I am doing research and proved to the professor the coefficient of Fourier Series. As you can see from the picture below.
I don't know how to prove the $S_n(x)$ pointwise convergence as $N$ goes to zero. How did he get from equation $(2)$ to $(3)$?
Here is what I know: $S_n$ goes to $f$ if and only if $\sup | S_n(x) - f(x)|$ goes to zero and $x$ belongs to $(-\pi , \pi)$.
Going from equation (2) to (3) is just playing plug and chug on the definitions that are right there: $$S_N(x) = \frac 12A_0 + \sum_{n=1}^N A_n\cos nx + B_n\sin nx\\ A_n = \int_{-\pi}^\pi f(y)\cos ny\frac{dy}\pi\\B_n = \int_{-\pi}^\pi f(y)\sin ny\frac{dy}\pi$$ So (since $\cos 0y = 1$)$$\begin{align}S_N(x)& = \frac12\int_{-\pi}^\pi f(y)\frac{dy}\pi + \sum_{n=1}^N \int_{-\pi}^\pi f(y)\cos ny\frac{dy}\pi\cos nx + \int_{-\pi}^\pi f(y)\sin ny\frac{dy}\pi\sin nx\\ &=\int_{-\pi}^\pi\left[1+2\sum_{n=1}^N\cos nx\cos ny+\sin nx\sin ny\right]f(y)\frac{dy}{2\pi}\\ &=\int_{-\pi}^\pi\left[1+2\sum_{n=1}^N\cos n(x-y)\right]f(y)\frac{dy}{2\pi}\\ &=\int_{-\pi}^\pi K_N(x-y)f(y)\frac{dy}{2\pi}\end{align}$$ Where $K_N$ is as defined in equation (4).