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Lecture 4: The Gibbs Phenomenon

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According to our theory, with a complete set of basis functions we can represent any function exactly. We furthermore know how to obtain the best approximation if we use only a finite set of functions. Interestingly, even the best approximation can still have some substantial errors. Consider the error in the square-wave series. Observe that there is a jump just before the point of discontinuity. As it turns out, no matter how large $n$ is, this error remains, and it has an amplitude of about 9% of the discontinuity. As $n$ gets larger and larger, this wiggle becomes narrower and moves closer to the point of the discontinuity, but it never goes away. This overshoot phenomenon is known as the Gibbs phenomenon .

One of the important ramifications of this is in how we define functions to be equal. It is true that

\begin{displaymath}\int_{T_0} (f(t) - \sum_{n=0}^{\infty} C_n \cos(n\omega_0 t +
\theta_n))^2 dt = 0 \end{displaymath}
that is, there is zero error between the function and the Fourier series, as defined by this squared integral criterion. But it does not mean that the functions are point-for-point equal. In this case, the error region simply becomes so small that the integral is zero. But this does not mean that

\begin{displaymath}f(t) = \sum_{n=0}^{\infty} C_n \cos(n\omega_0 t + \theta_n)
\end{displaymath}
at every point. The mathematicians who like to leave no such stones unturned have made this an object of tremendous study, and consider such qualifications as equal ``almost everywhere'' (a.e.) or equal ``with probability one''. These are both in distinction to equal ``everywhere'' or equal ``always.'' We have to be careful what we mean when we say two things are equal! (Ayn Rand would probably have trouble with this: perhaps it is not the case that $A=A$ is always true!)
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 23). Lecture 4: The Gibbs Phenomenon. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/4_6node6.html. This work is licensed under a Creative Commons License Creative Commons License