Convolution theorem for the Fourier transform

Prove that if $f, g \in L^{2}(\partial \mathbf{D})$, then

for every $n \in \mathbf{Z}$.

Prove that if $f \in L^{1}(\mathbb {R})$ and $g \in L^{2}(\mathbb {R})$, then $\mathcal{F}(f * g) = \widehat{f} \mathcal{F} g$.

Prove that if $f, g \in L^{2}(\mathbb {R})$, then $\widehat{f g} = (\mathcal{F} f) *(\mathcal{F} g)$.

Prove that for all $f, g \in L_{1}(\mathbb {R})$ and $y \in \mathbb {R}$

Let $f \in L_{1}(\mathbb {R})$ and $g \in L_{2}(\mathbb {R})$. Prove that

Prove that if $f, g \in L_{1}(\mathbb {R})$, then $\widehat{f * g} = \sqrt{2 \pi } \hat{f} \cdot \hat{g}$.

Compute the convolution $\frac{1}{x^{2}+a^{2}} * \frac{1}{x^{2}+b^{2}}, a, b > 0$.

Prove that the convolution operator $S_{f}$ in $L_{2}(\mathbb {R})$ is unitarily equivalent to the operator of multiplication by the function $\sqrt{2 \pi } \hat{f}: \hat{F} S_{f} \hat{F}^{-1} h = \sqrt{2 \pi } \hat{f} h$ for every function $h \in L_{2}(\mathbb {R})$.

Find the spectrum of the convolution operator with the function $e^{-\left|x\right|}$ in $L_{2}(\mathbb {R})$.

Find the spectrum of the operator $A: L_{2}(\mathbb {R}) \rightarrow L_{2}(\mathbb {R})$ of the following form:

Is this operator compact?

Prove that if one of the generalized functions $G$ and $H$ is an element of $\mathcal{E}^{\prime }$, and the other an element of $\mathcal{S}^{\prime }$, then $\hat{F}[G * H] = \sqrt{2 \pi } \hat{F}[G] \cdot \hat{F}[H]$.

Let $S^1$ denote the unit circle (the set of complex numbers with modulus one, or the real numbers modulo $2\pi$). The convolution of two functions on $S^1$ is

Suppose that $f$ is an element of $L^2(S^1)$ with the property that its Fourier coefficient

is non-zero for all $n \in \mathbb {Z}$. Show that the linear space $\{ f * k \mid k \in L^2(S^1)\}$ is dense in $L^2(S^1)$.

This exercise provides the details needed for the proof of Theorem 9.21. Let $f, g \in L^{1}(\mathbb {R})$ be fixed, and prove the following facts.

Define the function $e_k$ on $\mathrm{T}^n$ by the formula $e_k(t)=e^{2\pi ikt}$ (here $k\in \mathbb {Z}^{n}$ is a multi-index and $kt=k_1t_1+\cdots +k_nt_n$). Prove the identity

A trigonometric polynomial on $\mathrm{T}^n$ is defined to be a linear combination of the functions $e_k$ defined in Problem 517. Prove that the collection of trigonometric polynomials forms an ideal in the algebra $L^{1}\left({\mathrm{T}^n,dt}\right)$.

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 $\chi_1,\chi_2\in \widehat G$

Let $f\in L^{1}\left({\mathbb {R},dx}\right)$. Find the spectrum of the convolution operator $S(f)$ on the space $L^{2}\left({\mathbb {R},dx}\right)$.

Let $f\in L^{1}\left({\mathrm T,dt}\right)$. Find the spectrum of the operator $S(f)$ of convolution with $f$ on the space $L^{2}\left({\mathrm T,dt}\right)$.

Reduce the convolution operator $S(f)$, $f\in L^{1}\left({\mathbb {R},dx}\right)$, to the form of multiplication by a function. For what functions $f$ is this operator selfadjoint?

Can the convolution operator $S(f)$, $f\in L^{1}\left({\mathbb {R},dx}\right)$, be unitary?

Let $G$ be a locally compact abelian group with invariant measure $\mu$. Determine conditions on a function $f\in L^{1}\left({G,\mu }\right)$ under which the convolution operator $S(f)$ is:

Let $f$ be a continuous function with period $2 \pi$. Express the Fourier coefficients $A_{n}, B_{n}$ of the convolution

in terms of the Fourier coefficients $a_{n}, b_{n}$ of the function $f$.

Find the expression for the Fourier coefficients $A_{n}, B_{n}$ of the Steklov function

in terms of the Fourier coefficients $a_{n}, b_{n}$ of a $2 \pi$-periodic function $f$, absolutely integrable over the period.

Let the functions $f$ and $g$ be continuous, bounded, and absolutely integrable on $R$. Prove that:

The Fourier transform of a function $f \in \mathscr {L}^{1}\left(\mathbb {R}^{m}\right)$ is the function $\widehat{f}$, defined by the equality

where $(s, t)$ is the scalar product of the vectors $s, t \in \mathbb {R}^{m}$. The convolution of two summable functions $g$ and $h$ on $\mathbb {R}^{m}$ is denoted by $g * h$: by definition $(g * h)(x)=\int_{\mathbb {R}^{m}} g(x-y) h(y) d y$ for almost all $x \in \mathbb {R}^{m}$.

Find the Fourier transform and the convolution $f_{c_{1}} * f_{c_{2}}$ of the functions in the following cases ($c>0$):

The convolution of two summable functions $g$ and $h$ on $\mathbb {R}^{m}$ is denoted by $g * h$: by definition $(g * h)(x)=\int_{\mathbb {R}^{m}} g(x-y) h(y) d y$ for almost all $x \in \mathbb {R}^{m}$. The Fourier transform of a function $f \in \mathscr {L}^{1}\left(\mathbb {R}^{m}\right)$ is the function $\widehat{f}(s)=\int_{\mathbb {R}^{m}} f(t) e^{-i(s, t)} d t$ ($s \in \mathbb {R}^{m}$); it is injective in the sense that if $\widehat{f}=\widehat{g}$, then $f(t)=g(t)$ for almost all $t \in \mathbb {R}^{m}$.

Prove that if the convolution of a function $f$ summable on $\mathbb {R}^{m}$ with the function $e^{-\left\| x\right\|^{2}}$ is identically zero, then $f(x)=0$ almost everywhere.

Compute the Fourier transform of the characteristic function of an interval. For $n = 1,2,3, \ldots$, let $g_{n}$ be the characteristic function of $[-n, n]$, let $h$ be the characteristic function of $[-1,1]$, and compute $g_{n} * h$ explicitly. (The graph is piecewise linear). Show that $g_{n} * h$ is the Fourier transform of a function $f_{n} \in L^{\mathbf{1}}$; except for a multiplicative constant,

Show that $\left\| f_{n}\right\|_{1} \rightarrow \infty$ and conclude that the mapping $f \rightarrow \hat{f}$ maps $L^{1}$ into a proper subset of $C_{0}$. Show, however, that the range of this mapping is dense in $C_{0}$.