I have a function $h\in L^1(\mathbb{T})$, and I want to show that:
$$\int_{\pi\geq |t|>\delta>0} h(x+t)D_N(t) dt/2\pi \leq \xi_N(h,\delta)$$
where $\xi_N(h,\delta) \rightarrow 0$ as $N\rightarrow \infty$.
($D_N$ is Dirichlet kernel).
This question arises from Riemann Localization principle where we have two $L^1$ functions that agree on a $\delta$ nbhd of $x$, and thus their partial Fourier sums, their difference converges to zero.
Best Answer
This is just a consequence of the following local convergence theorem for Fourier series, taken from Fourier Analysis, an Introduction by Stein & Shakarchi, page 81:
Theorem 2.1 Let $f$ be an integrable function on the circle which is differentiable at a point $\theta_0$. Then $S_N(f)(\theta_0)\to f(\theta_0)$ as $N$ tends to in infinity.
Here $S_N(f)$ is he $N$-th partial sum of the Fourier series of $f$. A key ingredient in the proof is the Riemann-Lebesgue lemma. It can be generalized to weaker conditions on $f$, like Dini's criterion.
Now to your question. We may assume without loss of generality hat $x=0$. let $h_\delta$ be defined by $$h_\delta(x)=\begin{cases} h(x) & \text{if }\delta<|x|\le\pi,\\ 0 & \text{if }|x|\le\delta. \end{cases}$$ Then $h_\delta$ is integrable, is differentiable in a neighborhood of $0$ and hence $$ S_N(h_\delta)(0)=\frac{1}{2\,\pi}\int_{\pi\ge|t|>\delta} h(t)D_N(t)\,dt\to0\text{ as }N\to\infty. $$