4
$\begingroup$

Plotting the series $$\displaystyle y = \sum_{k} \frac{\sin kx }{k}$$

In the limit it would look like

enter image description here

Taking a finite number of terms, I want to understand what is the reason for the jiggling at the extremes, while there the jiggling in the middle is so small its not noticable.

enter image description here

I truncated the sum to $1,2,3\; \mbox{and}\;4$ terms but cannot deduce much of a reason. enter image description here

The "jiggling" was noticeable here because the sum is linear in the limit, however, for an expression like $$p(x) = x\prod_k\Big(1-\frac{x^2}{k^2\pi^2}\Big) $$

Does the truncated expression oscillate back and forth the limit?

  • 10
    http://en.wikipedia.org/wiki/Gibbs_phenomenon2011-08-06
  • 0
    @Qiaochu so for the second part, the answer is no? As $\sin$ is a continouous function being approximated by continuous functions, the Gibbs phenomenon should not occur.2011-08-06
  • 0
    I don't understand the second question.2011-08-06
  • 0
    Hmm. The Gibbs phenomenon wouldn't apply to the second question, which is if the "partial products" would jiggle around the limiting value (of $\sin x$) (Gibbs phenomenon is for Fourier series apparently).2011-08-06
  • 1
    In the full product, $p(x)=\sin(x)$. The partial products are not periodic, and are unbounded as $x\to\infty$. They get better for a wider range of x as more terms are incorporated into the product, but there are no discontiuities at which to observe anything like a Gibbs phenomenon.2011-08-06
  • 0
    IIRC, Gibbs _explained_ why this happens after Albert Michelson _observed_ it. The same Michelson of Michaelson--Morley fame. But I could be wrong.....2011-08-07
  • 1
    It is gratifying to see cited a Wikipedia article whose initial version I created in November 2003---a time when the few sensible people who'd heard of Wikipedia knew that it would never be of any value.2011-08-07
  • 0
    @Michael Thanks for creating the article, though as far as I remember Wikipedia was already appearing in the first two results on google in 2003 for most topics. (I was in 9th grade and had my first computer) Pagerank had already judged it to be a credible place by then :)2011-08-07
  • 0
    I remember Wikipedia showing up in similar positions in Google searches then too. I think it might have reached that point in 2002. I think in those days, Google was used mostly be literate people rather than by everyone.2011-08-07

2 Answers 2