Algebraic properties of convolution

Show that if $f, g \in L^{1}(\partial \mathbf{D})$ then

for those $t \in \mathbb {R}$ such that $(f * g)\left(e^{i t}\right)$ makes sense; here $(f * g)^{\sim }, \widetilde{f}$, and $\widetilde{g}$ denote the transfers to the real line as defined in 11.24.

Prove that $f_{1} * f_{2} = f_{2} * f_{1}$ and $\left(f_{1} * f_{2}\right) * f_{3} = f_{1} *\left(f_{2} * f_{3}\right)$ for all $f_{1}, f_{2}, f_{3} \in L_{1}(\mathbb {R})$.

Prove that if $f, g, h \in L_{1}(\mathbb {R})$, then their convolution has the following properties:

Prove that $\operatorname {supp}(f * g) \subseteq \operatorname {supp} f+\operatorname {supp} g$.

Let $p(x) = \sum_{k = 0}^{M} a_{k} x^{k}$ and $q(x) = \sum_{k = 0}^{N} b_{k} x^{k}$ be two polynomials. Their pointwise product is a polynomial of the form $p(x) q(x) = \sum_{n = 0}^{M+N} c_{n} x^{n}$. Find an explicit formula for $c_{n}$, and explain its relation to the convolution of the sequences $\left(a_{k}\right)_{k \in \mathbb {Z}}$ and $\left(b_{k}\right)_{k \in \mathbb {Z}}$, where we take $a_{k} = 0$ if $k \neq 0, \ldots , M$ and similarly for $b_{k}$.

Prove Lemma 13.11, and establish analogous properties for the convolution of sequences.

Let $\delta = \delta_{0} = \left(\delta_{0 n}\right)_{n \in \mathbb {Z}}$. Show that $\delta$ is an identity for convolution on $\ell^{p}(\mathbb {Z})$, i.e., $c * \delta = c$ for every $c \in \ell^{p}(\mathbb {Z})$, where $1 \leq p \leq \infty$.

Remark We will see in Corollary 13.15 that there is no element of $L^{1}(\mathbf{T})$ that is an identity for convolution on $L^{p}(\mathbf{T})$.

Suppose that the function $\varphi$ on $\mathbb {R}^{n}$ coincides inside the ball of radius $R$ with some polynomial and equals zero outside this ball, and that the function $\psi \in L^{1}\left({\mathbb {R}^{n},dx}\right)$ has support in the ball of radius $r<R$. Prove that the convolution $\varphi *\psi$ has support in the ball of radius $R+r$ and coincides with some polynomial in the ball of radius $R-r$. (All these balls have center at $0$.)

Let $f_1$ and $f_2$ be compactly supported continuous functions on the half-line $[0,\infty )$. Define $F_i=f_i(\sqrt{x^2+y^2})$. Prove that the convolution $F=F_1*F_2$ also has the form $F(x,y)=f(\sqrt{x^2+y^2})$, where $f$ is a compactly supported continuous function on $[0,\infty )$, and give an explicit expression for $f$ in terms of $f_1$ and $f_2$.

Let $f, g, h$ be integrable functions over $(\mathbb {R})$ and let

Prove that:

Put $\varphi (t) = 1-\cos t$ if $0 \leq t \leq 2 \pi , \varphi (t) = 0$ for all other real $t$. For $-\infty < x < \infty$, define

Verify the following statements about convolutions of these functions:

(i) $(f * g)(x) = 0$ for all $x$.

(ii) $(g * h)(x) = (\varphi * \varphi )(x) > 0$ on $(0,4 \pi )$.

(iii) Therefore $(f * g) * h = 0$, whereas $f *(g * h)$ is a positive constant.

But convolution is supposedly associative, by Fubini's theorem (Exercise 5(c)). What went wrong?

Let $f, g, h \in L^{1}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$.

If $f$ and $g$ are elements of $L^{1}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ that depend only on $\left|x\right|$, then $f * g$ only depends on $\left|x\right|$.