Justifying Operations on Abel Summability? In the text Stein and Shakarchi, Fourier Analysis, Abel summability is applied in the context of Fourier Series as the following summation: $$A_r(f)\theta=\sum_{n=-\infty}^{\infty}r^{n}a_ne^{{in\theta}}$$ The author metions a key fact the Abel Means can be written as convolutions, as seen in 1.) $$1. \, \, \, \, \, \, A_r(f)(\theta)=(f * P_r)(\theta)$$ 
Remark: One can note that $P_r$ denotes the Poisson kernel as seen in 2.) $$2. \, \, \, \, \,  \, \, \, \, P_r(\theta) = \sum_{n=-\infty}^{\infty}r^{|n|}e^{in\theta}$$
The author makes the following observation that the partial sums of the series in 0.) can be written as Convolution's as seen in 3.). This observation proceeds as following:
$$3. \, \, \,A_r(f)\theta=\sum_{n=-\infty}^{\infty}r^{n}a_ne^{{in\theta}} = \sum_{n=-\infty}^{\infty}r^{|n|}(\frac{1}{2\pi}\int_{-\pi}^{pi}f(\varphi)e^{in\varphi}d\varphi)e^{in\theta} = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(\varphi)(\sum_{n=-\infty}^{\infty}r^{|n|}e^{-in(\varphi-\theta)})d\varphi$$
Where my initial understanding breaks down is in 3.) how can the interchange of summation and integrals be justified ?
 A: In general, if $f_n$ is a sequence of Riemann integrable functions on $[-\pi,\pi]$ and $f_n \to f$ uniformly, then $ \lim_n \int f_n = \int f$.
In our case, $| \sum_n r^{|n|} f(x) e^{-in(x-\theta)}| \le \sum_n r^{|n|} |f(x)|$
Since $f$ is Riemann integrable, $f$ is bounded.  So this sum is bounded by $c \sum_n r^{|n|}$ for some constant $c > 0$, hence the sum converges uniformly in $x$.
So, $\lim_{N \to \infty} \sum_{|n| \le N} \int r^{|n|} f(x) e^{-in(x-\theta)}dx = \lim_{N \to \infty} \int \sum_{|n| \le N} r^{|n|}f(x) e^{-in(x-\theta)} dx $
$= \int \lim_{N \to \infty} \sum_{|n| \le N} r^{|n|} f(x) e^{-in(x-\theta)}dx$
Edit In some sense, this is the appeal of Abel summability.  The object you started with is $\sum_n a_n e^{in\theta}$, with $a_n = \int f(x) e^{-inx} dx$.  For this sum to be manageable, we need $e^{in\theta}a_n$ to decay rapidly.  But $|e^{in\theta}| = 1$, so any straightforward estimate we can come up with isn't going to get us anywhere.  The summability of the Fourier series is going to depend strongly on how much cancelation we can get out of these $e^{-inx}$ terms for $n$ large.  
By replacing the coefficients with $r^{|n|} a_n e^{in\theta}$, we introduce a factor that will force convergence and then we can hope to take a limit as $r \to 1^{-}$.  I haven't read Stein's book recently, but he probably covers convergence in the case that $f$ is smooth as well (which controls the amount of oscillation $f$ can have and hence we can hope for a lot of cancellation).
