⧼exchistory⧽
2 exercise(s) shown, 0 hidden
BBy Bot
May 08'24
[math]
\newcommand{\R}{\mathbb{R}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Rbar}{\overline{\mathbb{R}}}
\newcommand{\Bbar}{\overline{\mathcal{B}}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\p}{\mathbb{P}}
\newcommand{\one}{\mathds{1}}
\newcommand{\0}{\mathcal{O}}
\newcommand{\mat}{\textnormal{Mat}}
\newcommand{\sign}{\textnormal{sign}}
\newcommand{\CP}{\mathcal{P}}
\newcommand{\CT}{\mathcal{T}}
\newcommand{\CY}{\mathcal{Y}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\mathds}{\mathbb}[/math]
Show that [math]g_\sigma *f\xrightarrow{\sigma\to 0}f[/math] uniformly on [math]\R[/math].
BBy Bot
May 08'24
[math]
\newcommand{\R}{\mathbb{R}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Rbar}{\overline{\mathbb{R}}}
\newcommand{\Bbar}{\overline{\mathcal{B}}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\p}{\mathbb{P}}
\newcommand{\one}{\mathds{1}}
\newcommand{\0}{\mathcal{O}}
\newcommand{\mat}{\textnormal{Mat}}
\newcommand{\sign}{\textnormal{sign}}
\newcommand{\CP}{\mathcal{P}}
\newcommand{\CT}{\mathcal{T}}
\newcommand{\CY}{\mathcal{Y}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\mathds}{\mathbb}[/math]
Show that if [math]\nu[/math] is a probability measure, then
[[math]]
\int_\R g_\sigma * fd\nu=\int_\R f(x)(g_\sigma *\nu)(x)dx=\int_\R f(x)\frac{1}{\sigma\sqrt{2\pi}}\int_\R e^{i\xi x}g_{\frac{1}{\sigma}}(\xi)\hat{\nu}(\xi)d\xi dx.
[[/math]]