Ross Ogilvie      7th October, 2024
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17.

Convoluted

The convolution of two functions $f,g:{ℝ}^n\to ℝ$ is defined by

$$(f\ast g)(x):={\int \; <!--\; nolimits\; -->}_{{ℝ}^n}f(y)g(x-y)\mathit{𝑑𝑦}.$$

(a)

Let $f_n(x)=0.5n$ for $x\in [-n^{-1},n^{-1}]$ and $0$ otherwise. Show that the following bounds hold

$$\underset{|y|\le n^{-1}}{\inf <!--\; FUNCTION\; APPLICATION\; -->}g(y)\le (g\ast f_n)(0)\le \underset{|y|\le n^{-1}}{\sup <!--\; FUNCTION\; APPLICATION\; -->}g(y).$$

(2 points)

(b)

Suppose now that $g$ is continuous. Show that $(g\ast f_n)(0)\to g(0)$ as $n\to \infty$. (2 points)

(c)

Show that the convolution of $C_0^{\infty }$-functions on ${ℝ}^n$ is a bilinear, commutative, and associative operation. (1+2+2 Points)

18.

Distributions

(a)

Choose any compact set $K\subset ℝ$. Since it is bounded, there exists $R>0$ with $K\subseteq [-R,R]$. Now choose any test function $\varphi \in C_0^{\infty }(ℝ)$ with compact support in $K$. Since it is continuous, $\underset{x\in K}{\sup <!--\; FUNCTION\; APPLICATION\; -->}|\varphi (x)|$ is finite. Prove the following inequality (1 point)

$$\left|{\int \; <!--\; nolimits\; -->}_0^{\infty }\varphi (x)dx\right|\le 2R\underset{x\in K}{\sup <!--\; FUNCTION\; APPLICATION\; -->}|\varphi (x)|.$$

(b)

Define the Heaviside function $H:ℝ\to ℝ$ by $H(x):=1$ for $x\ge 0$ and $H(x):=0$ for $x<0$. Show that the distribution associated to the Heaviside function

$$F_H:C_0^{\infty }(ℝ)\to ℝ,\varphi \mapsto {\int \; <!--\; nolimits\; -->}_0^{\infty }\varphi (x)dx$$

is in fact a distribution on $ℝ$ using part (a) and Definition 2.14 directly. (1 point)

(c)

Calculate the first and second derivatives of $H$ as a distribution. If they are regular distributions, describe the corresponding function. (2 points)

(d)

Consider the circle $C=\{x^2+y^2=1\}\subset {ℝ}^2$. Show that

$$G(\phi ):={\int \; <!--\; nolimits\; -->}_C\phi \mathit{𝑑𝜎}$$

defines a distribution in ${𝒟}^{\prime }({ℝ}^2)$. Note that the $\mathit{𝑑𝜎}$ indicates this is an integration over the submanifold $C$. Does there exist a locally integrable function $g:{ℝ}^2\to ℝ$ with

$$G(\phi )={\int \; <!--\; nolimits\; -->}_{{ℝ}^2}g\phi dx$$

for all $\phi \in C_0^{\infty }(ℝ)$? (Hint. Use Lemma 2.15) (2 Points + 2 Bonus Points)

19.

Delta Quadrant

(a)

Prove that the support of the delta distribution $\delta$ is $\{0\}$. (2 points)

(b)

Argue from Lemma 2.12 that $\delta$ is the limit of the standard mollifier ${({\lambda }_{𝜖})}_{𝜖>0}$ as $𝜖\downarrow 0$ as a sequence of distributions. (1 point)

(c)

Prove for any distribution $F$ that the convolution with $\delta$ is again $F$. (2 points)