Math Problems Vault

Problem 9.1

Prove that the sinc function $d_{\pi }(\xi ) = \frac{\sin \pi \xi }{\pi \xi }$ does not belong to $L^{1}(\mathbb {R})$ .

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Problem 9.2

(a)(b)

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(a)

Prove that every function $f \in C_{0}(\mathbb {R})$ is uniformly continuous, and show that uniform continuity is equivalent to the statement

\lim _{a \rightarrow 0}\left\| T_{a} f-f\right\| _{L^{\infty }} = 0 \tag {9.21}

Show that equation (9.21) can fail if we only assume $f \in C_{b}(\mathbb {R})$ .

(b)

Show that if $1 \leq p < \infty$ and $f \in L^{p}(\mathbb {R})$ , then

\lim _{a \rightarrow 0}\left\| T_{a} f-f\right\| _{L^{p}} = 0

Because of this, we say that translation is strongly continuous on $L^{p}(\mathbb {R})$ for finite $p$ .

Problem 9.3

Prove that the equalities in equations (9.3) and (9.4) hold for all functions $f \in L^{1}(\mathbb {R})$ .

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Problem 9.4

Prove Theorem 9.15.

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Remark.

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Theorem 9.15: Given $f \in L^{1}(\mathbb {R})$ , if $x^{m} f(x) \in L^{1}(\mathbb {R})$ for some $m \in \mathbb {N}$ then $\widehat{f} \in C_{0}^{m}(\mathbb {R})$ , i.e., $\widehat{f}$ is $m$ -times differentiable and $\widehat{f}, \widehat{f}^{\prime }, \ldots , \widehat{f}^{(m)} \in C_{0}(\mathbb {R})$ . Furthermore, in this case $x^{k} f(x) \in L^{1}(\mathbb {R})$ for $k = 0, \ldots , m$ , and the $k$ th derivative of $\widehat{f}$ is the Fourier transform of $(-2 \pi i x)^{k} f(x)$ : $\widehat{f}^{(k)} = \frac{d^{k}}{d \xi^{k}}\widehat{f} = \widehat{\left((-2 \pi i x)^{k} f(x)\right)}$ for $k = 0, \ldots , m$ .

Problem 9.5

This exercise provides the details needed for the proof of Theorem 9.21. Let $f, g \in L^{1}(\mathbb {R})$ be fixed, and prove the following facts.

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(a)

The Fourier transform of the involution $\widetilde{f}(x) = \overline{f(-x)}$ is

(\tilde{f})^{\wedge }(\xi ) = \overline{\widehat{f}(\xi )}, \quad \xi \in \mathbb {R}

(b)

The convolution

(f * g)(x) = \int _{-\infty }^{\infty } f(y) g(x-y) d y

exists for almost every $x$ , and $f * g \in L^{1}(\mathbb {R})$ with $\left\| f * g\right\|_{L^{1}} \leq \left\| f\right\|_{L^{1}}\left\| g\right\|_{L^{1}}$ . Further, $f * g = g * f$ .

(c)

The Fourier transform of $f * g$ is

(f * g)^{\wedge }(\xi ) = \widehat{f}(\xi ) \widehat{g}(\xi ), \quad \xi \in \mathbb {R}

(d)

If $f, g$ are compactly supported, then so is $f * g$ .

(e)

If $g$ is differentiable and $g^{\prime }$ is integrable and bounded, then $f * g$ is differentiable and $(f * g)^{\prime } = f * g^{\prime }$ .

Problem 9.6

(a)(b)

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(a)

Show that equations (9.3) and (9.4) hold for every $f \in C_{c}^{2}(\mathbb {R})$ .

(b)

Use part (a), together with the fact that $L^{1}(\mathbb {R}) \cap L^{2}(\mathbb {R})$ is dense in $L^{2}(\mathbb {R})$ , to show that equations (9.3) and (9.4) hold for every $f \in L^{2}(\mathbb {R})$ , in the sense of pointwise almost everywhere equality of functions.

Problem 9.7

(a)(b)

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(a)

Prove that if $f \in L^{1}(\mathbb {R})$ is real and even, then $\widehat{f}$ is real and even.

(b)

Show that part (a) also holds if we assume $f \in L^{2}(\mathbb {R})$ .

Problem 9.8

(a)(b)

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(a)

Prove that if $f \in L^{1}(\mathbb {R})$ and $\widehat{f} \in L^{1}(\mathbb {R})$ , then $f^{\wedge \wedge }(\xi ) = f(-\xi )$ and $f^{\wedge \wedge \wedge \wedge }(\xi ) = f(\xi )$ for all $\xi \in \mathbb {R}$ .

(b)

Prove that if $f \in L^{2}(\mathbb {R})$ , then $f^{\wedge \wedge }(\xi ) = f(-\xi )$ and $f^{\wedge \wedge \wedge \wedge }(\xi ) = f(\xi )$ for almost every $\xi \in \mathbb {R}$ .

Problem 9.9

Prove Theorem 9.28.

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Remark.

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Theorem 9.28 (Integration by Parts): If $f, g$ are absolutely continuous on $[a, b]$ , then $\int_{a}^{b} f^{\prime }(x) g(x) d x = f(b) g(b)-f(a) g(a)-\int_{a}^{b} f(x) g^{\prime }(x) d x$ .