Convolution

Prove that if $f, g, h \in L_{1}(\mathbb {R})$, then their convolution has the following properties:

Prove that if $f, g \in L_{1}(\mathbb {R})$, then their convolution has the following properties:

Prove that $\operatorname {supp}(f * g) \subseteq \operatorname {supp} f+\operatorname {supp} g$.

Compute the convolution of the characteristic functions $\chi_{[1,2]}(x) * \chi_{[3,4]}(x)$.

Compute the convolution $f(x) * \chi_{[2,4]}(x)$, where

Prove that the convolution $\omega_{a, b} * \chi_{[c, d]}$, where $\omega_{a, b}$ is the "bump" from problem 15.1, is a function of the "hat" type from problem 15.2.

Prove that if $f, g \in L_{1}(\mathbb {R}), f \in C^{1}(\mathbb {R})$ and $f^{\prime } \in L_{1}(\mathbb {R})$, then $f * g \in C^{1}(\mathbb {R})$ and $(f * g)^{\prime } = f^{\prime } * g$.

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 if $f \in L_{1}(\mathbb {R})$ and $g \in L_{\infty }(\mathbb {R})$, then the convolution $f * g$ is uniformly continuous on $\mathbb {R}$.

Prove that if $f \in L_{1}(\mathbb {R}) \cap L_{\infty }(\mathbb {R})$ and $g \in L_{1}(\mathbb {R})$, then $f * g \in C_{0}(\mathbb {R}) \cap L_{1}(\mathbb {R})$ and $\left\| f * g\right\|_{C_{0}(\mathbb {R})} \leqslant \left\| f\right\|_{L_{\infty }(\mathbb {R})} \cdot \left\| g\right\|_{L_{1}(\mathbb {R})}$.

Prove that if $f \in L_{2}(\mathbb {R})$ and $g \in L_{2}(\mathbb {R})$, then the convolution $f * g$ is defined and belongs to the space $C_{0}(\mathbb {R})$.

Give an example of two functions $f, g \in L_{1}(\mathbb {R})$ such that $f * g$ is discontinuous.

(E. Titchmarsh, 1926). Let $f, g \in L_{1}(\mathbb {R}), f \equiv g \equiv 0$ for $t < 0$ and $f * g = 0$. Prove that then one of the functions $f$ or $g$ is zero almost everywhere.

Let the functions $f, g \in L_{1}(\mathbb {R})$ have support on the segment $[0,1]$, be continuous on this segment, and $f(1) \neq 0$. Prove that if $\operatorname {supp}(f * g) \subseteq [0,1]$, then $g \equiv 0$.

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?

Find the spectrum of the Hilbert operator in $L_{2}(\mathbb {R})$:

(N. Wiener, 1933). Let $f \in L_{2}(\mathbb {R})$. Prove that the system $\left\{ f(x-a): a \in \mathbb {R}\right\}$ of shifts of this function is complete in $L_{2}(\mathbb {R})$ if and only if the measure of the set $\left\{ y \in \mathbb {R}: \hat{f}(y) = 0\right\}$ equals zero.

Prove that for a generalized function $G \in \mathcal{E}^{\prime }$ the map $\varphi \rightarrow G * \varphi$ defines a continuous operator $S_{G}: \mathcal{D} \rightarrow \mathcal{D}$ and the inclusion holds:

Prove that if a regular generalized function $G = G_{\varphi }$ is generated by a test function $\varphi$, then for any generalized function $F$ the convolution $F * G$ is a regular function, generated by the convolution $F * \varphi$ (which is computed by Definition 18.2).

Definition 18.2: For any test function $\varphi \in \mathcal{D}$ (or $\mathcal{S}$, or $\mathcal{E}$) its convolution with a generalized function $F \in \mathcal{D}^{\prime }$ (respectively, $\mathcal{S}^{\prime }$, $\mathcal{E}^{\prime }$) as a function of the variable $x$ is defined as follows: $(F * \varphi )(x) := \left\langle F(t), \varphi (x-t) \right\rangle$.

Prove that if one of the generalized functions $F$ and $G$ is an element of $\mathcal{E}^{\prime }$, and the other an element of $\mathcal{D}^{\prime }$, then the convolution $F * G$ is defined, is an element of the space $\mathcal{D}^{\prime }$, and the relation $F * G = G * F$ holds.

Prove that if one of the generalized functions $F$ and $G$ is an element of $\mathcal{E}^{\prime }$, and the other an element of $\mathcal{S}^{\prime }$, then the convolution $F * G$ is defined, is an element of the space $\mathcal{S}^{\prime }$, and the relation $F * G = G * F$ holds.

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 $f \in L_{2}[-\pi , \pi ]$. For every $a \in [0,2 \pi ]$ and $t \in [-\pi , \pi ]$ set

Prove that the system of functions $\left\{ f_{a}\right\}_{a \in [0,2 \pi ]}$ is complete in $L_{2}[-\pi , \pi ]$ if and only if $c_{n}(f) = \frac{1}{2 \pi } \int_{-\pi }^{\pi } f(t) e^{-i n t} d t \neq 0$ for all $n \in \mathbb {Z}$.

Denote by $\mathcal{D}_{+}^{\prime }$ the set of generalized functions $F$ with the property $\operatorname {supp} F \subset [0,+\infty )$. Prove that if $F, G \in \mathcal{D}_{+}^{\prime }$, then the convolution $F * G$ is defined and $F * G \in \mathcal{D}_{+}^{\prime }$.

Prove that in $\mathcal{D}_{+}^{\prime }$ convolution is associative.

Give an example of generalized functions $F, G$ and $H$ for which the expressions $F *(G * H)$ and $(F * G) * H$ are defined and different.

Let the function $g \in L_{1}(\mathbb {R})$ and $\int_{\mathbb {R}} g d x = 1$. Let $F_{n} \in \mathcal{D}^{\prime }$ be the regular generalized functions generated by the functions $n g(n x)$. Prove that $F_{n} * \varphi \rightarrow \varphi$ in $\mathcal{D}$ for every function $\varphi \in \mathcal{D}$ (cf. problems 16.20 and 18.31 а)).

Compute the convolutions:

Show that for a linear continuous operator $A: \mathcal{D} \rightarrow \mathcal{E}$ the following properties are equivalent:

1. $A$ commutes with shifts: $(A \varphi )(x-a) = A(\varphi (x-a))$ for all $a \in \mathbb {R}$;

2. А commutes with differentiation: $(A \varphi )^{\prime } = A\left(\varphi^{\prime }\right)$;

3. $A \varphi = F * \varphi$ for all $\varphi$, where $F \in \mathcal{D}^{\prime }$ is some generalized function.

Compute the antiderivative of order $1 / 2$ of

Verify that $I\left(\frac{1}{2}\right) I\left(\frac{1}{2}\right) \theta (x) = x_{+}$.

Prove that $I(\alpha ) I(\beta ) f = I(\alpha +\beta ) f$ for $f \in \mathcal{D}_{+}^{\prime }, \alpha , \beta \in \left(-\frac{1}{2}, \frac{1}{2}\right)$.

Compute the derivative of order $1 / 2$ of $\theta (x)$.

Prove that the Abel integral equation

with known differentiable function $g$ has the solution $f(x) = \frac{1}{\Gamma (\alpha )} \int_{0}^{x} \frac{g^{\prime }(t) d t}{(x-t)^{1-\alpha }}$ for every $\alpha \in (0,1)$.