Convolution, Smooth Approximation, and a Smooth Urysohn's Lemma

Define $f: \mathbb {R} \rightarrow \mathbb {R}$ by $f(0) = 0$ and otherwise $f(x) = \exp \left(-1 / x^{2}\right)$. Use the Taylor expansion for $\exp (x)$ at $x = 0$ to show that for each polynomial $p(x)$,

Use this to show that $f \in C^{\infty }(\mathbb {R})$, and $f^{(k)}(0) = 0$ for all $k$. For what $x \in \mathbb {R}^{n}$ does the Taylor expansion of $f$ at $x = 0$ agree with $f(x)$ ?

Use the preceding problem to verify that the function $w$ belongs to $C_{c}^{\infty }(\mathbb {R})$.

For $1 \leq p < \infty$, let $f \in L^{p}\left(\mathbb {R}^{n}\right)$. Show that if $\int_{\mathbb {R}^{n}} f(x) g(x) d x = 0$ for all $g \in C_{c}^{\infty }\left(\mathbb {R}^{n}\right)$, then $f = 0$.

For the function $f: \mathbb {R} \rightarrow \mathbb {R}$ defined by $f(x) = x^{-1 / 2} \cdot \chi_{(0,1)}(x)$, show that $(f * f)(x)$ is defined for all $x$ and is finite for $x \neq 0$.

Show that if $f: \mathbb {R}^{n} \rightarrow \mathbb {R}$ is bounded and continuous, then, as $\epsilon \rightarrow 0,\left\{ f * w_{\epsilon }\right\} \rightarrow f$ pointwise, uniformly on compact sets.

Let $k \in L^{2}\left(\mathbb {R}^{n} \times \mathbb {R}^{n}\right)$, and define the integral operator $T: L^{2}\left(\mathbb {R}^{n}\right) \rightarrow L^{2}\left(\mathbb {R}^{n}\right)$ by

Show that the linear operator $T$ is properly defined, and determine a $c \geq 0$ for which