Integral equality $\int_{-\pi}^\pi\dots = \int_{|t|\le \delta}\dots+\int_{\delta\le |t|\le \pi}\dots$

Question

This is an excerpt from here (page 6, bottom)

I don't know if this is a typo or not, but what exactly happened to the integral of $\int_{-\pi}^{-\delta}$ for the $|\sigma_n(x) - f(x)|$? I don't understand the equality.

Answer 1

The two integrals at the bottom are over $|t|<\delta$ and $\delta\le|t|\le\pi$ respectively. The first inequality restricts $t$ to the range $(-\delta,\delta)$, and the second restricts $t$ to a union of two ranges: $[-\pi,-\delta]\cup[\delta,\pi]$. The integrals shown would correspond to three integrals, if upper and lower limits of integration were indicated: $\displaystyle\int_{-\delta}^{\delta}\cdots dt $ for the first one, and $\displaystyle \int_{-\pi}^{-\delta}\cdots dt $ + $\displaystyle \int_\delta^{\pi}\cdots dt$ for the second one.