mathcal{F}: f \mapsto \widehat{f}$ is an isometric isomorphism of $A(\mathbf{T})$ onto $L^{1}(\mathbf{T})$ .

(b)

$A(\mathbf{T})$ is closed under pointwise products, and $\left\| f g\right\|_{A} \leq \left\| f\right\|_{A}\left\| g\right\|_{A}$ (hence $A(\mathbf{T})$ is a Banach algebra with respect to pointwise products).

(c)

$C^{2}(\mathbf{T}) \subseteq A(\mathbf{T})$ , and therefore $A(\mathbf{T})$ is dense in $C(\mathbf{T})$ .

(d)

$L^{2}(\mathbf{T}) * L^{2}(\mathbf{T}) = A(\mathbf{T})$ .

(e)

$A(\mathbf{T})$ is a meager subset of $C(\mathbf{T})$ .

Problem 14.5 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 14 "Basis Properties of Fourier Series"

Prove Theorem 14.6.

Remark.

Theorem 14.6: The Fourier algebra $A(\mathbf{T})$ admits conjugation, and if $f \in A(\mathbf{T})$ , then the twisted partial sums $S_{N}^{\mathrm{t}} f$ converge to $H f$ in $L^{p}$ -norm for each $1 \leq p \leq \infty$ : for all $f \in A(\mathbf{T})$ and $1 \leq p \leq \infty$ , $\lim_{N \rightarrow \infty }\left\| H f-S_{N}^{\mathrm{t}} f\right\|_{L^{p}} = 0$ .

Problem 3.11.2 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.11 "Commutative Banach Algebras"

State and prove an analogue of Wiener's lemma 11.6 for power series that converge absolutely on the closed unit disc.

Problem 3.11.17 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.11 "Commutative Banach Algebras"

Suppose $\left\{ \omega_{n}\right\} (-\infty < n < \infty )$ are positive numbers such that $\omega_{0} = 1$ and

\omega _{m+n} \leq \omega _{m} \omega _{n}

for all integers $m$ and $n$ . Let $A = A\left\{ \omega_{n}\right\}$ be the set of all complex functions $f$ on the integers for which the norm

\left\| f\right\| = \sum _{-\infty }^{\infty }\left|f(n)\right| \omega _{n}

is finite. Define multiplication in $A$ by

(f * g)(n) = \sum _{k = -\infty }^{\infty } f(n-k) g(k).

(a)

Show that each $A\left\{ \omega_{n}\right\}$ is a commutative Banach algebra.

(b)

Show that $R_{+} = \lim_{n \rightarrow \infty }\left(\omega_{n}\right)^{1 / n}$ exists and is finite, by showing that $R_{+} = \inf_{n \geq 0}\left(\omega_{n}\right)^{1 / n}$ .

(c)

Show similarly that $R_{-} = \lim_{n \rightarrow \infty }\left(\omega_{-n}\right)^{-1 / n}$ exists and that $R_{-} \leq R_{+}$ .

(d)

Put $\Delta = \left\{ \lambda \in \mathcal{C}: R_{-} \leq \left|\lambda \right| \leq R_{+}\right\}$ . Show that $\Delta$ can be identified with the maximal ideal space of $A\left\{ \omega_{n}\right\}$ and that the Gelfand transforms are absolutely convergent Laurent series on $\Delta$ .

(e)

Consider the following choices for $\left\{ \omega_{n}\right\}$ :

$\omega_{n} = 1$ .
$\omega_{n} = 2^{n}$ .
$\omega_{n} = 2^{n}$ if $n \geq 0, \omega_{n} = 1$ if $n < 0$ .
$\omega_{n} = 1+2 n^{2}$ .
$\omega_{n} = 1+2 n^{2}$ if $n \geq 0, \omega_{n} = 1$ if $n < 0$ . For which of these is $\Delta$ a circle? For which choices is $A\left\{ \omega_{n}\right\}$ self-adjoint, in the sense that $\hat{A}$ is closed under complex conjugation?

(f)

Is $A\left\{ \omega_{n}\right\}$ always semisimple?

(g)

Is there an $A\left\{ \omega_{n}\right\}$ , with $\Delta$ the unit circle, such that $\hat{A}$ consists entirely of infinitely differentiable functions?

References

[Hei11] Heil C. A Basis Theory Primer. Birkhäuser, 2011.

[Rud91] Rudin W. Functional Analysis. 2nd ed., McGraw-Hill, 1991.