proof that a fourier series converges to a pi periodic function

Question

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so given the information on the left, how did they on the right hand conclude the first equality that $$S_{N}(f;x)-f(x) = \frac{1}{2pi}\int_{-\pi}^{\pi}g(t)sin(N+\frac{1}{2})tdt$$

Answer 1

We have that $S_N(f,x) = \frac{1}{2\pi}\int_{-\pi}^\pi f(t-x)D(t)dt$. By definition of $D(t)$, $$\frac{1}{2\pi}\int_{-\pi}^\pi D(x)dx = 1,$$

so (note that $f(x)$ is constant with respect to $t$):,

$$S_N(f,x)-f(x) = \frac{1}{2\pi}\int_{-\pi}^\pi [f(x-t)D(t)-f(x)D(t)]dt.$$

Now just look at the definition of $g(t)$.