Abel-Poisson summation of Fourier series

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Problem 11A.4 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 11A "Fourier Series and Poisson Integral"

Suppose $f \in L^{1}(\partial \mathbf{D}), z \in \partial \mathbf{D}$ , and $f$ is continuous at $z$ . Prove that

\lim _{r \uparrow 1}\left(\mathcal{P}_{r} f\right)(z) = f(z)

Remark.

The result in this exercise differs from 11.18 because here we are assuming continuity only at a single point and we are not even assuming that $f$ is bounded, as compared to 11.18, which assumed continuity at all points of $\partial \mathrm{D}$ .

Problem 11A.9 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 11A "Fourier Series and Poisson Integral"

Find a formula (in closed form, not as an infinite sum) for $\mathcal{P}_{r} f$ , where $f$ is the function in the second bullet point of Example 11.8.

Remark.

Example 11.8 (second bullet point): define $f: \partial D \rightarrow \mathbb {R}$ by $f(z) = \frac{1}{\left|3-z\right|^{2}}$ . Then for $z \in \partial D$ , $f(z) = \frac{1}{8} \sum_{n=-\infty }^{\infty } \frac{z^{n}}{3^{\left|n\right|}}$ , and the Fourier coefficients are $\widehat{f}(n) = \frac{1}{8} \cdot \frac{1}{3^{\left|n\right|}}$ for all $n \in \mathbb {Z}$ .

Problem 11A.12 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 11A "Fourier Series and Poisson Integral"

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

Define $f: \partial \mathbf{D} \rightarrow \mathbb {R}$ by

f(z) = \begin{cases} 1 & \text{ if } \operatorname {Im} z > 0 \\ -1 & \text{ if } \operatorname {Im} z < 0 \\ 0 & \text{ if } \operatorname {Im} z = 0\end{cases}

(a)

Show that if $n \in \mathbf{Z}$ , then

\widehat{f}(n) = \begin{cases} -\frac{2 i}{n \pi } & \text{ if } n \text{ is odd } \\ 0 & \text{ if } n \text{ is even }\end{cases}

(b)

Show that

\left(\mathcal{P}_{r} f\right)(z) = \frac{2}{\pi } \arctan \frac{2 r \operatorname {Im} z}{1-r^{2}}

for every $r \in [0,1)$ and every $z \in \partial \mathbf{D}$ .

(c)

Verify that $\lim_{r \uparrow 1}\left(\mathcal{P}_{r} f\right)(z) = f(z)$ for every $z \in \partial \mathbf{D}$ .

(d)

Prove that $\mathcal{P}_{r} f$ does not converge uniformly to $f$ on $\partial \mathbf{D}$ as $r \uparrow 1$ .

Problem 2.22.99 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Prove that for the function $s(x ; r)$ (see (21)) the formula holds

s(x ; r) = \frac{1}{2 \pi } \int _{-\pi }^{\pi } f(t) \frac{1-r^{2}}{1-2 r \cos (t-x)+r^{2}} d t

(Poisson).

Problem 2.22.100 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Prove that if a $2 \pi$ -periodic function $f$ , absolutely integrable over the period, has left- and right-hand limits $f(x \pm 0)$ at the point $x$ , then

\begin{aligned} & \lim _{r \rightarrow 1-0} s(x ; r) = \lim _{r \rightarrow 1-0} \frac{1}{2 \pi } \int _{-\pi }^{\pi } f(t) \frac{1-r^{2}}{1-2 r \cos (t-x)+r^{2}} d t = \\ & = \frac{f(x+0)+f(x-0)}{2} \end{aligned}

i.e., the Fourier series of the function $f$ is summable by the Poisson-Abel method to the value $\frac{f(x+0)+f(x-0)}{2}$ .

Problem 2.22.101 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Prove that if $f$ is a $2 \pi$ -periodic, continuous function, then $s(x ; r)$ tends uniformly, on the entire real line, to the function $f$ as $r \rightarrow 1-0$ .

References

[Axl19] Axler S. Measure, Integration & Real Analysis. 3rd ed., Springer, 2019.

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Интегралы. Ряды. 2-е изд., ФИЗМАТЛИТ, 2003.