Questions related to Fourier Analysis

1. Jul 24, 2008

xalvyn

Dear all

I have recently taken up the study of Fourier analysis. My background knowledge is limited - some basic notions of analysis, including the Riemann integral, as well as uniform and pointwise convergence of series of functions. These are not exactly homework problems, but questions that I have been thinking about during the course of my self-study - and to which I have so far not found any satisfactory answers. I would gladly appreciate any help rendered.

1. The problem statement, all variables and given/known data

1. Is it true that, if $$0 < \theta _0 < \pi$$, then $$|\sin n\theta _0|$$ diverges as $$n$$ tends to infinity? If so, how should we prove it generally?

2. Suppose that the Fourier series of a continuous function $$f(x)$$ converges pointwise at $$x=x_0$$. Is it necessarily true that $$\sum_{i=-\infty}^\infty \hat{f} (n) e^{in x_0} = f(x_0)$$?

2. Relevant equations

3. The attempt at a solution

1. This seems much easier to prove for individual values of $$\theta _0$$, but I find it difficult to extend the argument to a general case. I could, however, prove that the value of $$|\sin \theta _0|$$ does not approach 0: we may show that there exists a $$c > 0$$ such that there are arbitrarily large values of $$n$$ satisfying $$|\sin n \theta _0| > c > 0.$$ Choose $$c$$ less than $$\frac{1}{2}r$$ by a very small amount, where $$k$$ is the largest positive integer satisfying $$\pi = k \theta _0 + r$$ with $$r > 0$$. If, then, $$|\sin p \theta _0| \leq c$$, it follows that $$|\sin (p+1) \theta _0| > c$$. Extending this sort of argument seems more difficult.

2. I could think of a discontinuous function for which the statement is false: the sawtooth function defined by

$$f(x) = -\frac{\pi}{2}-\frac{x}{2}$$ if $$-\pi < x \le 0$$;

$$\frac{\pi}{2} - \frac{x}{2}$$ if $$0 < x < \pi$$.

In this case, while the Fourier series of f converges to 0 at $$x=0$$, the value of $$f(0)$$ is $$-\frac{\pi}{2}$$. I cannot, however, construct an example of a continuous function for which the statement is not true.

Thanks for any help.