Convolution theorem for Fourier series

Where does the proof of 11.42 fail when $p = \infty$?

Suppose $g \in L^{1}(\partial \mathbf{D})$ is such that $\widehat{g}(n) \neq 0$ for infinitely many $n \in \mathbf{Z}$. Prove that if $f \in L^{1}(\partial \mathbf{D})$, then $f * g \neq g$.

Prove that if $f, g \in L^{2}(\partial \mathbf{D})$, then

for every $n \in \mathbf{Z}$.

Let $S^1$ denote the unit circle (the set of complex numbers with modulus one, or the real numbers modulo $2\pi$). The convolution of two functions on $S^1$ is

Suppose that $f$ is an element of $L^2(S^1)$ with the property that its Fourier coefficient

is non-zero for all $n \in \mathbb {Z}$. Show that the linear space $\{ f * k \mid k \in L^2(S^1)\}$ is dense in $L^2(S^1)$.

Is the Fourier series of a product the same as the formal product of the Fourier series?

Define the function $e_k$ on $\mathrm{T}^n$ by the formula $e_k(t)=e^{2\pi ikt}$ (here $k\in \mathbb {Z}^{n}$ is a multi-index and $kt=k_1t_1+\cdots +k_nt_n$). Prove the identity

A trigonometric polynomial on $\mathrm{T}^n$ is defined to be a linear combination of the functions $e_k$ defined in Problem 517. Prove that the collection of trigonometric polynomials forms an ideal in the algebra $L^{1}\left({\mathrm{T}^n,dt}\right)$.

Let $f$ be a continuous function with period $2 \pi$. Express the Fourier coefficients $A_{n}, B_{n}$ of the convolution

in terms of the Fourier coefficients $a_{n}, b_{n}$ of the function $f$.

Find the expression for the Fourier coefficients $A_{n}, B_{n}$ of the Steklov function

in terms of the Fourier coefficients $a_{n}, b_{n}$ of a $2 \pi$-periodic function $f$, absolutely integrable over the period.

Let the functions $f$ and $g$ have integrable squares on the segment $[-\pi ; \pi ]$. Prove that the Fourier series of the product $f g$ of the functions $f$ and $g$ can be obtained by term-by-term multiplication of the Fourier series of these functions.

Show that $L^{1}(T)$ is a commutative Banach algebra (without unit) if multiplication is defined by

Find all complex homomorphisms of $L^{1}(T)$, as in Theorem 9.23. If $E$ is a set of integers and if $I_{E}$ is the set of all $f \in L^{1}(T)$ such that $\hat{f}(n) = 0$ for all $n \in E$, prove that $I_{E}$ is a closed ideal in $L^{1}(T)$, and prove that every closed ideal in $L^{1}(T)$ is obtained in this manner.