The Fourier Integral

Let $D_k=\partial /\partial x_k$, and $M_k$ the operator of multiplication by $x_k$. Define the operators $A_k=iD_k+M_k$, $A_k^*=iD_k-M_k$, $k=1,2,\ldots ,n$ (the so-called creation and annihilation operators in quantum field theory) in the space $S(\mathbb {R}^{n})$.

Prove that every continuous operator in $S(\mathbb {R}^{n})$ that commutes with the operators $M_k$, $1\leq k\leq n$, (see Problem 580) is the operator of multiplication by a function.

Prove that every continuous operator in $S(\mathbb {R}^{n})$ that commutes with the operators $M_k$ and $D_k$, $1\leq k\leq n$, (see Problem 580) is a scalar operator.

Prove that the direct and inverse Fourier transformations preserve the space $S(\mathbb {R}^{n})$ and are mutually inverse continuous transformations in it.

What can be said about the Fourier transform of a function $f$ if it is known that $f$

The functions $f$ and $g$ on $\mathbb {R}^{n}$ are connected by the relation $f(x)=g(Ax+b)$, where $A$ is an invertible linear operator on $\mathbb {R}^{n}$, and $b\in \mathbb {R}^{n}$. How are the Fourier transforms $\tilde f(\lambda )$ and $\tilde g(\lambda )$ related?

Prove that if $f\in L^{1}\left({\mathbb {R}^{n},dx}\right)$ and $\tilde f(\lambda )\equiv 0$, then $f(x)=0$ almost everywhere.

The space $H_s(\mathbb {R}^{n})$ is defined for $s\geq 0$ as the space of Fourier transforms of the functions in $L^{2}\left({\mathbb {R}^{n},(1+\left\| \lambda \right\|^{s/2})\, d\lambda }\right)$. Prove that for $s>n/2$ each function $f\in H_s(\mathbb {R}^{n})$ coincides almost everywhere with some bounded continuous function.

Prove that the operators $D_k:H_s(\mathbb {R}^{n})\to H_{s-1}(\mathbb {R}^{n})$, $1\leq k\leq n$, $s\geq 1$, are continuous (see Problem 580 and Problem 587).

Prove that the convolution of two functions in $S(\mathbb {R}^{n})$ is also in $S(\mathbb {R}^{n})$.

Prove that the convolution of functions $f_1\in H_{s_1}(\mathbb {R}^{n})$ and $f_2\in H_{s_2}(\mathbb {R}^{n})$ (see Problem 587) belongs to $BC^k(\mathbb {R}^{n})$ if $s_1+s_2\geq k$. ($BC^k(\mathbb {R}^{n})$ denotes the space of functions on $\mathbb {R}^{n}$ with continuous bounded derivatives up to order $k$, endowed with the norm

Let $P$ be a polynomial on $\mathbb {R}$ of degree $2m$ without real roots.

Let $f\in L^{1}\left({\mathbb {R},dx}\right)$ be a rational function. Prove that $\left|\tilde f(\lambda )\right|\leq ce^{-\varepsilon \left|\lambda \right|}$, $\lambda \in \mathbb {R}$, for some constants $c>0$ and $\varepsilon >0$.

Prove that every continuous positive-definite function $f$ on the real line has the form $f(x)=\int_{\mathbb {R}} e^{2\pi i\lambda x}\, d\mu (\lambda )$, where $\mu$ is some finite Borel measure on $\mathbb {R}$.

Let $\left\{ U(t)\right\}$, $t\in \mathbb {R}$, be a one-parameter group of unitary operators on a Hilbert space $H$ (i.e., $U(t)U(s)=U(t+s)$) that is continuous with respect to $t$ in the strong operator topology. Prove that for any vector $\xi \in H$ the function $f(t)= \left\langle U(t)\xi , \xi \right\rangle$ is positive-definite.

Under the conditions of Problem 595 assume that the vector $\xi$ is cyclic for $U(t)$ (i.e., the linear span of the vectors $U(t)\xi$, $t\in \mathbb {R}$, is dense in $H$). Construct an isomorphism between the spaces $H$ and $L^{2}\left({\mathbb {R},\mu }\right)$ under which the operator $U(t)$ passes into the operator of multiplication by $e^{2\pi i\lambda t}$.

Prove that the Fourier transforms of the functions in $\mathcal{D}(\mathbb {R})$ form the space of entire functions $g$ of $\lambda \in \mathbb {C}$ having the property that there exists a number $a>0$ and constants $c_k$ such that

Let $f$ be a continuous function on $\mathbb {R}^{n}$ that decreases like $O(\left\| x\right\|^{-n})$ at infinity. Then for any affine submanifold $L\subset \mathbb {R}^{n}$ of dimension $n-1$ the restriction of $f$ to $L$ is integrable over $L$ with respect to the natural Lebesgue measure $\mu_L$ on $L$.

Find a function $f\in S(\mathbb {R}^{3})$ if the integrals of this function over all lines intersecting a given line $l\subset \mathbb {R}^{3}$ are known.