Showing that $P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}\rightarrow 0$ uniformly on $[-\pi,-\delta]\cup[\delta,\pi]$ as $r\uparrow 1$

Question

Let $0<r<1$ and consider the series $$s = \sum_{n=-\infty}^\infty r^{|n|}e^{inx}.$$

I have shown that the series converges uniformely to $$P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}$$ on all of $\mathbb{R}$. Now I am asked to...

Show that for every $0<\delta<\pi$, $P_r(x)\rightarrow 0$ uniformely on the intervals $[-\pi,-\delta,]\cup[\delta,\pi]$ as $r\uparrow 1$.

Now, first off I am not entirely sure what this means: In order to talk about uniform convergence, do we not need a sequence of functions? Should I construct such a sequence to work with, e.g. $\{P_{1-\frac{1}{n}}(x)\}_n$? If so, does the problem now become: Given $\epsilon>0$, find $N\in\mathbb{N}$ such that if $n\ge N$ then $|P_{1-\frac{1}{n}}(x)-0|<\epsilon$ on all of $[-\pi,-\delta]\cup[\delta,\pi]$?

Answer 1

One does not need a sequence of functions to discuss uniform convergence. The concept applies to continuous variables also. Here, there are two variables, $x$ and $r$. If $x=0$, then note that $P(r,x=0)=\frac{1+r}{1-r}$, which is undefined for $r=1$. If $x\ne 0$, then clearly $P(r,x \ne 0) \to 0$ as $r \to 1$.

So, when is this convergence uniform? To prove uniform convergence on a set, given $\epsilon >0$, one needs to find a $\eta > 0$ independent on $x$, such that $|P-1|<\epsilon$ whenever $|r-1|< \eta $ for all $x$ is a set. (Note, we used $\eta$ here so as to avoid confusion with the $\delta$ specified in the problem statement.)

Now, we can write

$$\left|\frac{1-r^2}{1-2r\cos x +r^2}\right|=\left|\frac{1-r^2}{(1-r)^2+4r\sin^2(x/2)}\right|\le \left|\frac{(r+1)(r-1)}{4r\sin^2(x/2)}\right|=\frac{r+1}{4r}\frac{\left|(r-1)\right|}{\sin^2(x/2)}$$

Note that provided $\pi\le |x|>\delta$, then

$$1 \ge \sin^2(x/2)>\sin^2(\delta /2)$$

To complete the proof, first take $0<\eta<1/2$. Then, certainly $(r+1)/4r < 3/4$.

Thus, given $\epsilon >0$,

$$\left| \frac{1-r^2}{1-2r\cos x +r^2} \right| \le \frac34 \frac{\left|r-1\right|}{\sin^2(\delta/2)}<\epsilon$$

whenever $|r-1|< \eta =\min (1/2, 4\epsilon \sin^2(\delta/2)/3 )$.

Answer 2

For the first question, we have $$s=\sum_{n=-\infty}^{+\infty}r^{|n|}e^{inx} =1+\sum_{n=1}^{+\infty}r^ne^{inx}+\sum_{n=1}^{+\infty}r^ne^{-inx} =1+\sum_{n=1}^{+\infty}\left(re^{ix}\right)^n+\sum_{n=1}^{+\infty}\left(re^{-ix}\right)^n$$ $$=1+\frac{re^{ix}}{1-re^{ix}}+\frac{re^{-ix}}{1-re^{-ix}} =1+\frac{2r\cos\left(x\right)-2r^2}{1-2r\cos\left(x\right)+r^2} =\frac{1-r^2}{1-2r\cos\left(x\right)+r^2}=P_r\left(x\right)$$ and the convergence is uniform for all $0<r<1$.

The second question has been discussed by John, Thibaut Dumont and Dr. MV.