Convolution of functions in $L^1(\mathbb{R})$

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 $g \equiv 0$.