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:

    “There must be a profound recognition that parents are the first teachers and that education begins before formal schooling and is deeply rooted in the values, traditions, and norms of family and culture.”
    —Sara Lawrence Lightfoot (20th century)

    “The most distinct and beautiful statement of any truth must take at last the mathematical form.”
    —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)

    “Culture relates to objects and is a phenomenon of the world; entertainment relates to people and is a phenomenon of life.”
    —Hannah Arendt (1906–1975)