Gibbs Phenomenon - Formal Mathematical Description of The Phenomenon

Formal Mathematical Description of The Phenomenon

Let be a piecewise continuously differentiable function which is periodic with some period . Suppose that at some point, the left limit and right limit of the function differ by a non-zero gap :

For each positive integer N ≥ 1, let SN f be the Nth partial Fourier series

S_N f(x) := \sum_{-N \leq n \leq N} \hat f(n) e^{2\pi i n x/L} = \frac{1}{2} a_0 + \sum_{n=1}^N \left( a_n \cos\left(\frac{2\pi nx}{L}\right) + b_n \sin\left(\frac{2\pi nx}{L}\right) \right),

where the Fourier coefficients are given by the usual formulae

Then we have

and

but

More generally, if is any sequence of real numbers which converges to as, and if the gap a is positive then

and

If instead the gap a is negative, one needs to interchange limit superior with limit inferior, and also interchange the ≤ and ≥ signs, in the above two inequalities.

Read more about this topic:  Gibbs Phenomenon

Famous quotes containing the words formal, mathematical, description and/or phenomenon:

    “The conviction that the best way to prepare children for a harsh, rapidly changing world is to introduce formal instruction at an early age is wrong. There is simply no evidence to support it, and considerable evidence against it. Starting children early academically has not worked in the past and is not working now.”
    —David Elkind (20th century)

    “All science requires mathematics. The knowledge of mathematical things is almost innate in us.... This is the easiest of sciences, a fact which is obvious in that no one’s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon.”
    —Roger Bacon (c. 1214–c. 1294)

    “He hath achieved a maid
    That paragons description and wild fame;
    One that excels the quirks of blazoning pens.”
    —William Shakespeare (1564–1616)

    “The teacher must derive not only the capacity, but the desire, to observe natural phenomena. In our system, she must become a passive, much more than an active, influence, and her passivity shall be composed of anxious scientific curiosity and of absolute respect for the phenomenon which she wishes to observe. The teacher must understand and feel her position of observer: the activity must lie in the phenomenon.”
    —Maria Montessori (1870–1952)