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For $n \in \mathbb{N}$, the Dirichlet kernel $D_n$ is the function on the circle $\mathbb{T} = \mathbb{R}/\mathbb{2\pi \mathbb{Z}}$ defined by $$ D_n(x) = \frac{ \sin [(n+1/2)x] }{ \sin(x/2) } \quad \text{for } x \in \mathbb{T} $$ where each $x \in \mathbb{T}$ is identified with a point in the interval $(-\pi, \pi]$. The Dirichlet kernel is important in Fourier analysis because the Fourier partial sum can be given as $S_nf = f * D_n$ for $f \in L^1(\mathbb{T})$ and $n \in \mathbb{N}$.

The $L^1$-norm of the Dirichlet kernel is given by

$$ ||D_n||_1 = \frac{1}{2\pi} \int_{-\pi}^\pi \left| \frac{ \sin [(n+1/2)x] }{ \sin(x/2) } \right| dx $$

The factor $1/2\pi$ is here to normalize the length of $\mathbb{T}$ to $1$.

With this definition of $L^1$-norm, it is known that

$$ ||D_n||_1 = \frac{4}{\pi^2} \ln n + O(1) $$

Thus, the limsup and liminf

$$ \overline{C} = \limsup_{n \to \infty} \left( ||D_n||_1 - \frac{4}{\pi^2} \ln n \right) \quad \text{and} \quad \underline{C} = \liminf_{n \to \infty} \left( ||D_n||_1 - \frac{4}{\pi^2} \ln n \right) $$

exist and are finite.

My question is, are the exact values of $\overline{C}$ and $\underline{C}$ known? If not, what are the current best known upper and lower bounds for $\overline{C}$ and $\underline{C}$?

I know that

$$ ||D_n||_1 \geq \frac{4}{\pi^2} \ln n + \frac{2}{\pi} \int_0^\pi \frac{\sin t}{t} dt \quad \text{for all } n \in \mathbb{N} $$

Thus, a trivial lower bound is

$$ \overline{C} \geq \underline{C} \geq \frac{2}{\pi} \int_0^\pi \frac{\sin t}{t} dt $$

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1 Answer 1

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$$f(x)=\frac1{|\sin(x/2)|}-\frac{1}{|x/2|}\in C^0[-\pi,\pi]$$ So the asymptotic follows easily $$\int_{-\pi}^\pi \left|\frac{\sin((n+1/2)x)}{\sin(x/2)}\right|dx=\int_{-\pi}^\pi |\sin((n+1/2)x)| f(x)dx+\int_{-\pi}^\pi \left|\frac{\sin((n+1/2)x)}{x/2}\right|dx $$ $$ =\int_{-\pi}^\pi \frac{2}\pi f(x)dx+o(1)+4\int_0^{(n+1/2)\pi} \left|\frac{\sin(x)}{x}\right|dx $$ $$ = C+o(1)+4\int_1^{(n+1/2)\pi} \frac{2/\pi}{|x|}dx=C+\frac8\pi\log \pi +o(1)+\frac8\pi\log n$$ where $$C=\int_{-\pi}^\pi \frac{2}\pi f(x)dx+ 4\int_0^1 \frac{|\sin(x)|}{|x|}dx+4\int_1^\infty\frac{|\sin(x)|-2/\pi}{|x|}dx$$

I think the $o(1)$ term can be improved to $\sum_{j=1}^J a_j n^{-j}+O(n^{-J-1})$

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