Operations on Generalized Functions

Prove that a generalized function $F\in \mathcal D'(\mathbb {R})$ with the property that $F'=0$ is a constant.

Prove that all the solutions of the equation $xF=0$ among the generalized functions $F\in \mathcal D'(\mathbb {R})$ are proportional to the $\delta$-function.

Prove that every generalized function on the line with support at a point $a\in \mathbb {R}$ has the form $P(d/dx)\delta_a(x)$, where $P$ is a polynomial.

Suppose that $g\in \mathcal E(\mathbb {R})$ determines a one-to-one mapping of the line onto itself, and let $h\in \mathcal E(\mathbb {R})$ be the inverse mapping. Express the function $\delta '(g(x))$ explicitly in terms of $\delta (x)$ and its derivatives.

A generalized function $F$ on the line is said to be homogeneous of degree $(\lambda ,\varepsilon )$, where $\lambda \in \mathbb {C}$ and $\varepsilon =0$ or $1$, if $F(tx)=\left|t\right|^{\lambda }(\operatorname {sgn}t)^\varepsilon F(x)$ for $t\ne 0$.

Prove that a homogeneous function of degree $(\lambda ,\varepsilon )$ on the line (see first-edition Problem 513) satisfies the differential equation $xF'=\lambda F$.

Prove that for all $\lambda \in \mathbb {C}$ and $\varepsilon =0,1$ there exists a homogeneous generalized function on the line of degree $(\lambda ,\varepsilon )$, and that it is unique to within a numerical factor.

Prove that the only generalized functions on the line that are translation invariant are the constants.

Suppose that $F$ is a generalized function on the plane that is invariant under translations along the axis $Ox$.

Suppose that $F$ is a generalized function on the plane with support the segment $[0,1]$ of the axis $Ox$.

Prove that the regular generalized function $f(x)=\exp (ie^x)$ is in $S'(\mathbb {R})$, but that its derivative in the sense of generalized functions does not coincide with $f'(x)=ie^xe^{ie^x}$.

Solve the equation $(\sin x)\cdot f(x)=0$ in the generalized functions on the line.

Suppose that the generalized function $F$ on $\mathbb {R}^{n}$ satisfies the relation $(\sum_i x_i^2-R^2)F(x)=0$ and is invariant under rotations of $\mathbb {R}^{n}$. Prove that $\langle F,\varphi \rangle =c\int_{S_R}\varphi (x)\, d\sigma (x)$, where $c$ is a constant and $d\sigma$ is the rotation invariant surface element on the sphere $S_R$ of radius $R$ in $\mathbb {R}^{n}$.

Suppose that the generalized function $F$ on the line has the following properties:

1. $F(x+1)=F(x)$;

2. $e^{2\pi ix}F(x)=F(x)$. Prove that $F(x)=c\sum_{k\in \mathbb {Z}}\delta (x-k)$.

Prove that every generalized function $F$ on the line satisfying the equation $F'(x)=a(x)F(x)+b(x)$, $a,b\in \mathcal E(\mathbb {R})$, is regular (and, hence, coincides with the ordinary solution of this equation).

Suppose that $F$ is a generalized function on $\mathbb {R}^{n}$ such that all the partial derivatives of the form $\partial^kF/\partial x_i^k$, $1\le i\le n$, $0\le k\le r$, belong to $L^{2}\left({\mathbb {R}^{n},dx}\right)$. Prove that for $r>n/2+l$ the function $F$ coincides almost everywhere with a function in the class $C^l(\mathbb {R}^{n})$.

Under what conditions on the coefficients $\left\{ c_n\right\}$ does the series $\sum_{n\in \mathbb {Z}}c_ne^{2\pi inx}$ converge in $\mathcal D'(\mathbb {R})$?

Is it possible to define a multiplication operation in $\mathcal D'(\mathbb {R})$ in such a way that it is continuous in each variable and coincides with ordinary multiplication on regular functions?