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<title>Gibbs phenomenon and Cesàro mean</title>
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<h1>Gibbs phenomenon and Cesàro mean</h1>
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<p><i>January 4, 2018 — Support my next blog post, <a href="https://www.paypal.com/donate/?hosted_button_id=UCLCSJLFL433E">buy me a coffee</a> ☕.</i></p>
<center>
<img src="/blog/gibbs1.jpg" class="img-responsive">
</center>
<p>In this post, I talk about a fascinating subject: the
<a href="http://en.wikipedia.org/wiki/Convergence_of_Fourier_series#Uniform_convergence">convergence of
Fourier series</a> and the
<a href="http://en.wikipedia.org/wiki/Gibbs_phenomenon">Gibbs phenomenon</a>.</p>
<p><a href="http://en.wikipedia.org/wiki/Periodic_function">Periodic</a>
<a href="http://en.wikipedia.org/wiki/Continuous_function">continuous</a>
functions \(f\)
of <a href="http://en.wikipedia.org/wiki/Bounded_variation">bounded variation</a>
have a unique and
<a href="http://en.wikipedia.org/wiki/Uniform_convergence">uniformly convergent</a>
<a href="http://en.wikipedia.org/wiki/Fourier_series">Fourier series</a>,
that is, the <i>partial sums</i>
$$
f_n(\theta) = \sum_{k=0}^{n}a_k\cos(k\theta) + \sum_{k=1}^{n}b_k\sin(k\theta),
$$
with <i>Fourier coefficients</i>
$$
a_k = \frac{1}{\pi}\int_0^{2\pi}f(\theta)\cos(k\theta)d\theta,
$$
and
$$
b_k = \frac{1}{\pi}\int_0^{2\pi} f(\theta) \sin(k\theta)d\theta,
$$
converge uniformly to \(f\). (Note that \(1/\pi\) is changed to \(1/(2\pi)\) for \(a_0\).) We write
$$
f(\theta) = \sum_{k=0}^{\infty}a_k\cos(k\theta) + \sum_{k=1}^{\infty}b_k\sin(k\theta).
$$
</p>
<p>For functions \(f\) that are merely of bounded variation, the convergence is only
<a href="http://en.wikipedia.org/wiki/Pointwise_convergence">pointwise</a>:
at every point \(\theta\), the partial sums converge to \(\frac{1}{2}[f(\theta^-) + f(\theta^+)]\);
in particular, these converge to \(f(\theta)\) at every point of continuity. What happens at points of
<a href="http://en.wikipedia.org/wiki/Classification_of_discontinuities">discontinuity</a>?</p>
<p>A classical example is the square wave
$$
f(\theta) = \frac{\pi}{4}\mathrm{sign}(\pi - \theta), \quad \theta\in[0,2\pi].
$$
This function is of bounded variation and discontinuous at \(0\), \(\pi\) and \(2\pi\).
Its Fourier coefficients can be computed analytically and are given by
$$
a_k = 0, \quad b_{2k} = 0, \quad b_{2k+1} = \frac{1}{2k+1}.
$$
When we compute the partial sums for \(n=16,32,64,128\) we obtain the picture at the top.
This is the so-called <i>Gibbs phenomenon</i>, which involves both the fact that the partial sums <i>overshoot</i> at a discontinuity and that this overshoot does not disappear as more terms are added to the sums. The overshoot at \(0\), \(\pi\) and \(2\pi\) is known exactly and is given by
$$
\frac{1}{2}\int_0^{\pi}\frac{\sin(\theta)}{\theta}d\theta - \frac{\pi}{4} = \frac{\pi}{2}\times(0.0895\ldots),
$$
or about \(9\%\) of the jump. Note that, as \(n\) increases, the position of the overshoot moves closer
to the discontinuity. The Gibbs phenomenon is a very standard result that most (applied) mathematicians know.</p>
<p>What is perhaps less known is a simple cure using the \((C,1)\)
<a href="http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation">Cesàro mean</a> of the partial sums, which is simply the
<a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a>,
$$
\sigma_n(\theta) = \frac{1}{n+1}\sum_{k=0}^{n}f_k(\theta).
$$
For integrable functions \(f\), one can show that \(\sigma_n(\theta)\)
converges to \(\frac{1}{2}[f(\theta^-) + f(\theta^+)]\); in particular,
this converges to \(f(\theta)\) at every point of continuity,
and for continuous functions the convergence is uniform—this is
<a href="http://en.wikipedia.org/wiki/Fej%C3%A9r%27s_theorem">Fejér's thereom</a>. Moreover, if \(m\leq f(\theta)\leq M\) then \(m\leq \sigma_n(\theta)\leq M\).
Therefore, the \((C,1)\) Cesàro mean does not show the Gibbs phenomenon, as the following figure illustrates:</p>
<center>
<img src="/blog/gibbs2.jpg" class="img-responsive">
</center>
<p>With <a href="http://home.cc.umanitoba.ca/~slevinrm/">Mikael Slevinsky</a>,
we are using the \((C,2)\) Cesàro mean for
<a href="http://en.wikipedia.org/wiki/Spherical_harmonics">spherical harmonic</a> expansions on the sphere to post-process numerical solutions of nonlocal
<a href="http://en.wikipedia.org/wiki/Partial_differential_equation">PDEs</a>—the paper is coming soon!</p>
<hr>
<h4>Blog posts about spectral methods</h4>
<p>2020 <a href="2020-05-19.html">Exponential integrators for stiff PDEs</a></p>
<p>2018 <a href="2018-12-05.html">Computer-assisted proofs for PDEs</a></p>
<p>2018 <a href="2018-02-27.html">Spherical caps in cell polarization</a></p>
<p>2018 <a href="2018-01-25.html">Solving nonlocal equations on the sphere</a></p>
<p>2018 <a href="2018-01-04.html">Gibbs phenomenon and Cesàro mean</a></p>
<p>2017 <a href="2017-10-26.html">Solving PDEs on the sphere</a></p>
<p>2017 <a href="2017-10-09.html">When planets dance</a></p>
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