Sampling theorem (Shannon-Whittaker-Kotelnikov)

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Problem 17.33 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 17.1 "Fourier transform of ordinary functions"

(V. A. Kotelnikov, 1933). Prove that the function gg from problem 17.31 satisfies the equality

g(y)=∑k∈Zg(πka)sin⁡(ay−πk)ay−πk g(y) = \sum _{k \in \mathbb {Z}} g\left(\frac{\pi k}{a}\right) \frac{\sin (a y-\pi k)}{a y-\pi k}

where the series on the right converges in L2(R).L_{2}(\mathbb {R}).1

Footnotes

  1. In other formulations this theorem was obtained by E. Whittaker (1915), H. Nyquist (1928) and C. Shannon (1949). ↩

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Problem 10.5 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 10 "Sampling, Weighted Exponentials, and Translations"

Prove Theorem 10.8.

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Remark.
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Theorem 10.8: Let {λn}n∈N\left\{ \lambda_{n}\right\}_{n \in \mathbb {N}} be any sequence of real numbers such that {eλn}n∈N\left\{ e_{\lambda_{n}}\right\}_{n \in \mathbb {N}} is a frame for L2(T)L^{2}(\mathbf{T}). Let {e~n}n∈N\left\{ \widetilde{e}_{n}\right\}_{n \in \mathbb {N}} be the canonical dual frame of {eλn}n∈N\left\{ e_{\lambda_{n}}\right\}_{n \in \mathbb {N}} in L2(T)L^{2}(\mathbf{T}), and define s~n=(e~nχ[−12,12])∨\widetilde{s}_{n} = \left(\widetilde{e}_{n} \chi_{\left[-\frac{1}{2}, \frac{1}{2}\right]}\right)^{\vee }. Then for any f∈PW(R)f \in \mathrm{PW}(\mathbb {R}) we have f=∑nf(λn)s~nf = \sum_{n} f(\lambda_{n}) \widetilde{s}_{n}, where the series converges in L2L^{2}-norm.

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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Problem 3.22 from Gallager R.G. "Stochastic Processes: Theory for Applications" [Gal13]More problems in Chapter 3 "Gaussian random vectors and processes"

Let h(t)h(t) be a real square-integrable function whose Fourier transform is 0 for ∣f∣>B\left|f\right| > B for some B>0B>0. Show that ∑nh2(t−n/2B)=(1/2B)∫h2(τ) dτ\sum_n h^2(t-n/2B) = (1/2B)\int h^2(\tau )\, d\tau for all t∈Rt \in \mathbb {R}.

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References

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.
[Hei11] Heil C. A Basis Theory Primer. Birkhäuser, 2011.
[Gal13] Gallager R.G. Stochastic Processes: Theory for Applications. Cambridge University Press, 2013.