Math Problems Vault

Problem 10.1

Show that the Fourier transform $\mathcal{F}$ and the inverse Fourier transform $\mathcal{F}^{-1}$ each map $L_{[-\Omega , \Omega ]}^{2}(\mathbb {R})$ unitarily onto $\mathcal{F} L_{[-\Omega , \Omega ]}^{2}(\mathbb {R})$ .

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Problem 10.2

Show that the dilation operator $D_{r}$ maps $\mathcal{F} L_{[-\Omega , \Omega ]}^{2}(\mathbb {R})$ unitarily onto $\mathcal{F} L_{[-r \Omega , r \Omega ]}^{2}(\mathbb {R})$ .

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Problem 10.3

(a)(b)

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(a)

Prove statement (a) in Theorem 10.4.

(b)

Show that $\mathrm{PW}(\mathbb {R}) \subseteq C^{\infty }(\mathbb {R})$ by justifying the statements made in the proof of Theorem 10.4(d).

Problem 10.4

(a)(b)(c)(d)

Given $f \in \mathrm{PW}(\mathbb {R})$ , prove the following statements.

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(a)

The $n$ th derivative of $f$ can be written as

f^{(n)}(x) = \int _{-1 / 2}^{1 / 2}(2 \pi i \xi )^{n} \widehat{f}(\xi ) e^{2 \pi i \xi x} d \xi

(b)

$\left\| f^{(n)}\right\|_{\infty } \leq \pi^{n}\left\| \widehat{f}\right\|_{L^{1}}$ for each $n \geq 0$ .

(c)

Given $a \in \mathbb {R}$ , the Taylor series for $f$ about the point $a$ converges to $f(x)$ for every $x$ , i.e.,

f(x) = \sum _{n = 0}^{\infty } \frac{f^{(n)}(a)}{n!}(x-a)^{n}, \quad x \in \mathbb {R}

Remark: This says that $f$ is a real analytic function on $\mathbb {R}$ . The Paley-Wiener Theorem implies more, in particular $f$ has an extension to a function that is complex analytic on $\mathbb {C}$ .

(d)

Use part (c) to give another proof of Corollary 10.6.

Problem 10.5

Prove Theorem 10.8.

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Remark.

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Theorem 10.8: Let $\left\{ \lambda_{n}\right\}_{n \in \mathbb {N}}$ be any sequence of real numbers such that $\left\{ e_{\lambda_{n}}\right\}_{n \in \mathbb {N}}$ is a frame for $L^{2}(\mathbf{T})$ . Let $\left\{ \widetilde{e}_{n}\right\}_{n \in \mathbb {N}}$ be the canonical dual frame of $\left\{ e_{\lambda_{n}}\right\}_{n \in \mathbb {N}}$ in $L^{2}(\mathbf{T})$ , and define $\widetilde{s}_{n} = \left(\widetilde{e}_{n} \chi_{\left[-\frac{1}{2}, \frac{1}{2}\right]}\right)^{\vee }$ . Then for any $f \in \mathrm{PW}(\mathbb {R})$ we have $f = \sum_{n} f(\lambda_{n}) \widetilde{s}_{n}$ , where the series converges in $L^{2}$ -norm.

Problem 10.6

Show that if $b > 1$ then a function $f \in \mathrm{PW}(\mathbb {R})$ is not uniquely determined by the sequence of sample values $\left\{ f(b n)\right\}_{n \in \mathbb {Z}}$ .

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Problem 10.7

Prove that the subspace $H_{\varphi }$ defined in equation (10.6) is a closed subspace of $L^{2}(\mathbf{T})$ .

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Problem 10.8

Complete the proof of Theorem 10.10: let $\varphi \in L^{2}(\mathbf{T})$ and let $\mathcal{E}(\varphi ) = \left\{ \varphi e_{n}\right\}_{n \in \mathbb {Z}}$ .

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(a)

$\mathcal{E}(\varphi )$ is complete in $L^{2}(\mathbf{T})$ if and only if $\varphi (x) \neq 0$ a.e.

(b)

$\mathcal{E}(\varphi )$ is minimal in $L^{2}(\mathbf{T})$ if and only if $1/\varphi \in L^{2}(\mathbf{T})$ (in which case $\mathcal{E}(\varphi )$ is exact, biorthogonal to $\mathcal{E}(\widetilde{\varphi })$ where $\widetilde{\varphi } = 1/\varphi$ ).

(c)

$\mathcal{E}(\varphi )$ is a Bessel sequence in $L^{2}(\mathbf{T})$ if and only if $\varphi \in L^{\infty }(\mathbf{T})$ .

(d)

$\mathcal{E}(\varphi )$ is a frame sequence in $L^{2}(\mathbf{T})$ if and only if there exist $A, B > 0$ such that $A \leq \left|\varphi (x)\right|^{2} \leq B$ for a.e. $x \notin Z_{\varphi }$ (with closed span $H_{\varphi } = \left\{ f: f = 0\text{ a.e. on }Z_{\varphi }\right\}$ ).

(e)

$\mathcal{E}(\varphi )$ is an unconditional basis for $L^{2}(\mathbf{T})$ if and only if there exist $A, B > 0$ such that $A \leq \left|\varphi (x)\right|^{2} \leq B$ a.e. (in which case it is a Riesz basis).

(f)

$\mathcal{E}(\varphi )$ is an orthonormal basis for $L^{2}(\mathbf{T})$ if and only if $\left|\varphi (x)\right| = 1$ a.e.

Problem 10.9

Let $\varphi (t) = \left|t-\frac{1}{2}\right|^{\alpha }$ where $0 < \alpha < 1 / 2$ , as in Example 5.13. Show that $\mathcal{E}(\varphi )$ is exact in $L^{2}(\mathbf{T})$ but cannot be an unconditional basis for $L^{2}(\mathbf{T})$ .

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Problem 10.10

Justify the equality in equation (10.9).

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Problem 10.11

Given a sequence $c = \left(c_{k}\right)_{k \in \mathbb {Z}} \in \ell^{2}(\mathbb {Z})$ , let $T_{n} c$ denote the translated sequence $T_{n} c = \left(c_{k-n}\right)_{k \in \mathbb {Z}}$ . Characterize those sequences $c$ such that $\left\{ T_{n} c\right\}_{n \in \mathbb {Z}}$ is complete, Bessel, minimal, a frame, a Schauder basis, a Riesz basis, or an orthonormal basis for $\ell^{2}(\mathbb {Z})$ .

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Problem 10.12

Let $g \in L^{2}(\mathbb {R})$ and $a > 0$ be given, and define $h(x) = g(a x)$ . Show that $\left\{ g(x-a k)\right\}_{k \in \mathbb {Z}}$ is a frame sequence in $L^{2}(\mathbb {R})$ if and only if $\left\{ h(x-k)\right\}_{k \in \mathbb {Z}}$ is a frame sequence in $L^{2}(\mathbb {R})$ . What is the relation between the closed spans of these two systems?

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Problem 10.13

Given a function $f \in L^{1}(\mathbb {R})$ and given $a > 0$ , we call the function

\varphi (x) = \sum _{n \in \mathbb {Z}} f(x+a n)

the $a$ -periodization of $f$ (or simply the periodization if $a = 1$ ). Show that the series defining $\varphi$ converges absolutely in $L^{1}[0, a]$ , and

\int _{0}^{a} \varphi (x) d x = \int _{-\infty }^{\infty } f(x) d x

In particular, the bracket product of $f, g \in L^{2}(\mathbb {R})$ is the function $[f, g] \in L^{1}(\mathbf{T})$ defined by

[f, g](x) = \sum _{n \in \mathbb {Z}} f(x+n) \overline{g(x+n)}

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Problem 10.14

Show that if $g \in L^{2}(\mathbb {R})$ is not the zero function, then $\left\{ T_{a} g\right\}_{a \in \mathbb {R}}$ is finitely linearly independent.

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Problem 10.15

Suppose that $g \in L^{2}(\mathbb {R})$ is such that $\mathcal{M}(\widehat{g}) = \left\{ e^{-2 \pi i k \xi } \widehat{g}(\xi )\right\}_{k \in \mathbb {Z}}$ is a Bessel sequence in $L^{2}(\mathbb {R})$ . By Theorem 7.2, if $\left(c_{k}\right)_{k \in \mathbb {Z}} \in \ell^{2}(\mathbb {Z})$ then the series $F(\xi ) = \sum_{k \in \mathbb {Z}} c_{k} e^{-2 \pi i k \xi } \widehat{g}(\xi )$ converges unconditionally in $L^{2}(\mathbb {R})$ . Also, since $\left\{ e^{-2 \pi i k \xi }\right\}_{k \in \mathbb {Z}}$ is an orthonormal basis for $L^{2}(\mathbf{T})$ the series $\hat{c}(\xi ) = \sum_{k \in \mathbb {Z}} c_{k} e^{-2 \pi i k \xi }$ converges unconditionally in $L^{2}(\mathbf{T})$ . Show that $F(\xi ) = \widehat{c}(\xi ) \widehat{g}(\xi )$ a.e.

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Problem 10.16

Prove the remaining statements in Theorem 10.19.

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(a)

Show that $\mathcal{T}(g)$ is a Bessel sequence in $V_{0}(g)$ if and only if $\Phi_{g} \in L^{\infty }(\mathbf{T})$ , and in this case $\Phi_{g} \leq B$ a.e. where $B$ is a Bessel bound.

(b)

Show that $\mathcal{T}(g)$ is a frame for $V_{0}(g)$ if and only if there exist $A, B > 0$ such that $A \leq \Phi_{g}(\xi ) \leq B$ a.e. $\xi \notin Z_{\Phi_{g}}$ . In this case $A, B$ are frame bounds for $\mathcal{T}(g)$ .

(c)

Show that $\mathcal{T}(g)$ is minimal in $V_{0}(g)$ if and only if $1 / \Phi_{g} \in L^{1}(\mathbf{T})$ , and in this case $\mathcal{T}(g)$ is exact in $V_{0}(g)$ .

(d)

Show that $\mathcal{T}(g)$ is an unconditional basis for $V_{0}(g)$ if and only if there exist $A, B > 0$ such that $A \leq \Phi_{g} \leq B$ a.e., and in this case it is a Riesz basis for $V_{0}(g)$ .

(e)

Show that $\mathcal{T}(g)$ is an orthonormal basis for $V_{0}(g)$ if and only if $\Phi_{g}(t) = 1$ a.e.

(f)

Show that, with respect to the ordering $\left\{ 0,-1,1,-2,2, \ldots \right\}$ of the index set $\mathbb {Z}$ , $\mathcal{T}(g)$ is a Schauder basis for $V_{0}(g)$ if and only if $\Phi_{g} \in \mathcal{A}_{2}(\mathbf{T})$ .

Problem 10.17

(a)(b)

Suppose that $g \in L^{2}(\mathbb {R})$ is such that $\mathcal{T}(g)$ is a frame sequence in $L^{2}(\mathbb {R})$ , and let $S: V_{0}(g) \rightarrow V_{0}(g)$ be its frame operator.

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(a)

Show that $S\left(T_{k} f\right) = T_{k}(S f)$ for each $f \in V_{0}(g)$ . Use this to show that the canonical dual frame in $V_{0}(g)$ is $\mathcal{T}(h)$ where $h = S^{-1} g$ .

(b)

Show that

f \in V_{0}(g) \Longleftrightarrow \exists m \in L^{2}(\mathbf{T}) \text{ such that } \widehat{f} = m \widehat{g} \text{ a.e. }

Problem 10.18

(a)(b)

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(a)

Suppose $g \in L^{2}(\mathbb {R})$ and $\widehat{g}$ is nonzero almost everywhere. Show that if $f \in V_{0}(g)$ then there is a 1 -periodic function $p$ such that $\widehat{f} = p \widehat{g}$ a.e.

(b)

Given an an arbitrary function $g$ in $L^{2}(\mathbb {R})$ , show that $\mathcal{T}(g)$ is incomplete in $L^{2}(\mathbb {R})$ .

Problem 10.19

(a)(b)

Suppose that $f, g \in L^{2}(\mathbb {R})$ are such that $\mathcal{T}(f)$ and $\mathcal{T}(g)$ are both orthonormal sequences.

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(a)

Show that $\mathcal{T}(f)$ and $\mathcal{T}(g)$ have the same closed spans if and only if $\widehat{f}(\xi ) = \alpha (\xi ) \widehat{g}(\xi )$ a.e., where $\alpha$ is 1 -periodic and $\left|\alpha (\xi )\right| = 1$ a.e.

(b)

Show that if $f$ and $g$ are each compactly supported, then $\overline{\operatorname {span}}(\mathcal{T}(f)) = \overline{\operatorname {span}}(\mathcal{T}(g))$ if and only $f = \alpha T_{k} g$ where $k \in \mathbb {Z}$ and $\alpha$ is a scalar with $\left|\alpha \right| = 1$ .

Problem 10.20

(a)(b)(c)

Let $g \in L^{2}(\mathbb {R})$ be given.

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(a)

Show that if $g$ is compactly supported (zero almost everywhere outside of some finite interval), then $\mathcal{T}(g)$ is a Bessel sequence but cannot be a redundant frame for its closed span $V_{0}(g)$ (compare Exercise 10.22).

(b)

Show that if $\widehat{g}$ is continuous and compactly supported, then $\mathcal{T}(g)$ is a Bessel sequence but cannot be a redundant frame for $V_{0}(g)$ .

(c)

Find a function $g \in L^{2}(\mathbb {R})$ such that $\mathcal{T}(g)$ is a redundant frame for $V_{0}(g)$ .

Problem 10.21

Suppose that $g \in L^{2}(\mathbb {R})$ is such that $\mathcal{T}(g)$ is a frame sequence in $L^{2}(\mathbb {R})$ , and let $g^{\sharp } \in L^{2}(\mathbb {R})$ be the function whose Fourier transform is

\widehat{g^{\sharp }}(\xi ) = \begin{cases} \widehat{g}(\xi ) \Phi _{g}(\xi )^{-1 / 2}, & \xi \notin Z_{\Phi _{g}}, \\ 0, & \xi \in Z_{\Phi _{g}}.\end{cases}

Show that $g^{\sharp } \in V_{0}(g) = \overline{\operatorname {span}}(\mathcal{T}(g))$ , and $\mathcal{T}\left(g^{\sharp }\right)$ is a Parseval frame for $V_{0}(g)$ . Show further that if $\mathcal{T}(g)$ is a Riesz basis for $V_{0}(g)$ then $\mathcal{T}\left(g^{\sharp }\right)$ is an orthonormal basis for $V_{0}(g)$ .

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Problem 10.22

Let $w(x) = \max \left\{ 1-\left|x\right|, 0\right\}$ be the "hat" or "tent" function on $[-1,1]$ . Use Lemma 10.20 to show that

\Phi _{w}(\xi ) = \frac{2+\cos 2 \pi \xi }{3} = \frac{1+2 \cos ^{2} \pi \xi }{3},

and conclude that $\mathcal{T}(w)$ is a Riesz basis for its closed span $V_{0}(g)$ . Find $\widehat{\widetilde{w}}, \Phi_{\widetilde{w}}$ , $\widehat{w^{\sharp }}$ , and $\left[\widehat{w^{\sharp }}, \widehat{\widetilde{w}}\right]$ .

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Problem 10.23

Suppose that $g \in L^{2}(\mathbb {R})$ and $a_{k} \in \mathbb {R}$ are such that $\left\{ g\left(x-a_{k}\right)\right\}_{k \in \mathbb {N}}$ is a Schauder basis for $L^{2}(\mathbb {R})$ . Show that the sequence $\left(a_{k}\right)_{k \in \mathbb {N}}$ must be uniformly separated, i.e., $\inf_{j \neq k}\left|a_{j}-a_{k}\right| > 0$ .

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