Applications of Orthogonal Polynomials on the Sierpinski Gasket

Questions

• Can we use polynomials (orthogonal or otherwise) to accurately interpolate functions?

• Can we obtain an analog of Gauss-Legendre quadrature on SG?

• Can we develop an algorithm for polynomial quadrature on SG and determine error estimates for it?

Interpolation

• Polynomials of degree \(n\) can have more than \(3n+3\) zeros on SG.

• We need to establish the invertibility of

  \[M_n = \begin{bmatrix} P_{1}(x_1) & P_{1}(x_{2}) & \dots & P_{1}(x_{3n+3})\\ \vdots & \vdots & \ddots & \vdots \\ P_{3n+3}(x_{1}) & P_{3n+3}(x_{2}) & \dots & P_{3n+3}(x_{3n+3}) \end{bmatrix}\]

If we choose a rotationally symmetric set of points, the interpolation matrix \(M_n\) becomes circulant-block. This enables us to prove the invertibility of \(M_2\).

• The interpolation matrix for the \(P_{jk}\) polynomials on \(V_1\) points is invertible (by computation)

• By choosing \(n+1\) points on the left side of \(V_n\) along with their rotations, the interpolation matrix becomes circulant-block. These matrices are computationally invertible up to at least \(n = 50\).

• The general case is unknown.

Note: The determinants were computed with exact rational artithmetic, so they are non-zero. The rapid decay in the determinants is due to the fact that the monomial values decay as their degree increases. As a result, solving the interpolation system computationally is not useful even though it is possible.

Quadrature

• Gauss-Legendre quadrature on \(\mathbb{R}\) requires the polynomial division algorithm on \(\mathbb{R}\). However, we do not have this on SG.

• We have a pseudo-division algorthim on SG, but the quotient is a linear combination of powers of the Green’s operator: \[\mathcal{Q}_f = \frac{1}{b_m} \sum\limits_{i = 0}^{n-m}c^{(i)}\mathcal{G}^{(n-m-i)}\]

• We can try to use n-Harmonic spline quadrature, but this comes back to the interpolation problem since we need to solve \[\begin{bmatrix} P_{1}(x_1) & P_{1}(x_{2}) & \dots & P_{1}(x_{3n+3})\\ \vdots & \vdots & \ddots & \vdots \\ P_{3n+3}(x_{1}) & P_{3n+3}(x_{2}) & \dots & P_{3n+3}(x_{3n+3}) \end{bmatrix} \begin{bmatrix} w_1\\ \vdots\\ w_{3n+3} \end{bmatrix} = \begin{bmatrix} \int_{SG}P_{01}\\ \vdots\\ \int_{SG}P_{n3} \end{bmatrix}\]

N-Harmonic Extension: The Solution to Many Polynomial Problems on SG

• Is there an algorithm to extend a function defined on \(3n+3\) vertices of \(V_n\) n-Harmonically?

• Given this algorithm, we have the following quadrature error estimate: \[\left|I_n^m(f) - \int_{SG}f\right|\leq c_15^{-(n+1)(m-n)}\|\Delta^{(n+1)}f\|_\infty\]

Outstanding Questions Regarding Polynomials on SG

1. Is \(\partial_nf_n \neq 0 \ \forall \ n\)?

2. Is \(P_{j1} > 0\) except at \(q_0\)?

3. Interpolation problem: Does sampling an \(n\) degree polynomial on any \(3n + 3\) points uniquely determine the polynomial?

Questions 2 and 3 can be solved given an n-Harmonic Extension Algorithm.