Harmonic analysis for star graphs and the spherical coordinate trapezoidal rule (Q629455)

Using an orthogonal basis of eigenfunctions for the Laplace operator in \(L^2([0,1]^2)\), the author construct an orthogonal basis of eigenfunctions for the graph operator. Further, integrals over the unit sphere of \({\mathbb R}^3\) are transformed into integrals over \([0,1]^2\) by spherical coordinates. These integrals over \([0,1]^2\) are computed by a product trapezoidal rule.