NPTEL IIT Guwahati

Gibbs Phenomenon: When is discontinuous, the graph of the partial sum $s_{n}(x)$ of the Fourier series of exhibits over and under shoots close to the discontinuities. This is called the Gibbs Phenomenon, and it persists for all values of . This behaviour reflects the way of continuous function $s_{n}(x)$ obtained from the Fourier series approximates the behaviour of at a point of discontinuity.

Definition: A function is said to satisfy a Lipschitz condition at if for some and $\delta > 0$ , whenever .
If is differentiable at , satisfies a Lipschitz condition at .

The following is a weaker form of the sufficient condition for the pointwise convergence of Fourier series of .
Theorem: Let be a periodic function with period defined on $[-p, \; p]$ . If is Riemann integrable on $[-p, \; p]$ and if satisfies Lipschitz condition at , then the Fourier series of converges to $f^{*}(x)$ .


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Gibbs Phenomenon: When is discontinuous, the graph of the partial sum $s_{n}(x)$ of the Fourier series of exhibits over and under shoots close to the discontinuities. This is called the Gibbs Phenomenon, and it persists for all values of . This behaviour reflects the way of continuous function $s_{n}(x)$ obtained from the Fourier series approximates the behaviour of at a point of discontinuity. Definition: A function is said to satisfy a Lipschitz condition at if for some and $\delta > 0$ , whenever . If is differentiable at , satisfies a Lipschitz condition at . The following is a weaker form of the sufficient condition for the pointwise convergence of Fourier series of . Theorem: Let be a periodic function with period defined on $[-p, \; p]$ . If is Riemann integrable on $[-p, \; p]$ and if satisfies Lipschitz condition at , then the Fourier series of converges to $f^{*}(x)$ .

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