Math Problems Vault

Problem 13.1

(a)(b)

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(a)

Prove that every function in $C(\mathbf{T})$ is uniformly continuous, and use this to prove that translation is strongly continuous on $C(\mathbf{T})$ , i.e.,

\forall f \in C(\mathbf{T}), \quad \lim _{a \rightarrow 0}\left\| T_{a} f-f\right\| _{\infty } = 0.

(b)

Use the fact that $C(\mathbf{T})$ is dense in $L^{p}(\mathbf{T})$ to prove that translation is strongly continuous on $L^{p}(\mathbf{T})$ when $1 \leq p < \infty$ , i.e.,

\forall 1 \leq p < \infty , \quad \forall f \in L^{p}(\mathbf{T}), \quad \lim _{a \rightarrow 0}\left\| T_{a} f-f\right\| _{L^{p}} = 0.

Problem 13.2

Prove Lemma 13.5, and also derive analogous formulas for the inverse Fourier transforms $\left(T_{a} f\right)^{\vee },\left(M_{m} f\right)^{\vee },\left(T_{m} c\right)^{\vee }$ , and $\left(M_{a} c\right)^{\vee }$ .

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Problem 13.3

This exercise will show how the Fourier transform transforms smoothness of a function $f$ into decay of its Fourier coefficients. Show that if $f \in C^{1}(\mathbf{T})$ then $\widehat{f^{\prime }}(n) = 2 \pi i n \widehat{f}(n)$ for $n \in \mathbb {Z}$ , and use this to show that

\left|\widehat{f}(n)\right| \leq \frac{\left\| f^{\prime }\right\| _{L^{1}}}{2 \pi \left|n\right|}, \quad n \neq 0

Extend to higher derivatives, and compare Theorem 9.16.

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Problem 13.4

This exercise will show how the Fourier transform transforms decay of a sequence $c$ into smoothness of its Fourier transform. Show that if $c = \left(c_{n}\right)_{n \in \mathbb {Z}}$ satisfies $\sum_{n \in \mathbb {Z}}\left|n c_{n}\right| < \infty$ , then $\hat{c}(\xi ) = \sum_{n \in \mathbb {Z}} c_{n} e^{-2 \pi i n x}$ is differentiable and

\widehat{c}^{\prime }(\xi ) = -2 \pi i \sum _{n \in \mathbb {Z}} n c_{n} e^{-2 \pi i n x} = \widehat{d}(\xi ),

where $d = \left(-2 \pi i n c_{n}\right)_{n \in \mathbb {Z}} \in \ell^{1}(\mathbb {Z})$ . Extend to higher derivatives, and compare Theorem 9.15.

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Problem 13.5

Let $p(x) = \sum_{k = 0}^{M} a_{k} x^{k}$ and $q(x) = \sum_{k = 0}^{N} b_{k} x^{k}$ be two polynomials. Their pointwise product is a polynomial of the form $p(x) q(x) = \sum_{n = 0}^{M+N} c_{n} x^{n}$ . Find an explicit formula for $c_{n}$ , and explain its relation to the convolution of the sequences $\left(a_{k}\right)_{k \in \mathbb {Z}}$ and $\left(b_{k}\right)_{k \in \mathbb {Z}}$ , where we take $a_{k} = 0$ if $k \neq 0, \ldots , M$ and similarly for $b_{k}$ .

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Problem 13.6

Show that $f, g \in L^{1}(\mathbf{T})$ does not imply $f g \in L^{1}(\mathbf{T})$ .

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Problem 13.7

Prove Theorem 13.8 for the cases $p = 1$ and $p = \infty$ .

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Remark.

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Theorem 13.8 (Young's Inequality): Fix $1 \leq p \leq \infty$ . If $f \in L^{p}(\mathbf{T})$ and $g \in L^{1}(\mathbf{T})$ then $f * g \in L^{p}(\mathbf{T})$ , and we have $\left\| f * g\right\|_{L^{p}} \leq \left\| f\right\|_{L^{p}}\left\| g\right\|_{L^{1}}$ .

Definition 13.6: The convolution of measurable 1-periodic functions $f$ and $g$ is the function $f * g$ given by $(f * g)(x) = \int_{0}^{1} f(y) g(x-y)\, d y$ , whenever this integral is well defined.

Problem 13.8

Prove Theorem 13.9.

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Remark.

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Theorem 13.9 (Young's Inequality): Fix $1 \leq p \leq \infty$ . If $f \in \ell^{p}(\mathbb {Z})$ and $g \in \ell^{1}(\mathbb {Z})$ then $f * g \in \ell^{p}(\mathbb {Z})$ , and we have $\left\| f * g\right\|_{\ell^{p}} \leq \left\| f\right\|_{\ell^{p}}\left\| g\right\|_{\ell^{1}}$ .

Problem 13.9

(a)(b)

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(a)

Justify the use of Fubini's Theorem in the proof of part (a) of Theorem 13.10.

(b)

Prove part (b) of that theorem.

Problem 13.10

Prove Lemma 13.11, and establish analogous properties for the convolution of sequences.

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Remark.

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Lemma 13.11: The following facts hold for $f, g, h \in L^{1}(\mathbf{T})$ .

Commutativity: $f * g = g * f$ .
Associativity: $(f * g) * h = f *(g * h)$ .
Distributive laws: $f *(g+h) = f * g+f * h$ .
Commutativity with translations: $f *\left(T_{a} g\right) = \left(T_{a} f\right) * g = T_{a}(f * g)$ , $a \in \mathbb {R}$ . Here $T_{a} f(x) = f(x-a)$ is translation, and the convolution of 1-periodic functions is $(f * g)(x) = \int_{0}^{1} f(y) g(x-y)\, d y$ (Definition 13.6). The convolution of sequences $a = \left(a_{k}\right)_{k \in \mathbb {Z}}$ and $b = \left(b_{k}\right)_{k \in \mathbb {Z}}$ is the sequence $a * b$ given by $(a * b)_{k} = \sum_{j \in \mathbb {Z}} a_{j} b_{k-j}$ , $k \in \mathbb {Z}$ (Definition 13.7).

Problem 13.11

Given $f \in L^{1}(\mathbf{T})$ , set $f^{*}(x) = f(-x)$ and show that $\widehat{f^{*}}(n) = \widehat{f}(-n) = \stackrel{\vee }{f}(n)$ . Conclude that if $f$ is even then $\widehat{f}$ is even and $\widehat{f} = \stackrel{\vee }{f}$ .

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Problem 13.12

Let $\delta = \delta_{0} = \left(\delta_{0 n}\right)_{n \in \mathbb {Z}}$ . Show that $\delta$ is an identity for convolution on $\ell^{p}(\mathbb {Z})$ , i.e., $c * \delta = c$ for every $c \in \ell^{p}(\mathbb {Z})$ , where $1 \leq p \leq \infty$ .

Remark We will see in Corollary 13.15 that there is no element of $L^{1}(\mathbf{T})$ that is an identity for convolution on $L^{p}(\mathbf{T})$ .

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Problem 13.13

(a)(b)

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(a)

Show that if $f \in L^{1}(\mathbf{T})$ and $g \in C(\mathbf{T})$ then $f * g \in C(\mathbf{T})$ .

(b)

Prove that convolution commutes with differentiation in the following sense: If $f \in L^{1}(\mathbf{T})$ and $g \in C^{1}(\mathbf{T})$ then $f * g \in C^{1}(\mathbf{T})$ , and $(f * g)^{\prime } = f * g^{\prime }$ . Extend to $g \in C^{m}(\mathbf{T})$ and $g \in C^{\infty }(\mathbf{T})$ .

Problem 13.14

Fix $1 \leq p \leq \infty$ . Show that if $f \in L^{p}(\mathbf{T})$ and $g \in L^{p^{\prime }}(\mathbf{T})$ then $f * g \in C(\mathbf{T})$ .

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Problem 13.15

Design an approximate identity $\left\{ k_{N}\right\}$ such that $k_{N} \in C(\mathbf{T}), k_{N} \geq 0$ everywhere, and $k_{N} = 0$ on $\left[\frac{1}{N}, 1-\frac{1}{N}\right]$ . How smooth can you make $k_{N}$ ?

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Problem 13.16

Prove Theorem 13.13 for $1 < p < \infty$ and for $C(\mathbf{T})$ .

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Problem 13.17

The existence of infinitely differentiable, compactly supported functions is demonstrated in Exercise 11.9. Combine Exercises 13.13 and 13.15 with Theorem 13.13 to show that $C^{\infty }(\mathbf{T})$ is dense in $L^{p}(\mathbf{T})$ for $1 \leq p < \infty$ , and is dense in $C(\mathbf{T})$ with respect to the uniform norm. Conclude that $C^{m}(\mathbf{T})$ is dense in these spaces for each integer $m \geq 0$ .

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Problem 13.18

Fix $0 < \alpha < 1$ . Show that if $f \in C(\mathbf{T})$ is Hölder continuous with exponent $\alpha$ (see Exercise 1.23) then

\left|\widehat{f}(n)\right| \leq \frac{1}{2}\left(\frac{1}{2\left|n\right|}\right)^{\alpha }, \quad n \neq 0

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Problem 13.19

Prove Lemma 13.17.

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Remark.

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Lemma 13.17: Let $\chi_{N}$ denote the discrete characteristic function of the set $\left\{ -N, \ldots , N\right\}$ , i.e., $\chi_{N}(n) = \sum_{m = -N}^{N}\delta_{m}(n)$ (so $\chi_{N}(n) = 1$ for $-N \leq n \leq N$ and $\chi_{N}(n) = 0$ for $\left|n\right| > N$ ). Then for each $N \in \mathbb {N}$ we have $d_{N}(x) = \sum_{n = -N}^{N} e^{2 \pi i n x} = \frac{\sin \left((2 N+1)\pi x\right)}{\sin \pi x} = \widehat{\chi }_{N}(x) = \left(\chi_{N}\right)^{\vee }(x)$ .

Problem 13.20

Prove Theorem 13.18.

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Remark.

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Theorem 13.18: $d_{N} \in L^{1}(\mathbf{T})$ and $\int_{0}^{1} d_{N} = 1$ for each $N \in \mathbb {N}$ , and for $N > 1$ we have $\frac{4}{\pi^{2}} \sum_{k = 1}^{N} \frac{1}{k} \leq \left\| d_{N}\right\|_{L^{1}} \leq 3+\frac{4}{\pi^{2}} \sum_{k = 1}^{N} \frac{1}{k}$ .

Problem 13.21

(a)(b)(c)

Given a sequence of scalars $a = \left(a_{k}\right)_{k \in \mathbb {Z}}$ , let $s_{N} = \sum_{k = -N}^{N} a_{k}$ .

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(a)

Let $\sigma_{N} = \left(s_{0}+\cdots +s_{N}\right) /(N+1)$ denote the Cesàro means, and show that

\sigma _{N} = \sum _{n = -N}^{N}\left(1-\frac{\left|n\right|}{N+1}\right) a_{n}.

(b)

Show that if the partial sums $s_{N}$ converge, then the Cesàro means $\sigma_{N}$ converge to the same limit, i.e.,

\lim _{N \rightarrow \infty } \sum _{n = -N}^{N}\left(1-\frac{\left|n\right|}{N+1}\right) a_{n} = \lim _{N \rightarrow \infty } s_{N} = \sum _{n = -\infty }^{\infty } a_{n}.

(c)

Set $a_{n} = (-1)^{n}$ for $n \geq 0$ and $a_{n} = 0$ for $n < 0$ . Show that the series $\sum_{n \in \mathbb {Z}} a_{n}$ is Cesàro summable even though the partial sums do not converge, and find the limit of the Cesàro means.

Problem 13.22

Prove the equality appearing in equation (13.18).

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Problem 13.23

(a)(b)

Prove Lemma 13.20.

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(a)

$w_{N}(x)$ has the closed form $w_{N}(x) = \frac{1}{N+1}\left(\frac{\sin \pi (N+1) x}{\sin \pi x}\right)^{2}$ .

(b)

$\left\{ w_{N}\right\}$ forms an approximate identity on $\mathbf{T}$ .

Problem 13.24

(a)(b)

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(a)

Show that if $f \in L^{1}(\mathbf{T})$ and $\widehat{f} \in \ell^{2}(\mathbb {Z})$ , then $f \in L^{2}(\mathbf{T})$ .

(b)

Use part (a) to show that the Plancherel Equality remains true if we assume that $f$ belongs to $L^{1}(\mathbf{T})$ instead of the smaller space $L^{2}(\mathbf{T})$ . In other words, show that if $f \in L^{1}(\mathbf{T})$ , then we have

\sum _{n \in \mathbb {Z}}\left|\widehat{f}(n)\right|^{2} = \left\| f\right\| _{L^{2}}^{2},

in the sense that one side is finite if and only if the other side is finite, and in this case they are equal; otherwise both sides are infinite.

Problem 13.25

Let $f(x) = \pi^{2}\left(x^{2}-x+\frac{1}{6}\right)$ for $x \in [0,1)$ . When extended 1 -periodically, $f$ is a continuous function on $\mathbf{T}$ . Compute $\widehat{f}$ and show that $\widehat{f} \in \ell^{1}(\mathbb {Z})$ . Use this to show that

\sum _{n = 1}^{\infty } \frac{\cos 2 \pi n x}{n^{2}} = \pi ^{2}\left(x^{2}-x+\frac{1}{6}\right), \quad x \in [0,1] \tag {13.20}

where the series converges uniformly on $[0,1]$ . What does the series converge to for other $x$ ? Take $x = 0$ to obtain Euler's formula:

\sum _{n = 1}^{\infty } \frac{1}{n^{2}} = \frac{\pi ^{2}}{6}

Compare Exercise 1.55.

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Problem 13.26

Prove the following facts about the Fourier algebra $A(\mathbf{T})$ .

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(a)

$A(\mathbf{T})$ is a Banach space with respect to the norm $\left\| f\right\|_{A} = \left\| \widehat{f}\right\|_{\ell^{1}}$ , and $\mathcal{F}: f \mapsto \widehat{f}$ is an isometric isomorphism of $A(\mathbf{T})$ onto $L^{1}(\mathbf{T})$ .

(b)

$A(\mathbf{T})$ is closed under pointwise products, and $\left\| f g\right\|_{A} \leq \left\| f\right\|_{A}\left\| g\right\|_{A}$ (hence $A(\mathbf{T})$ is a Banach algebra with respect to pointwise products).

(c)

$C^{2}(\mathbf{T}) \subseteq A(\mathbf{T})$ , and therefore $A(\mathbf{T})$ is dense in $C(\mathbf{T})$ .

(d)

$L^{2}(\mathbf{T}) * L^{2}(\mathbf{T}) = A(\mathbf{T})$ .

(e)

$A(\mathbf{T})$ is a meager subset of $C(\mathbf{T})$ .