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Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering

Ole Christensen

Chapter 5

The Lp-spaces - all with Video Answers

Educators

Chapter Questions

Problem 1

This exercise concerns the proof of Lemma 5.1.4. Consider the functions $\left\{f_k\right\}_{k=1}^{\infty}$ defined in (5.5).
(i) Make a draft of the functions $f_1, f_2$, and $f_3$.
(ii) Make a draft of the functions $f_2-f_1$ and $f_3-f_2$.
(iii) Show that $\left\{f_k\right\}_{k=1}^{\infty}$ is a Cauchy sequence in $C_c(\mathbb{R})$.

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15:42

Problem 2

Consider the functions
$$
f_1(x)=e^{-|x|}, \quad f_2(x)=\min \left(0, x^2-1\right) .
$$
Make a sketch of the functions $f_1$ and $f_2$, and check for each of the functions whether it belongs to the vector spaces $C_c(\mathbb{R}), C_0(\mathbb{R})$, or $L^1(\mathbb{R})$.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:11

Problem 3

Consider the following functions defined on $\mathbb{R}$ :
$$
\begin{aligned}
& f_1(x)=e^{-x^2}, \\
& f_2(x)=e^{-x}, \\
& f_3(x)=x^3+2 x+4, \\
& f_4(x)=\sin (x), \\
& f_5(x)=\sin (x) \chi_{[-2,2]}(x), \\
& f_6(x)=\sin (x) \chi_{[-2 \pi, 2 \pi]}(x), \\
& f_7(x)=\frac{1}{1+x^2}, \\
& f_8(x)= \begin{cases}x & \text { if } x \in]-1,1], \\
2-x & \text { if } x \in[1,3], \\
0 & \text { otherwise. }\end{cases}
\end{aligned}
$$
(i) Make a rough sketch of the graph of each of the functions.
(ii) Determine the support for each of the functions. Which functions have compact support?
(iii) Which functions belong to $C_0(\mathbb{R})$ ?
(iv) Which functions belong to $C_c(\mathbb{R})$ ?
(v) Which functions belong to $L^1(\mathbb{R})$ ?

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:32

Problem 4

We consider the vector space $C_c(\mathbb{R})$, equipped with the $\|\cdot\|_2$-norm.
(i) Show that
$$
\langle f, g\rangle=\int_{-\infty}^{\infty} f(x) \overline{g(x)} d x
$$
defines an inner product on $C_c(\mathbb{R})$.
(ii) Show that $C_c(\mathbb{R})$ does not form a Hilbert space with respect to the norm $\|f\|_2=\sqrt{\langle f, f\rangle}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:19

Problem 5

Verify that $L^1(\mathbb{R})$ is a vector space.

Anthony Ramos
Anthony Ramos
Numerade Educator
00:55

Problem 6

A set $\Gamma \subset \mathbb{R}$ is countable if its elements can be written as a list,
$$
\Gamma=\left\{x_1, x_2, \ldots\right\} .
$$
Note that in order to show that a set $\Gamma$ is countable, we need to specify a procedure guaranteeing that each element in $\Gamma$ appears somewhere in the list. Show that the sets $\mathbb{N}, \mathbb{Z}$, and $\mathbb{Q}$ are countable.

Angelo Rendina
Angelo Rendina
Numerade Educator
01:20

Problem 7

Prove the result in Example 5.3.4.

Manik Pulyani
Manik Pulyani
Numerade Educator

Problem 8

Show that for $p \in[1, \infty[$, the expression (5.23) defines a norm on $L^p(\mathbb{R})$. (Hint: use Minkowski's inequality, see Theorem 1.7.1.)

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02:16

Problem 9

This exercise relates $L^2(\mathbb{R})$ and $L^1(\mathbb{R})$.
(i) Show that $L^1(\mathbb{R})$ is not a subspace of $L^2(\mathbb{R})$ (Hint: find a concrete function belonging to $L^1(\mathbb{R})$ but not to $L^2(\mathbb{R})$.)
(ii) Show that $L^2\left(\mathbb{R}\right.$ ) is not a subspace of $L^1(\mathbb{R})$
(iii) Assume that $f \in L^2(\mathbb{R})$ has compact support. Show that $f \in L^1(\mathbb{R})$; in particular, this shows that
$$
L^2(\mathbb{R}) \cap C_c(\mathbb{R}) \subset L^1(\mathbb{R}) .
$$

Uma Kumari
Uma Kumari
Numerade Educator

Problem 10

This exercise relates $L^1(0,1)$ and $L^2(0,1)$.
(i) Show that $L^2(0,1) \subset L^1(0,1)$.
(ii) Show that if a sequence of functions $f_k$ in $L^2(0,1)$ converges to 0 in $L^2(0,1)$ as $k \rightarrow \infty$, then $f_k \rightarrow 0$ in $L^1(0,1)$ as $k \rightarrow \infty$.

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02:18

Problem 11

Consider the functions $f_k, k \in \mathbb{N}$, defined by
$$
f_k(x)= \begin{cases}1 / k & \text { if } x \in[0, k], \\ 0 & \text { otherwise. }\end{cases}
$$
(i) Make a sketch of the first few functions $f_k$.
(ii) Show that $f_k \rightarrow 0$ in $L^{\infty}(\mathbb{R})$ as $k \rightarrow \infty$.
(iii) Show that $f_k \in L^1(\mathbb{R})$ for all $k$.
(iv) Does it hold that
$$
f_k \rightarrow 0 \text { in } L^1(\mathbb{R}) \text { as } k \rightarrow \infty \text { ? }
$$

Dwijendra Rao
Dwijendra Rao
Numerade Educator

Problem 12

Consider a subinterval $] a, b\left[\subseteq \mathbb{R}\right.$, and functions $f, f_1, f_2, f_3, \ldots$ defined on $] a, b[$. Assume that
$$
\left\|f-f_k\right\|_{L^{\infty}(a, b)} \rightarrow 0 \text { as } k \rightarrow \infty .
$$
(i) Assume that the interval $] a, b[$ is finite. Show that for all $p \in[1, \infty[$, the assumption (5.29) implies that
$$
\left\|f-f_k\right\|_{L^p(a, b)} \rightarrow 0 \text { as } k \rightarrow \infty .
$$
(ii) Assume that $] a, b[=\mathbb{R}$. Show that regardless of the choice of $p \in[1, \infty[$, the assumption (5.29) does not imply that (5.30) holds.

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02:52

Problem 13

Consider the following functions defined on $\mathbb{R}$ :
$$
\begin{aligned}
f_1(x) & =e^x \chi_{[0, \infty[}(x), \\
f_2(x) & =e^{-x} \chi_{[0, \infty[}(x), \\
f_3(x) & =\frac{1}{\sqrt{x}} \chi_{[1, \infty[}(x), \\
f_4(x) & =\frac{1}{\sqrt{x}} \chi_{] 0,1[}(x) .
\end{aligned}
$$
(i) Which functions belong to $L^{\infty}(\mathbb{R})$ ?
(ii) For each of the functions $f_k, k=1, \ldots, 4$, determine the exact range of parameters $p \in\left[1, \infty\left[\right.\right.$ for which $f_k \in L^p(\mathbb{R})$.

Rae Xin
Rae Xin
Numerade Educator
01:15

Problem 14

The purpose of the exercise is show how certain functions in $L^1(\mathbb{R})$ can be approximated by functions in $C_c(\mathbb{R})$.
(i) Let $f(x):=\chi_{[0,1]}(x)$. Argue that for any $\epsilon>0$ there exists a function $g \in C_c(\mathbb{R})$ such that $\|f-g\|_1 \leq \epsilon$.
(ii) The function $f(x):=\left(1+x^2\right)^{-1}$ belongs to $L^1(\mathbb{R})$ by the result in Exercise 5.3. Show that for any $\epsilon>0$ there exists a function $g \in C_c(\mathbb{R})$ such that $\|f-g\|_1 \leq \epsilon$.
(iii) Argue (in words) how any bounded piecewise continuous function in $L^1(\mathbb{R})$ can be approximated by a function in $C_c(\mathbb{R})$.

Carson Merrill
Carson Merrill
Numerade Educator

Problem 15

This exercise relates the spaces $C_0(\mathbb{R}), C_c(\mathbb{R})$, and $L^p(\mathbb{R})$.
(i) Show that $C_c(\mathbb{R})$ is a subspace of $L^p(\mathbb{R})$ for all $p \in[1, \infty[$.
(ii) Show that $C_0(\mathbb{R})$ is not a subspace of $L^p(\mathbb{R})$ for any $p \in[1, \infty[$.

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Problem 16

Let $\left.w: \mathbb{R} \rightarrow\right] 0, \infty[$ be a continuous function, and define for $p \in] 1, \infty\left[\right.$ the vector space $L_w^p(\mathbb{R})$ by
$$
L_w^p(\mathbb{R})=\left\{f:\left.\mathbb{R} \rightarrow \mathbb{C}\left|\int_{-\infty}^{\infty}\right| f(x)\right|^p w(x) d x<\infty\right\} .
$$
(i) Show that
$$
\|f\|_{L_w^p(\mathbb{R})}:=\left(\int_{-\infty}^{\infty}|f(x)|^p w(x) d x\right)^{1 / p}
$$
defines a norm on $L_w^p(\mathbb{R})$.
(ii) Use the fact that $L^p(\mathbb{R})$ is a Banach space to show that $L_w^p(\mathbb{R})$ equipped with the norm in (5.31) is a Banach space.

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Problem 17

Let $p \in[1, \infty[$ and consider the mapping
$$
T: L^P(\mathbb{R}) \rightarrow L^P(\mathbb{R}),(T f)(x):=f(3 x+2) .
$$
(i) Show that $T$ indeed maps $L^p(\mathbb{R})$ into $L^p(\mathbb{R})$.
(ii) Show that $T$ is linear and bounded.
(iii) Consider the function
$$
f(x):=\frac{1}{x^2} \chi_{[1, \infty]}(x),
$$
and show that $f \in L^p(\mathbb{R})$ for all $p \in[1, \infty[$.

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Problem 18

Let $p \in[1, \infty[$ and consider the dilation operator
$$
D: L^p(\mathbb{R}) \rightarrow L^p(\mathbb{R}),(D f)(x)=2^{1 / 2} f(2 x) .
$$
(i) Show that $D$ actually maps $L^p(\mathbb{R})$ into $L^p(\mathbb{R})$.
(ii) Show that $D$ is linear and bounded.
(iii) Find, as a function of $p$, the norm of the operator $D$.

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Problem 19

Consider the mapping
$$
T: L^1(0,2) \rightarrow L^1(0,2),(T f)(x):=\int_0^x t f(t) d t .
$$
(i) Show that $T$ indeed maps $L^1(0,2)$ into $L^1(0,2)$.
(ii) Show that $T$ is linear and bounded.

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Problem 20

Let $p \in[1, \infty[$ and consider the mapping
$$
T: L^p(-2,2) \rightarrow L^p(-2,2),(T f)(x):=x f(x) .
$$
(i) Show that $T$ indeed maps $L^p(-2,2)$ into $L^p(-2,2)$.
(ii) Show that $T$ is linear and bounded.
(iii) Calculate the norm of the operator $T$.
$$
\frac{\|T f\|_p}{\|f\|_p}
$$
as $\epsilon \rightarrow 2$.

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Problem 21

Let $c>0$ be given and consider the mapping
$$
D_c: C_c(\mathbb{R}) \rightarrow C_c(\mathbb{R}),\left(D_c f\right)(x):=\frac{1}{\sqrt{c}} f\left(\frac{x}{c}\right), x \in \mathbb{R} .
$$
(i) Argue that $D_c$ actually maps $C_c(\mathbb{R})$ into $C_c(\mathbb{R})$.
(ii) Show that $D_c$ is linear.
(iii) Show that $D_c$ is bounded as operator from $C_c(\mathbb{R})$ into $C_c(\mathbb{R})$ (as usual, $C_c(\mathbb{R})$ is equipped with the $\|\cdot\|_{\infty}$-norm).
In Section 6.2 the operator $D_c$ will be considered on $L^2(\mathbb{R})$.

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