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)

    “It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.”
    —Henry David Thoreau (1817–1862)

    “Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.”
    —Willard Van Orman Quine (b. 1908)

    “When the ice is covered with snow, I do not suspect the wealth under my feet; that there is as good as a mine under me wherever I go. How many pickerel are poised on easy fin fathoms below the loaded wain! The revolution of the seasons must be a curious phenomenon to them. At length the sun and wind brush aside their curtain, and they see the heavens again.”
    —Henry David Thoreau (1817–1862)