Convolutions of Generalized Functions

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.