The Fourier Transform of Generalized Functions

Let $f$ be a homogeneous generalized function of degree $(\lambda ,\varepsilon )$. Prove that $\tilde f$ is also homogeneous, and find its degree.

Let $f$ be a regular generalized function on the circle $\mathrm{T}$, and $F$ the generalized periodic function on the line connected with $f$ by the relation $\langle F,\varphi \rangle =\int_{\mathbb {R}} f(e^{2\pi it})\varphi (t)\, dt$. How are the Fourier transforms of $F$ and $f$ connected?

A continuous function $f$ on $\mathbb {R}^{n}$ is said to be quasi-periodic with period $R$ if its integral over any ball of radius $R$ does not depend on the location of the center of the ball.

Find the Fourier transform of the generalized function $f(x)=e^{-\pi \left\langle Ax, x \right\rangle }$ on $\mathbb {R}^{n}$, where $A$ is a symmetric matrix with positive-definite real part.

Find the Fourier transform of the generalized function $f(x)=e^{i\pi \left\langle Ax, x \right\rangle }$ on $\mathbb {R}^{n}$, where $A$ is a real symmetric nonsingular matrix.

Prove that the image of $\mathcal{E}'(\mathbb {R})$ under the Fourier transformation is the collection of entire functions $g(\lambda )$, $\lambda \in \mathbb {C}$, satisfying an estimate

where $C,N,R$ are constants (depending on $g$). On what properties of the function being transformed do the constants $R$ and $N$ depend?

Prove that the equation $\Delta f=f$ does not have nonzero solutions in $S'(\mathbb {R}^{n})$. Here $\Delta =\sum_{k=1}^n(\partial^2/\partial x_k^2)$ is the Laplace operator.)

Let $u(t,x)$ be the solution of the heat conduction equation $\partial u/\partial t=\partial^2u/\partial x^2$ with initial data $u(0,x)=v(x)$, $v\in L^{1}\left({\mathbb {R},dx}\right)$. Show that $u(t,x)$ has the form $v*f_t(x)$, and find the function $f_t$.

Let $T(a)$ be the operator of translation by the vector $a\in \mathbb {R}^{n}$, and $M(a)$ the operator of multiplication by $2\pi iax$ in $S'(\mathbb {R}^{n})$. Derive the commutation relations

where $\mathcal{F}$ is the Fourier transformation.

The generalized function $f\in S'(\mathbb {R}^{3})$ is regular and depends only on the radius $r=\left\| x\right\|$, i.e., $f(x)=\varphi (\left\| x\right\| )$. Prove that its Fourier transform is given by the formula

and find the function $k$.