# On the uniform convergence of a fourier series

Claim: Let $$f\left( x \right)$$ be $$2\pi$$-Periodic and continuously differentiable on the real line. Then the Fourier series of $$f(x)$$ converges to $$f(x)$$ uniformly on $$\left[ { - \pi ,\pi } \right]$$.

I've seen the above claim in numerous places and even with weakened conditions (that the function is periodic, continuous but only piecewise smooth). I'm also very familiar with the proof (it makes use of the Fourier series of the derivative, Cauchy Schwarz inequality and the Weierstrass M-test).

Yet my professor insisted throughout the entire semester that the above claim is false. So either I have missed something or he's wrong (which I doubt since he's one of the smartest people in the math department)

Is the claim true or false?

• It's certainly true. Maybe he was simply wrong, or maybe it was actually some other similar statement that he was saying is false. Ask him for a counterexample... – David C. Ullrich Jan 13 at 13:15
• Thanks! I will pursue this matter further but I am relieved to know that it is not something I have missed – zokomoko Jan 13 at 21:56

If $$f$$ is periodic, continuous and of bounded variation on $$[0,2\pi]$$ the the Fourier series of $$f$$ converges to $$f$$ uniformly on $$[0,2\pi]$$. In particulr this is true for continuously differentiable periodic functions. So your professor is wrong. It is easy to find references. One reference is https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=2ahUKEwi9ne2j2urfAhWBXysKHfWCAVoQFjADegQICBAC&url=http%3A%2F%2Fcourses.mai.liu.se%2FGU%2FTATA57%2FDokument%2FFourierSeries2.pdf&usg=AOvVaw1RY7-Tn-XSApbGIaIcR_uo