Convolution algebras

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Problem 11B.13 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 11B "Fourier Series and $L^{p}$ of Unit Circle"

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 $f * g \neq g$ .

Problem VII.9.1 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VII.9 "The Group Algebra of a Locally Compact Abelian Group"

Prove that if $L^1(G)$ has an identity, then $G$ is discrete.

Problem 516 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.1.1 "Convolutions of Test Functions"

(a)(b)

Define an operation $*$ in the space of functions on the group $G$ by the equation $f^*(x)=\overline{f(-x)}$ . Prove that

(a)

$f_1^**f_2^*=(f_1*f_2)^*$ for $f_1,f_2\in L^{1}\left({G,\mu }\right)$ ;

(b)

$(f_1*f_2^*)(0)= \left\langle f_1, f_2 \right\rangle$ for $f_1,f_2\in L^{2}\left({G,\mu }\right)$ .

Problem 2.8.5 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.8 "Integration on Product Spaces"

Let $M$ be the Banach space of all complex Borel measures on $\mathbb {R}^{1}$ . The norm in $M$ is $\left\| \mu \right\| = \left|\mu \right|\left(\mathbb {R}^{1}\right)$ . Associate to each Borel set $E \subset \mathbb {R}^{1}$ the set

E_{2} = \left\{ (x, y): x+y \in E\right\} \subset \mathbb {R}^{2}

If $\mu$ and $\lambda \in M$ , define their convolution $\mu * \lambda$ to be the set function given by

(\mu * \lambda )(E) = (\mu \times \lambda )\left(E_{2}\right)

for every Borel set $E \subset \mathbb {R}^{1} ; \mu \times \lambda$ is as in Definition 8.7.

(a)

(b)

Prove that $\mu * \lambda$ is the unique $v \in M$ such that

\int f d v = \iint f(x+y) d \mu (x) d \lambda (y)

for every $f \in C_{0}\left(\mathbb {R}^{1}\right)$ . (All integrals extend over $\mathbb {R}^{1}$ ).

(c)

Prove that convolution in $M$ is commutative, associative, and distributive with respect to addition.

(d)

Prove the formula

(\mu * \lambda )(E) = \int \mu (E-t) d \lambda (t)

for every $\mu$ and $\lambda \in M$ and every Borel set $E$ . Here

E-t = \left\{ x-t: x \in E\right\}

(e)

Define $\mu$ to be discrete if $\mu$ is concentrated on a countable set; define $\mu$ to be continuous if $\mu (\left\{ x\right\} ) = 0$ for every point $x \in \mathbb {R}^{1}$ ; let $m$ be Lebesgue measure on $\mathbb {R}^{1}$ (note that $m \notin M$ ). Prove that $\mu * \lambda$ is discrete if both $\mu$ and $\lambda$ are discrete, that $\mu * \lambda$ is continuous if $\mu$ is continuous and $\lambda \in M$ , and that $\mu * \lambda \ll m$ if $\mu \ll m$ .

(f)

Assume $d \mu = f d m, \quad d \lambda = g d m, \quad f \in L^{1}\left(\mathbb {R}^{1}\right)$ , and $\quad g \in L^{1}\left(\mathbb {R}^{1}\right)$ , and prove that $d(\mu * \lambda ) = (f * g) d m$ .

(g)

Properties (a) and (c) show that the Banach space $M$ is what one calls a commutative Banach algebra. Show that $(e)$ and $(f)$ imply that the set of all discrete measures in $M$ is a subalgebra of $M$ , that the continuous measures form an ideal in $M$ , and that the absolutely continuous measures (relative to $m$ ) form an ideal in $M$ which is isomorphic (as an algebra) to $L^{1}\left(\mathbb {R}^{1}\right)$ .

(h)

Show that $M$ has a unit, i.e., show that there exists a $\delta \in M$ such that $\delta * \mu = \mu$ for all $\mu \in M$ .

(i)

Only two properties of $\mathbb {R}^{1}$ have been used in this discussion: $\mathbb {R}^{1}$ is a commutative group (under addition), and there exists a translation invariant Borel measure $m$ on $\mathbb {R}^{1}$ which is not identically 0 and which is finite on all compact subsets of $\mathbb {R}^{1}$ . Show that the same results hold if $\mathbb {R}^{1}$ is replaced by $\mathbb {R}^{k}$ or by $T$ (the unit circle) or by $T^{k}$ (the $k$ -dimensional torus, the cartesian product of $k$ copies of $T$ ), as soon as the definitions are properly formulated.

Problem 2.18.9 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.18 "Elementary Theory of Banach Algebras"

Prove that a closed linear subspace of the Banach algebra $L^{1}\left(\mathbb {R}^{1}\right)$ (see Sec. 9.19) is translation invariant if and only if it is an ideal.

Problem 2.18.10 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.18 "Elementary Theory of Banach Algebras"

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

(f * g)(t) = \frac{1}{2 \pi } \int _{-\pi }^{\pi } f(t-s) g(s) d s

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.

Problem 5.3.10 from Saxe K. "Beginning Functional Analysis" [Sax02]More problems in Section 5.3 "Banach Algebras and Spectra; Compact Operators"

Let $\ell^{1}(\mathbb {Z})$ denote the collection of all (doubly ended) sequences $\left\{ a_{i}\right\}_{i=-\infty }^{\infty }$ of complex numbers such that $\sum_{i=-\infty }^{\infty }\left|a_{i}\right|<\infty$ . This space is very much like $\ell^{1}=\ell^{1}(\mathbb {N})$ . With the norm of $a=\left\{ a_{i}\right\}_{i=-\infty }^{\infty }$ given by

\left\| a\right\| =\sum _{i=-\infty }^{\infty }\left|a_{i}\right|

$\ell^{1}(\mathbb {Z})$ becomes a Banach space. The point of this exercise is to see that we can define a multiplication to make $\ell^{1}(\mathbb {Z})$ into a Banach algebra. For $a=\left\{ a_{i}\right\}_{i=-\infty }^{\infty }$ and $b=\left\{ b_{i}\right\}_{i=-\infty }^{\infty }$ in $\ell^{1}(\mathbb {Z})$ , define the product $a * b$ by

(a * b)_{i}=\sum _{j=-\infty }^{\infty } a_{i-j} b_{j}

This multiplication is called convolution. Show that $\ell^{1}(\mathbb {Z})$ is a Banach algebra.

Problem 10.5.4 from Vestrup E.M. "The Theory of Measures and Integration" [Ves03]More problems in Section 10.5 "Convolutions"

$L^{1}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ has no element $u$ such that $u * f = f$ $\bar{\lambda }$ -a.e. on $\mathbb {R}$ for every $f$ in $L^{1}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ .

Remark.

Suppose such a $u$ exists. Find $\delta > 0$ with $\int_{[-2\delta , 2\delta ]} \left|u\right| \, d\bar{\lambda } < 1$ . Let $f = \; \mathbb {1}_{[-\delta , \delta ]}$ . For $\bar{\lambda }$ -a.e. $x \in \mathbb {R}$ we have

\begin{aligned} f(x) = u * f(x) & = \int _{\mathbb {R}} u(x - y) f(y) \, d\bar{\lambda }(y) \\ & = \int _{[-\delta , \delta ]} u(x - y) \, d\bar{\lambda }(y) \\ & = \int _{[x - \delta , x + \delta ]} u(t) \, d\bar{\lambda }(t) . \end{aligned}

Since $\bar{\lambda }\left([-\delta , \delta ]\right) > 0$ , there is $x_{0} \in [-\delta , \delta ]$ with $1 = f\left(x_{0}\right) = \int_{[x_{0} - \delta , x_{0} + \delta ]} u(t) \, d\bar{\lambda }(t)$ . Now $[x_{0} - \delta , x_{0} + \delta ] \subseteq [-2\delta , 2\delta ]$ , so

1 = \left|\int _{x_{0} - \delta }^{x_{0} + \delta } u(t) \, d\bar{\lambda }(t)\right| \leq \int _{x_{0} - \delta }^{x_{0} + \delta } \left|u(t)\right| \, d\bar{\lambda }(t) \leq \int _{-2\delta }^{2\delta } \left|u(t)\right| \, d\bar{\lambda }(t) < 1 .

This gives a contradiction.

References

[Axl19] Axler S. Measure, Integration & Real Analysis. 3rd ed., Springer, 2019.

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.

[Sax02] Saxe K. Beginning Functional Analysis. Springer, 2002.

[Ves03] Vestrup E.M. The Theory of Measures and Integration. Wiley-Interscience, 2003.