Convolutions on an Abelian Group

Let $G$ be a finite group, and $K$ some field.

Let $C_n$ be the cyclic group of order $n$.

Prove that $\mathbb {R}{}[S_3]$ is isomorphic to $\mathbb {R}+\mathbb {R}+\operatorname {Mat}_2\mathbb {R}$.

Prove that the natural imbedding of $G$ into $K[G]$ is the universal object for the category of mappings $\varphi$ of the group $G$ into associative $K$-algebras with unit such that $\varphi (g_1g_2)=\varphi (g_1)\varphi (g_2)$, $\varphi (1)=1$ (a morphism of an object $\varphi :G\to A$ to an object $\psi :G\to B$ is defined to be a homomorphism $\chi :A\to B$ such that the diagram

commutes). Can the condition $\varphi (1)=1$ in this definition be omitted?

Let $G=\mathbb {R}^{n}$ or $\mathrm{T}^n$. Prove that the convolution of two bounded functions in $L^{1}\left({G,\mu }\right)$ is a continuous function.

Suppose that the function $\varphi$ is compactly supported and $k$ times continuously differentiable in $\mathbb {R}^{n}$, and that the function $f$ belongs to $L^{1}\left({\mathbb {R}^{n},dx}\right)$. Prove that the convolution $\varphi *f$ has continuous derivatives up to order $k$.

Suppose that $L$ is a Banach space of functions on a locally compact abelian group $G$ with invariant measure $\mu$, with a norm $\left\| \cdot \right\|_{L}$ having the properties:

1. $\left\| T(a)f\right\|_{L}=\left\| f\right\|_{L}$, where $T(a)$ is the translation operator $T(a)f(x)=f(x+a)$;

2. $\left\| T(a)f-f\right\|_{L}\to 0$ as $a\to 0$ in $G$. Prove that the operator $S(\varphi )$, $\varphi \in L^{1}\left({G,\mu }\right)$, carries $L$ into itself and has a norm in this space not exceeding $\left\| \varphi \right\|_{L}$.

A sequence $\left\{ f_k\right\}$ of functions on a topological space $X$ with a Borel measure $\mu$ is said to be $\delta$-shaped for a point $a\in X$ if:

1. $f_k(x)\geq 0$ on $X$;

2. $\int_X f_k(x)\, d\mu (x)=1$;

3. $\int_{X\setminus U}f_k(x)\, d\mu (x)\to 0$ for any neighborhood $U$ of $a$ as $k\to \infty$. Prove that the following sequences are $\delta$-shaped for the point $0\in \mathbb {R}^{n}$:

Let $G$ be a topological abelian group with invariant measure $\mu$, $\left\{ f_k\right\}$ a $\delta$-shaped sequence for the point $a\in G$ (see Problem 510), and $L$ a Banach space of functions on $G$ satisfying the conditions of Problem 509. Prove that $S(f_k)\to T(a)$ as $k\to \infty$.

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 $0$.)

Prove that every continuous function on $\mathbb {R}^{n}$ can be approximated uniformly by polynomials on any compact set.

Let $f\in L^{1}\left({G,\mu }\right)$. Prove that $S(f)$ is a bounded operator in the Hilbert space $L^{2}\left({G,\mu }\right)$. Compute the adjoint operator $S(f)^*$.

Let $f_1$ and $f_2$ be in $L^{2}\left({G,\mu }\right)$. Prove that the convolution $f_1*f_2$ is defined and belongs to $L^{\infty }\left({G,\mu }\right)$.

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

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 $L^{1}\left({\mathrm{T}^n,dt}\right)$.

Construct a $\delta$-shaped sequence consisting of trigonometric polynomials on the torus $\mathrm{T}^n$.

Prove that the trigonometric polynomials form a dense set in $C^k(\mathrm{T}^n)$ for any $k$.

Let $f_1$ and $f_2$ be locally integrable functions on $\mathbb {R}$ whose supports are bounded to the left. Then the convolution $f_1*f_2$ can be defined, and it has the same property. Compute explicitly the convolutions:

Suppose that the numbers $p\geq 1$, $q\geq 1$, $r\geq 1$ are related by the equation $1/p+1/q-1/r=1$. Prove that

Express the differential operator with constant coefficients $L:f\to \sum_{k=0}^n c_k f^{(k)}$ as a convolution.

Prove the identity $f*1=\langle f,1\rangle \cdot 1$ for any generalized function $f\in \mathcal{E}'(\mathbb {R})$.

Let $\check f(x)=f(-x)$. Prove the identity $(f_1*f_2)=\check f_1*\check f_2$, where $f_1$ and $f_2$ are generalized functions, one of which has compact support.

Let $f\in S'(\mathbb {R}^{n})$, $\varphi \in S(\mathbb {R}^{n})$. The convolution $f*\varphi$ is defined by the formula $f*\varphi =S(\check\varphi )'f$, i.e., $\langle f*\varphi ,\psi \rangle =\langle f,\varphi *\psi \rangle$. Prove that:

Let $f\in \mathcal{E}'(\mathbb {R}^{n})$, $\varphi \in \mathcal{E}(\mathbb {R}^{n})$. The convolution $f*\varphi$ is defined by the formula $f*\varphi =S(\check\varphi )f$ (cf. Problem 526). Prove that:

Let $f\in \mathcal{D}'(\mathbb {R}^{n})$. Prove that the operator $S(f):\mathcal{D}(\mathbb {R}^{n})\to \mathcal{E}(\mathbb {R}^{n})$ is continuous.

Let $\mathcal{E}(\mathrm{T}^n)$ be the space of infinitely differentiable functions on the torus $\mathrm{T}^n$ (with the topology of uniform convergence of all derivatives), and $\mathcal{E}'(\mathrm{T}^n)$ the dual space of generalized functions. Define a convolution operation in $\mathcal{E}'(\mathrm{T}^n)$ and prove that $\mathcal{E}'(\mathrm{T}^n)*\mathcal{E}(\mathrm{T}^n)\subset \mathcal{E}(\mathrm{T}^n)$.

Prove that the space of trigonometric polynomials is dense in $\mathcal{E}(\mathrm{T})$.

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 $\mathcal{E}_\pm (\mathbb {R})$ be the subspaces of $\mathcal{E}(\mathbb {R})$ consisting of the functions with support bounded to the left or to the right, and $\mathcal{D}'_\pm (\mathbb {R})$ the analogous subspaces of $\mathcal{D}'(\mathbb {R})$.

Let $f_\alpha (x)=[1/\Gamma (\alpha )]x^{\alpha -1}\theta (x)$ for $\alpha >0$.

Construct a family of operators $I(\alpha )$, $\alpha \in \mathbb {R}$, in $\mathcal{D}'_+(\mathbb {R})$ with the properties:

1. $I(\alpha )I(\beta )=I(\alpha +\beta )$, $I(0)=1$;

2. $I(1)\varphi (x)=\int_0^x\varphi (t)\, dt$ for $\varphi \in \mathcal{E}'_+(\mathbb {R})$;

3. $I(-1)\varphi (x)=\varphi '(x)$ for $\varphi \in \mathcal{E}'_+(\mathbb {R})$;

4. $I(\alpha )f_\beta =f_{\alpha +\beta }$ for $\beta >-1$, $\alpha +\beta >-1$ (the $f_\alpha$ are defined in Problem 534). The operator $I(\alpha )$ is called the operator of fractional integration of order $\alpha$ (or of fractional differentiation of order $-\alpha$) and denoted sometimes by $(d/dx)^{-\alpha }$.

Suppose that the generalized function $f$ on $\mathbb {R}^{2}$ has the form

Prove that $f*f$ is a regular generalized function, and compute it.

Compute the convolution $f*f$, where $f$ is the generalized function on $\mathbb {R}^{3}$ given by the formula

where $S$ is the sphere $\left\| x\right\| =1$ and $d\sigma$ is the area element of the sphere.