Bandlimited functions and the Paley-Wiener space

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Problem 10.1 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 10 "Sampling, Weighted Exponentials, and Translations"

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

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Problem 10.2 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 10 "Sampling, Weighted Exponentials, and Translations"

Show that the dilation operator DrD_{r} maps FL[−Ω,Ω]2(R)\mathcal{F} L_{[-\Omega , \Omega ]}^{2}(\mathbb {R}) unitarily onto FL[−rΩ,rΩ]2(R)\mathcal{F} L_{[-r \Omega , r \Omega ]}^{2}(\mathbb {R}).

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Problem 10.3 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 10 "Sampling, Weighted Exponentials, and Translations"
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(a)

Prove statement (a) in Theorem 10.4.

(b)

Show that PW(R)⊆C∞(R)\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 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 10 "Sampling, Weighted Exponentials, and Translations"

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

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(a)

The nnth derivative of ff can be written as

f(n)(x)=∫−1/21/2(2πiξ)nf^(ξ)e2πiξxdξ 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)

∥f(n)∥∞≤πn∥f^∥L1\left\| f^{(n)}\right\|_{\infty } \leq \pi^{n}\left\| \widehat{f}\right\|_{L^{1}} for each n≥0n \geq 0.

(c)

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

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

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

(d)

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

Problem 10.6 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 10 "Sampling, Weighted Exponentials, and Translations"

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

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References

[Hei11] Heil C. A Basis Theory Primer. Birkhäuser, 2011.