Poisson summation formula

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Problem 17.22 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"

For functions φ∈D\varphi \in \mathcal{D} prove the Poisson summation formula (cf. problem 16.48):

2π∑k=−∞+∞φ(2πk)=∑k=−∞+∞φ^(k). \sqrt{2 \pi } \sum _{k = -\infty }^{+\infty } \varphi (2 \pi k) = \sum _{k = -\infty }^{+\infty } \hat{\varphi }(k).
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Problem 17.32 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"

Prove that for the functions ff and gg of problem 17.31 the equality holds

f(x)=1aπ2∑k∈Zg(πka)e(ikπx)/a f(x) = \frac{1}{a} \sqrt{\frac{\pi }{2}} \sum _{k \in \mathbb {Z}} g\left(\frac{\pi k}{a}\right) e^{(i k \pi x) / a}

where the series converges in L2[−a,a]L_{2}[-a, a].

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Problem 4897 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Prove the equalities:

∑n=−∞∞e−πa(n+x)2=1a∑n=−∞∞e−πn2acos⁡2πnx,a>0. \sum _{n=-\infty }^{\infty } e^{-\pi a(n+x)^{2}} = \frac{1}{\sqrt{a}} \sum _{n=-\infty }^{\infty } e^{-\frac{\pi n^{2}}{a}} \cos 2\pi nx, \quad a > 0 .
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Problem 4898 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Prove the equalities:

12+∑n=1∞e−πn2x=1x(12+∑n=1∞e−πn2x),x>0. \frac{1}{2}+\sum _{n=1}^{\infty } e^{-\pi n^{2} x} = \frac{1}{\sqrt{x}}\left(\frac{1}{2}+\sum _{n=1}^{\infty } e^{-\frac{\pi n^{2}}{x}}\right), \quad x > 0 .
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Problem 4899 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Prove the equalities:

lim⁡x→1−01−x (x+x4+x9+…)=12π. \lim _{x \rightarrow 1-0} \sqrt{1-x}\, (x+x^{4}+x^{9}+\ldots ) = \frac{1}{2}\sqrt{\pi } .
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Problem 4900 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Prove the equalities:

lim⁡x→1−0(x−x4+x9−x16+…)=12. \lim _{x \rightarrow 1-0} (x-x^{4}+x^{9}-x^{16}+\ldots ) = \frac{1}{2} .
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Problem 4901 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Prove the equality (Poisson's formula):

∑n=abf(n)= \sum _{n=a}^{b} f(n) = =f(a)+f(b)2−∫abf(x) dx+2∑n=1∞∫abf(x)cos⁡2πnx dx. = \frac{f(a)+f(b)}{2}-\int _{a}^{b} f(x)\, dx+2\sum _{n=1}^{\infty } \int _{a}^{b} f(x)\cos 2\pi nx\, dx .
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Problem 5162 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 12.4 "Miscellaneous Problems"

Prove Poisson's formula:

∑n=abf(n)=f(a)+f(b)2+∫abf(x) dx+2∑n=1∞∫abf(x)cos⁡2πnx dx. \sum _{n=a}^{b} f(n) = \frac{f(a)+f(b)}{2} + \int _{a}^{b} f(x)\, dx + 2\sum _{n=1}^{\infty } \int _{a}^{b} f(x)\cos 2\pi nx\, dx.
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Problem 5163 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 12.4 "Miscellaneous Problems"

Prove the equality 12+∑n=1∞e−πn2x=1x(12+∑n=1∞e−πn2x)\frac{1}{2}+\sum_{n=1}^{\infty } e^{-\pi n^{2}x} = \frac{1}{\sqrt{x}}\left(\frac{1}{2}+\sum_{n=1}^{\infty } e^{-\frac{\pi n^{2}}{x}}\right).

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Problem 5165 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 12.4 "Miscellaneous Problems"

Prove the equality ∑n=−∞∞e−πa(n+x)2=1a∑n=−∞∞e−πn2acos⁡2πnx\sum_{n=-\infty }^{\infty } e^{-\pi a(n+x)^{2}} = \frac{1}{\sqrt{a}}\sum_{n=-\infty }^{\infty } e^{-\frac{\pi n^{2}}{a}}\cos 2\pi nx.

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Problem 2.9.11 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.9 "Fourier Transforms"

Find conditions on ff and/or ff which ensure the correctness of the following formal argument: If

φ(t)=12π∫−∞∞f(x)e−itxdx \varphi (t) = \frac{1}{2 \pi } \int _{-\infty }^{\infty } f(x) e^{-i t x} d x

and

F(x)=∑k=−∞∞f(x+2kπ) F(x) = \sum _{k = -\infty }^{\infty } f(x+2 k \pi )

then FF is periodic, with period 2π2 \pi, the nnth Fourier coefficient of FF is φ(n)\varphi (n), hence F(x)=∑φ(n)einxF(x) = \sum \varphi (n) e^{i n x}. In particular,

∑k=−∞∞f(2kπ)=∑n=−∞∞φ(n) \sum _{k = -\infty }^{\infty } f(2 k \pi ) = \sum _{n = -\infty }^{\infty } \varphi (n)

More generally,

∑k=−∞∞f(kβ)=α∑n=−∞∞φ(nα) if α>0,β>0,αβ=2π(*) \sum _{k = -\infty }^{\infty } f(k \beta ) = \alpha \sum _{n = -\infty }^{\infty } \varphi (n \alpha ) \quad \text{ if } \alpha > 0, \beta > 0, \alpha \beta = 2 \pi \tag {*}

What does () say about the limit, as α→0\alpha \rightarrow 0, of the right-hand side (for " nice " functions, of course)? Is this in agreement with the inversion theorem? [() is known as the Poisson summation formula].

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Problem 2.9.12 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.9 "Fourier Transforms"

Take f(x)=e−∣x∣f(x) = e^{-\left|x\right|} in Exercise 11 and derive the identity

e2πα+1e2πα−1=1π∑n=−∞∞αα2+n2 \frac{e^{2 \pi \alpha }+1}{e^{2 \pi \alpha }-1} = \frac{1}{\pi } \sum _{n = -\infty }^{\infty } \frac{\alpha }{\alpha ^{2}+n^{2}}
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Problem 2.9.13 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.9 "Fourier Transforms"

If 0<c<∞0 < c < \infty, define fc(x)=exp⁡(−cx2)f_{c}(x) = \exp \left(-c x^{2}\right).

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(a)

Compute f^c\hat{f}_{c}. Hint: If φ=f^c\varphi = \hat{f}_{c}, an integration by parts gives 2cφ′(t)+tφ(t)=02 c \varphi^{\prime }(t)+t \varphi (t) = 0.

(b)

Show that there is one (and only one) cc for which f^c=fc\hat{f}_{c} = f_{c}.

(c)

Show that fa∗fb=γfcf_{a} * f_{b} = \gamma f_{c}; find γ\gamma and cc explicitly in terms of aa and bb.

(d)

Take f=fcf = f_{c} in Exercise 11. What is the resulting identity?

References

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.
[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.
[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.