Basic Fourier analysis - Summability kernels over $\mathbb{R}$

Question

I'm working through some practice problems and am stuck on the following.

Let $f_N :\mathbb{R} \rightarrow \mathbb{R}$ be a sequence of continuous, periodic functions so that $\int_{0}^{2\pi} f_N(t)dt=1$ and $\int_{0}^{2\pi} |f_N(t)|dt\leq M<\infty$ for any natural $N$, and $\lim_{N\to\infty} \int_{\delta}^{2\pi-\delta} |f_N(t)|dt =0$ for any $0<\delta<2\pi$. If $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and periodic, show that $\lim_{N\to\infty} \int_{0}^{2\pi} g(x-t)f_N(t)dt =g(x)$ uniformly for $x\in \mathbb{R}$.

The following is what I have so far, but it seems that I've hit a dead end. I'm particularly unsure of where the assumption that $\int_{0}^{2\pi} |f_N(t)|dt\leq M<\infty$ should be used. I'd appreciate help finishing or reworking what I have.

Let $\epsilon >0$. As $[0, 2\pi]$ is compact, $g$ is uniformly continuous on it. Choose $\delta \in (0, 2\pi)$ so that if $|(x-t)-x|=|t|<\delta$ then $|g(x-t)-g(x)|<\frac{\epsilon}{2}$ for any $x\in [0,2\pi]$. This works because if $x-t\notin [0, 2\pi]$, then WLOG $x-t \in [2\pi, 4\pi]$ so that by periodicity $|g(x-t)-g(x)|\leq |g(x-t)-g(2\pi)|+|g(2\pi)-g(x)|<2\frac{\epsilon}{2} = \epsilon$.

Now, we have $|\int_{0}^{2\pi} g(x-t)f_N(t)dt -g(x)|\leq \int_{0}^{2\pi} |(g(x-t) -g(x))||f_N(t)|dt$. Where do I go from here?

Note that I have not shown any general properties of summability kernels, so that the only properties I have to work with are those in the hypotheses.

Answer 1

Let $h_x(t) = g(x-t)-g(x)$ and $ \sup_t |g(t)|= C > 0$. For $\epsilon > 0$, since $g(t)$ is uniformly continuous we can take $\delta$ small enough such that $\sup_{t\in [-\delta,\delta]} \sup_x |h_x(t)| < \epsilon / M$ : $$|\int_{[-\delta,\delta]} f_N(t) h_x(t)dt| \le (\sup_{t\in [-\delta,\delta]} |h_x(t)|) \int_{[-\delta,\delta]} |f_N(t)|dt < \epsilon $$ this is independent of $N$.

Next, choose $N$ large enough such that $|\int_{[-\pi,-\delta]\cup[\delta,\pi]} f_N(t) | < \epsilon/C$ : $$|\int_{[-\pi,-\delta]\cup[\delta,\pi]} f_N(t) h_x(t)dt| \le (\sup_{t} |h_x(t)|) \int_{[-\pi,-\delta]\cup[\delta,\pi]} |f_N(t)|dt < 2\epsilon$$

$\implies $ For any $\epsilon > 0$ there is $\delta$ small enough and $N$ large enough such that $$|g(x)-\int_{[-\pi,\pi ]} f_N(t) g(x-t) dt| =|\int_{[-\pi,\pi ]} f_N(t) h_x(t) dt| < 3\epsilon$$