Fourier Series - proving function is continuous

[ramdayal9]

Homework Statement

Let f be an integrable, periodic function whose Fourier coefficients satisfy [itex] \sum_{-\infty}^{\infty} n^6 |\hat{f}(n)|^2 < \infty [/itex]. Prove that f is continuous.

Homework Equations

Looking at my notes, the only relevant things i have for this question (i think) are Bessel's inequality (but that requires f to be continuous...), the Riemann-Lebesgue Theorem, Parseval's equality and also the decay and regularity of Fourier series. The statement that the uniform limit of a sequence of continuous functions is continuous, maybe I can use this?

The Attempt at a Solution

I thought this question would be simple, but I'm finding that its not. I have tried 2 methods:
1) [itex]\sum_{-\infty}^{\infty} n^6 |\hat{f}(n)|^2 < \infty [/itex] implies [itex] n^6 |\hat{f}(n)|^2 \rightarrow 0 [/itex] hence [itex]|\hat{f}(n)| \rightarrow 0 [/itex]. I need to show that f is continuous i.e. the converse of the Riemann Lebesgue Lemma. How would I proceed from here?
2) using L^2 convergence theorem (parseval's), [itex]\sum_{-\infty}^{\infty} |c_n|^2 \< \infty [/itex] hence there exists function g such that [itex]\hat{g}(n)=c_n=n^3 |\hat{f}(n)| [/itex]. Is this the route I should proceed?

The hypothesis also states that f is integrable, but i don't know how to use this here! I am also thinking of the M test, but don't know if I can apply it to a sum like this... Any help will be appreciated!

 

[mighty2000]

Thank you for posting this interesting problem. I would approach this question by first looking at the properties of the Fourier coefficients and how they relate to the continuity of a function. In particular, the fact that \sum_{-\infty}^{\infty} n^6 |\hat{f}(n)|^2 < \infty tells us that the Fourier coefficients decay very quickly as n increases. This suggests that the function f is very smooth and regular, and therefore likely to be continuous.

To prove this, we can use the fact that the uniform limit of a sequence of continuous functions is continuous. As you mentioned, we can construct a sequence of functions g_n such that \hat{g_n}(n) = n^3 |\hat{f}(n)|, and then use Parseval's equality to show that this sequence converges uniformly to f. Since each g_n is continuous, and the uniform limit of continuous functions is continuous, we can conclude that f is continuous.

Alternatively, we can also use the Riemann-Lebesgue lemma to show that the Fourier coefficients of f must decay at infinity. This, combined with the fact that they also decay very quickly, implies that f must be continuous.

I hope this helps guide you towards a solution. Good luck!