# Are the two functions equal? If not, what relation are they?

We have

$$A=\frac{1}{2\pi}\int_{-\infty}^{\infty}\prod_{i=1}^{\infty}\frac{\sin(v2^{-k})}{v2^{-k}}e^{ivx}dx$$

$$B=\frac{1}{2}\sum_{k=1}^{n}\cos (\pi kx)\prod_{i=1}^{m}\frac{\sin(\pi k2^{-i})}{\pi k2^{-i}}$$ where B is the Fourier approximation of A (However, I don't know how to verify this).

My question is:Are the two functions equal(update:if $n,m \to \infty$)? If not, what relation are they, such as $A\approx B$?

They're certainly not equal since $B$ depends on $n,m$ and $A$ does not. – Nate Eldredge Dec 2 '13 at 16:36
I guess the question is whether they become equal in the limit $n,m\rightarrow\infty$, which at least formally seems to be the case. – Carlo Beenakker Dec 2 '13 at 17:05