# Padua-type pointset for functions singular on line $x=y$

The Padua points $\mathrm{Pad}_{n} \subset [-1,1]^{2}$ are a unisolvent pointset with optimal growth of Lebesgue constant, described in detail here. With some work they can be used to generate a multivariate Chebyshev series representation of the function they attempt to interpolate.

I would like to use the Padua points to generate a multivariate Chebyshev series of a function singular along the line $x=y$, but this is impossible because two of the Padua points are consecutive vertices of the square.

Is there a way to generate a multivariate Chebyshev polynomial interpolant of a function $f\colon [-1,1]^2\to \mathbb{R}$ which is singular on the line $x=y$, perhaps using a modification of the Padua points?

Update: The goal is to write a piece of code which projects as many 2D functions defined on rectangular regions as possible onto Chebyshev polynomials and allow for interpolation and recovery of the coefficients for perhaps other uses. This is similar to the goals of chebfun in 2D.

• Is not a bit odd trying to interpolate a function with a discontinuity? – nicoguaro Mar 30 '18 at 17:01
• @nicoguaro: Gaussian quadrature of integrands with singular endpoints is the same thing-interpolating with a polynomial. Delta sequence approximations to Green's functions act in a similar way. It is delicate, and requires adaptivity, but it can be done. (Whether, in my case, it will actually work, is another question.) – user14717 Mar 31 '18 at 1:50
• Gaussian quadrature for functions with singularities is commonly done using orthogonal polynomials with respect to some weight function. – nicoguaro Mar 31 '18 at 2:00
• That works better, but it cannot be totally black boxed. – user14717 Mar 31 '18 at 2:02
• Gaussian quadrature of functions with singular endpoints still works, depending on the strength of the singularity. – user14717 Mar 31 '18 at 2:03