§3.6.2 The Poisson Integral Formula in Harmonic Analysis

3.6.2 The Poisson Integral Formula in Harmonic Analysis

Consider $𝑅 = 1$ , $𝜁 = e^{i 𝜏}$ , and $𝑧 = 𝜌 e^{i 𝜃}$ in (3.6.5) $(3.6.5) (3.6.5)$ :

\begin{aligned} 𝑢 (𝑧) & = \frac{1}{2 π} \int_{0}^{2 π} 𝑢 (𝜁) \frac{1 - {| 𝑧 |}^{2}}{{| 𝜁 - 𝑧 |}^{2}} d 𝜏 \\ = \frac{1}{2 π} \int_{0}^{2 π} \frac{(1 - 𝜌^{2}) 𝑢 (e^{i 𝜏}) d 𝜏}{(e^{i 𝜏} - 𝜌 e^{i 𝜃}) (e^{- i 𝜏} - 𝜌 e^{- i 𝜃})} \\ = \frac{1}{2 π} \int_{0}^{2 π} \frac{(1 - 𝜌^{2}) 𝑢 (e^{i 𝜏}) d 𝜏}{1 + 𝜌^{2} - 2 𝜌 cos (𝜃 - 𝜏)} . (3.6.2 . 1) \end{aligned}

Since $𝑢 (𝑧)$ is continuous on $𝜕 𝔻$ and $𝑢 (e^{i 𝜃})$ is periodic with period $2 π$ , it admits a Fourier series representation with coefficients

𝑎_{𝑛} = \frac{1}{2 π} \int_{0}^{2 π} 𝑢 (e^{i 𝜏}) e^{- i 𝑛 𝜏} d 𝜏, (3.6.2 . 2)

so that the corresponding Fourier series is

\sum_{𝑛 = - \infty}^{\infty} 𝑎_{𝑛} e^{i 𝑛 𝜃} .

This series may diverge. Observe that continuity of $𝑢$ on the compact set $𝜕 𝔻$ implies boundedness: $\exists 𝑀 > 0$ such that $| 𝑢 (e^{i 𝜃}) | \leq 𝑀$ for all $𝜃$ (Theorem 1.2.22 $Theorem 1.2.22 Theorem 1.2.22$ ). Consequently, $| 𝑎_{𝑛} | \leq 𝑀$ . Introducing factors $𝜌^{| 𝑛 |}$ with $| 𝜌 | < 1$ yields a convergent series:

\sum_{𝑛 = - \infty}^{\infty} 𝑎_{𝑛} e^{i 𝑛 𝜃} 𝜌^{| 𝑛 |}, | \sum_{𝑛 = - \infty}^{\infty} 𝑎_{𝑛} e^{i 𝑛 𝜃} 𝜌^{| 𝑛 |} | \leq \sum_{𝑛 = - \infty}^{\infty} | 𝑎_{𝑛} | 𝜌^{| 𝑛 |} \leq 𝑀 \frac{1 + | 𝜌 |}{1 - | 𝜌 |} .

Substituting the coefficients gives

\begin{aligned} \sum_{𝑛 = - \infty}^{\infty} 𝑎_{𝑛} e^{i 𝑛 𝜃} 𝜌^{| 𝑛 |} & = \sum_{𝑛 = - \infty}^{\infty} (\frac{1}{2 π} \int_{0}^{2 π} 𝑢 (e^{i 𝜏}) e^{- i 𝑛 𝜏} d 𝜏) e^{i 𝑛 𝜃} 𝜌^{| 𝑛 |} \\ = \frac{1}{2 π} \sum_{𝑛 = - \infty}^{\infty} \int_{0}^{2 π} 𝜌^{| 𝑛 |} 𝑢 (e^{i 𝜏}) e^{i 𝑛 (𝜃 - 𝜏)} d 𝜏 . \end{aligned}

By Theorem 2.3.3 $Theorem 2.3.3 Theorem 2.3.3$ and Theorem 2.3.7 $Theorem 2.3.7 Theorem 2.3.7$ ,

\frac{1}{2 π} \sum_{𝑛 = - \infty}^{\infty} \int_{0}^{2 π} 𝜌^{| 𝑛 |} 𝑢 (e^{i 𝜏}) e^{i 𝑛 (𝜃 - 𝜏)} d 𝜏 = \frac{1}{2 π} \int_{0}^{2 π} 𝑢 (e^{i 𝜏}) \sum_{𝑛 = - \infty}^{\infty} 𝜌^{| 𝑛 |} e^{i 𝑛 (𝜃 - 𝜏)} d 𝜏 . (3.6.2 . 3)

The summation simplifies as follows:

\begin{aligned} \sum_{𝑛 = - \infty}^{\infty} 𝜌^{| 𝑛 |} e^{i 𝑛 (𝜃 - 𝜏)} & = \sum_{𝑛 = 0}^{\infty} 𝜌^{𝑛} e^{i 𝑛 (𝜃 - 𝜏)} + \sum_{𝑛 = 1}^{\infty} 𝜌^{𝑛} e^{- i 𝑛 (𝜃 - 𝜏)} \\ = 1 + 2 \sum_{𝑛 = 1}^{\infty} 𝜌^{𝑛} cos (𝑛 (𝜃 - 𝜏)) \\ = 1 + 2 ℜ𝔢 (\sum_{𝑛 = 1}^{\infty} 𝜌^{𝑛} e^{i 𝑛 (𝜃 - 𝜏)}) \\ = 1 + 2 ℜ𝔢 (\frac{𝜌 e^{i (𝜃 - 𝜏)}}{1 - 𝜌 e^{i (𝜃 - 𝜏)}}) \\ = \frac{1 - 𝜌^{2}}{1 + 𝜌^{2} - 2 𝜌 cos (𝜃 - 𝜏)} . \end{aligned}

Substituting into (3.6.2.3) $(3.6.2.18) (3.6.2.3)$ yields

\sum_{𝑛 = - \infty}^{\infty} 𝑎_{𝑛} e^{i 𝑛 𝜃} 𝜌^{| 𝑛 |} = \frac{1}{2 π} \int_{0}^{2 π} \frac{(1 - 𝜌^{2}) 𝑢 (e^{i 𝜏})}{1 + 𝜌^{2} - 2 𝜌 cos (𝜃 - 𝜏)} d 𝜏 = 𝑢 (𝜌 e^{i 𝜃}) .

Furthermore, by the proof of Theorem 3.6.1.1 $Theorem 3.6.1.1 Theorem 3.6.1.1$ (specifically (3.6.1.2) $(3.6.1.12) (3.6.1.2)$ ),

lim_{𝜌 \to 1^{-}} \sum_{𝑛 = - \infty}^{\infty} 𝑎_{𝑛} e^{i 𝑛 𝜃} 𝜌^{| 𝑛 |} = 𝑢 (e^{i 𝜃}) .

Thus, for any continuous function $𝑢$ on $𝜕 𝔻$ , its Fourier series is Abel summable to $𝑢$ .

We now establish that real-valued continuous functions satisfying the local mean-value property are harmonic. Here, local means that for every $𝑧_{0} \in 𝑈$ , there is a disk $\overline{𝐷 (𝑧_{0}, 𝜆)} ⋐ 𝑈$ such that the identity holds on every concentric disk $\overline{𝐷 (𝑧_{0}, 𝜀)}$ with $0 < 𝜀 \leq 𝜆$ . As noted above, this use of local is informal terminology adopted for clarity rather than necessarily standard classical terminology.

Theorem 3.6.2.1.

Let $𝑈 \subseteq ℂ$ be an open region and $𝑓 : 𝑈 \to ℝ$ continuous. Suppose for every $𝑧_{0} \in 𝑈$ , there exists $𝜆 > 0$ with $\overline{𝐷 (𝑧_{0}, 𝜆)} ⋐ 𝑈$ such that for all $0 < 𝜀 \leq 𝜆$ ,

𝑓 (𝑧_{0}) = \frac{1}{2 π} \int_{0}^{2 π} 𝑓 (𝑧_{0} + 𝜀 e^{i 𝑡}) d 𝑡 .

Then

𝑓

is harmonic on

𝑈

Proof.

Fix $𝑧_{0} \in 𝑈$ arbitrarily and choose $𝜆 > 0$ such that $\overline{𝐷 (𝑧_{0}, 𝜆)} ⋐ 𝑈$ . Because $𝑓 \in 𝐶 (𝜕 𝐷 (𝑧_{0}, 𝜆))$ , Theorem 3.6.1.1 $Theorem 3.6.1.1 Theorem 3.6.1.1$ guarantees the existence of a unique harmonic function $𝑢$ on $𝐷 (𝑧_{0}, 𝜆)$ satisfying $𝑢 \equiv 𝑓$ on $𝜕 𝐷 (𝑧_{0}, 𝜆)$ with

𝑢 (𝑧) = \int_{0}^{2 π} 𝑓 (𝜁) 𝑃 (𝜁, 𝑧) d 𝜏 .

Define

𝜓 = 𝑓 - 𝑢

\overline{𝐷 (𝑧_{0}, 𝜆)}

. Then

𝜓

is continuous, satisfies the local mean-value property in

𝐷 (𝑧_{0}, 𝜆)

, and vanishes on

𝜕 𝐷 (𝑧_{0}, 𝜆)

. By Theorem 3.4.3

Theorem 3.4.3 Theorem 3.4.3

𝜓 \equiv 0

\overline{𝐷 (𝑧_{0}, 𝜆)}

. Thus,

𝑓 \equiv 𝑢

\overline{𝐷 (𝑧_{0}, 𝜆)}

, implying

𝑓

is harmonic at

𝑧_{0}

. The arbitrariness of

𝑧_{0}

establishes harmonicity on

𝑈

. ⁠

□