I'm studying Fourier analysis now and learned the concept of approximate identity. $$h_n\ge 0,\quad \int_{\mathbb{T}}h_n=1,\quad \lim_{n\to\infty}\int_{\mathbb{T}\setminus[-\delta,\delta]}h_n=0\quad (0<\delta<\pi)$$
If some sequence $\{h_n\}$ satisfies the condition above, then
$$\lim_{n\to\infty}\|f-h_n*f\|_1 =0 $$
holds for any integrable function $f$ on $(-π, π]$.
However, I saw an exercise problem that in fact {h_n} works as an approximate identity in the $L^p$ norm. Here $p$ is a positive number larger than $1$.
Hence,
$$\lim_{n\to\infty}\|f-h_n*f\|_p =0$$
for any function $f$ such that $f^p$ is integrable on $(-π, π]$.
The exercise said that the key is using Jensen inequality. So, first I thought of $y=x^{1/p}$ but found out that this is not a convex function. So, I am just stuck at this point. Could anyone show me how to solve this problem?
There may be a misunderstanding here; I don't see how one would use Jensen's inequality to control $L^p$ norm by $L^1$ norm. Anyway, the proof of $$ \lim_{n\to\infty}\|f-h_n*f\|_p =0 $$ is essentially the same for $1<p<\infty$ as it is for $p=1$.
For continuous functions $g$ we have $h_n*g\to g$ uniformly: just split the integral in $$g(t)-(h_n*g)(t)=\int h_n(s)(g(t)-g(t-s))\,ds$$ into $|s|<\delta$ and $|s|\ge \delta$, and show that each integral is small.
Continuous functions are dense in $L^p$, so for a given $f\in L^p$ there is a continuous $g$ such that $\|f-g\|_p<\epsilon$.
By the triangle inequality, $$\|f-h_n*f\|_p\le \|g-h_n*g\|_p+\|f-g\|_p+\|h_n*(f-g)\|_p$$ Use Young's convolution inequality to control the last term: $$\|h_n*(f-g)\|_p\le \|h_n\|_1 \|f-g\|_p= \|f-g\|_p$$