I'm reading Baby Rudin, and I don't understand his proof of theorem 8.14. He starts by defining an auxiliary function and then asserts that $\frac{1}{2π}$ times the integral of the Dirichlect kernel on $[-π, π]$ is 1. Good so far. But then I don't know what that integral has to do with the rest of the proof. Furthermore, he finds an expression for $s_N(f; x)-f(x)$ and says that it comes from line 78 (picture included) but I don't know where it came from. What's going on?

start of Theorem and some relevant equations. Theorem is ar bottom of page

second page of Theorem, with offending equation


He is using that $$ \frac1{2\pi}\int_{-\pi}^\pi f(x)\,D_N(t)\,dt =f(x). $$ Note the variables.

  • $\begingroup$ Ah! Because f(x) is a constant with regards to the integral $\endgroup$ – Zachary F Feb 9 '17 at 18:24
  • $\begingroup$ Exactly. A very common and useful trick. $\endgroup$ – Martin Argerami Feb 9 '17 at 18:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.