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 formal Washington dinner party has all the spontaneity of a Japanese imperial funeral.”
    —Simon Hoggart (b. 1946)

    “What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
    —Boris Pasternak (1890–1960)

    “I fancy it must be the quantity of animal food eaten by the English which renders their character insusceptible of civilisation. I suspect it is in their kitchens and not in their churches that their reformation must be worked, and that Missionaries of that description from [France] would avail more than those who should endeavor to tame them by precepts of religion or philosophy.”
    —Thomas Jefferson (1743–1826)

    “This phenomenon is more exhilarating to me than the luxuriance and fertility of vineyards.”
    —Henry David Thoreau (1817–1862)