= Sine expansion and Rayleigh quotient of a parabolic Dirichlet trial
{title2=$x(1-x)=\frac8{\pi^3}\sum_{n\geq1,\ n\text{ odd}}\frac{\sin(n\pi x)}{n^3}$}
On $(0,1)$, integration by parts gives the displayed <Fourier sine series>. For normalized <Dirichlet eigenfunctions> $y_i=\sqrt2\sin[(i+1)\pi x]$, indexed from $i=0$, it has the form $A[y_0+\sum_{i\geq1}c_i y_i]$, with $A=4\sqrt2/\pi^3$ and $c_i=[1-(-1)^{i+1}]/[2(i+1)^3]$. Its <Rayleigh quotient> for $-d^2/dx^2$ is $10$, since $\int_0^1(u')^2dx=1/3$ and $\int_0^1u^2dx=1/30$. Reflection symmetry forces $c_1=0$, because that first excited eigenfunction is antisymmetric about $1/2$. Thus the first excited state's contribution to the error is exactly zero; the first nonzero correction is $c_2=1/27$.
Back to article page
