The Fourier Transform

Find the explicit form of the characters on the cyclic group $C_n$ of order $n$.

Prove that every finite abelian group $G$ is isomorphic (not canonically) to its dual group $\widehat G$.

A generalized or nonunitary character on a group $G$ is defined to be a homomorphism of it into the multiplicative group of the field of complex numbers. Prove that for a compact group $G$ the generalized characters are all ordinary characters. Find the generalized characters of the groups:

Prove that if $G$ is a compact group, then the dual group $\widehat G$ is discrete.

Prove that if $G$ is a discrete group, then the dual group $\widehat G$ is compact.

Let $\chi$ be a character on $\mathbb {R}$, and regard it as an element of the space $\mathcal{D}'(\mathbb {R})$. Prove that $\chi$ satisfies the differential equation $\chi '=c\chi$, where $c$ is some constant.

Let $\chi$ be a character on the group $G$, and $f\in L^{1}\left({G,\mu }\right)$. Prove that $\chi *f=cf$, where

Let $G$ be a compact group with invariant measure $\mu$ normalized by the condition $\mu \left(G\right)=1$. Prove that for any two characters $\chi_1,\chi_2\in \widehat G$

Prove that the collection of characters on the group $\mathrm{T}^n$ is exhausted by the collection of functions $e_k(t)=e^{2\pi ikt}$ (cf. Problem 517).

Prove that the correspondence $G\mapsto \widehat G$ defines a contravariant functor in the category of topological abelian groups.

Let $L$ be an LTS over the field $\mathbb {R}$, and regard it as a topological abelian group. Find the group $\widehat L$ dual to $L$.

Let $\mathbb {Q}_p$ be the field of $p$-adic numbers (see Problem 31), and $\mathbb {Z}_p$ the subring of $p$-adic integers. Find the dual groups of the following groups:

Let $G_0$ be a closed subgroup of $G$, and $G_1=G/G_0$ the corresponding factor group. This can be expressed briefly in the form of the exact sequence

where $0$ is the trivial group with one element. Prove that the dual sequence

is also exact.

Find the dual group $\widehat G$ if $G=\mathbb {Q}/\mathbb {R}$. ($G$ can be identified in a natural way with the group of all roots of unity with the aid of the mapping $x\bmod \mathbb {Z}\to e^{2\pi ix}$.)

Let $G=\prod_{n=1}^\infty C_2$ be the group of all sequences of $0$'s and $1$'s (the group operation is addition modulo $2$, and the topology is defined by coordinatewise convergence).

Let $\alpha$ be an irrational number, and $f\in L^{1}\left({\mathrm{T},dt}\right)$ a function having the property $f(t+\alpha )=f(t)$ almost everywhere. Prove that $f$ is a constant almost everywhere.

Let $G=\mathbb {Q}_p^+$ be the additive group of the field of $p$-adic numbers. Denote by $\mathcal{D}(G)$ the space of compactly supported locally constant functions on $G$. Prove that the Fourier transformation carries the space $\mathcal{D}(G)$ into itself.

Let $S(\mathbb {Z})$ be the space of two-sided sequences $\left\{ c_n\right\}$ having the property that $c_n=O(n^{-k})$ for all $k$. The topology in $S(\mathbb {Z})$ is defined by the family of norms

Prove that the Fourier transformation establishes an isomorphism between the linear topological spaces $\mathcal{E}(\mathrm{T})$ and $S(\mathbb {Z})$.

A continuous function $f$ on a group $G$ is said to be positive-definite if for any finite set $x_1,\ldots ,x_n$ of elements in $G$ the matrix $A$ with elements $a_{kj}=f(x_k-x_j)$ is positive-definite. Prove the following relations for a positive-definite function $f$:

Let $G$ be a finite group. Prove that $f$ is positive-definite on $G$ if and only if $\tilde f$ is nonnegative on $\widehat G$.

Let $\varphi \in L^{1}\left({G,\mu }\right)$ and $\varphi \geq 0$. Prove that $\tilde\varphi$ is positive-definite on $\widehat G$.

What can be said about the Fourier coefficients of a function $f$ on $\mathrm{T}$ if it is known that $f$

Suppose that a function $f$ on $\mathrm{T}$ has a piecewise-differentiable $k$th derivative. What is the maximal number $l$ for which it is possible to ensure the estimate $\left|c_n\right|=o(\left|n\right|^{-l})$ for the Fourier coefficients of $f$?

Prove that the image of the space $C^k(\mathrm{T})$ under the Fourier transformation is contained in the set of sequences having the property $\left|c_n\right|=o(\left|n\right|^{-k})$ and contains the set of sequences having the property $\left|c_n\right|=o(n^{-k-1-\varepsilon })$, $\varepsilon >0$.

Let $W^k(\mathrm{T})$ be the collection of functions on $\mathrm{T}$ for which the $k$th generalized derivative belongs to $L^{2}\left({\mathrm{T},dt}\right)$. Give a description of this space in terms of the Fourier coefficients.

Express the following properties of a function $f$ in terms of the Fourier coefficients:

The exact sequence $0\to C_n\xrightarrow {i}\mathrm{T}\xrightarrow {p}\mathrm{T}\to 0$ is given by the imbedding $i:C_n\to \mathrm{T}$ defined by the formula $i(k\bmod n)=e^{2\pi ik/n}$ and the projection $p:\mathrm{T}\to \mathrm{T}$ defined by the formula $p(z)=z^n$. Describe the dual exact sequence.

Let the function $f$ be integrable on $[0,1/4]$. Extend $f$ to $[0,1]$ in such a way that its Fourier coefficients satisfy the relations $c_{2k}=0$, $c_{2k-1}=-c_{1-2k}$.

Let $\left\{ c_n\right\}$ be the Fourier coefficients of a function $f\in L^{1}\left({\mathrm{T},dt}\right)$. Find the Fourier coefficients $\left\{ c_n(h)\right\}$ for the smoothed function (or Steklov function) $f_h(x)=(1/2h)\int_{x-h}^{x+h}f(\xi )\, d\xi$.

What properties characterize the sequence $\left\{ c_n\right\}$ of Fourier coefficients

Prove that a continuous function $f$ on $\mathrm{T}$ is positive-definite (see Problem 557) if and only if its Fourier coefficients are nonnegative.

A sequence $\left\{ c_n\right\}$, $n\in \mathbb {Z}$, is said to be positive-definite if $\sum_{n,m}c_{n-m}z_n\overline{z_m}\geq 0$ for any finitely nonzero sequence $\left\{ z_n\right\}$, $n\in \mathbb {Z}$. Prove that a positive-definite sequence is bounded and has the properties:

Prove that every positive-definite sequence (see Problem 571) is the Fourier transform of a finite Borel measure $\mu$ on $\mathrm{T}$: $c_n=\int_0^1e^{-2\pi int}\, d\mu (t)$.

Let $U$ be a unitary operator on a Hilbert space $H$, and $\xi \in H$. Prove that the sequence $c_n= \left\langle U^n\xi , \xi \right\rangle$ is positive-definite.

Under the conditions of Problem 573 suppose that $\xi$ is a cyclic vector for $U$ (i.e., the linear span of the vectors $U^n\xi$, $n\in \mathbb {Z}$, is dense in $H$). Construct an isomorphism between $H$ and $L^{2}\left({\mathrm{T},\mu }\right)$, where $\mu$ is the Fourier transform of the sequence $\left\{ c_n\right\}$, under which $U$ passes into multiplication by $e^{2\pi it}$.

Suppose that $f$ is a piecewise differentiable real function on $\mathrm{T}$, $S_n=\sum_{k=-n}^n c_ke^{2\pi ikt}$ is the $n$th partial sum of its Fourier series, and $\Gamma_n\subset \mathrm{T}\times \mathbb {R}$ is the graph of the function $S_n$. Find the limit set for $\left\{ \Gamma_n\right\}$, i.e., the collection of all limit points of sequences $\left\{ \gamma_n\right\}$, $\gamma_n\in \Gamma_n$.

Prove that a generalized function on $\mathrm{T}$ is uniquely determined by its Fourier coefficients.

A generalized function $f$ on $\mathrm{T}$ is said to be positive-definite if $\langle f,\varphi *\varphi^*\rangle \geq 0$ for any function $\varphi \in \mathcal{E}(\mathrm{T})$. Characterize the positive-definite generalized functions in terms of their Fourier coefficients.

Let $\alpha$ be an irrational real number, and $X$ a measurable subset of $\mathrm{T}$ that is invariant under translation by $\alpha$. Prove that either $\mu \left(X\right)=0$ or $\mu \left(X\right)=1$, where $\mu$ is Haar measure. (This property of a transformation of a measure space is called ergodicity: the transformation is said to be ergodic if every measurable invariant subset either is of measure zero itself or has complement of measure zero.)

Use the Fourier method to solve the heat conduction equation $\partial u/\partial t=\partial^2u/\partial x^2$ on the circle $\mathrm{T}$ with initial data $u(0,x)=v(x)$ (see Problem 607).

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.

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 $k$.