A difficulty in understanding Theorem 4.1 in Stein & Shakarachi Fourier Analysis

Question

Before proving the theorem the writer said that:"The weight distribution concentrates its mass at y=0 as n becomes large" but I do not know why? could anyone explain this for me please?

Answer 1

According to the definition of good kernel, the sequence $(K_n)_n$ satisfies \begin{eqnarray} \frac{1}{2\pi}\int_{-\pi}^\pi K_n(y) dy = 1 \text{ for all }n \end{eqnarray} and \begin{eqnarray} \lim_{n\to\infty}\int_{\delta\leq|y|\leq\pi} |K_n(y)| dy = 0 \text{ for all } \delta>0. \end{eqnarray}

This is what the author describes as "The weight distribution concentrates its mass at $y=0$ as $n$ becomes large".