Sobolev Orthogonal Polynomials on the Sierpinski Gasket

Origin

  • Idea (Lewis): Given \(f \in L^2[-1,1]\), find polynomial \(g\) with deg \(g\le n\) such that \(g\) minimizes the quantity \(\|f-g\|_{H}\), where \(\|h\|_{H}^2:=\int (h^2(x)dx+h'{^2}(x))dx\)

  • Solution: \(\widetilde g=\sum\limits^{n}_{i=0} \langle f,e_i\rangle_H\,e_i\), \(\{e_i\}\) orthonormal basis

  • Core idea: Approximate a function with polynomials as “close” as possible.

Generalization to SG

  • Unless specified we will consider the Sobolev inner product \(\langle f,g\rangle_H:=\langle f,g\rangle_{L^2}+\lambda\langle \Delta f,\Delta g\rangle_{L^2}\) for some nonnegative constant \(\lambda\), where \(\langle f,g\rangle_{L^2}:=\int fg\,d\mu\) for a regular Borel probability measure \(\mu\) that is symmetric with respect to the line passing through \(q_0\) and the midpoint of the side opposing \(q_0\)

  • the monic Sobolev polynomials \(\{S_{nk}(x;\lambda)\}_{n=0}^{\infty}\)(For simplicity, write as \(\{S_{n}\}_{n=0}^{\infty}\)), is obtained by Gram-Schmidt Process.

Recurrence Relations (Case 1: \(k = 2\) or \(3\))

Theorem:/ When \(k = 2\) or \(3\) we have the following recurrence relation for \(n\geq-1\), where \(S_{-1}:=0\). \[S_{n+2} - a_nS_{n+1}-b_nS_n = f_{n+2}\] where \(f_{n+2} := \mathcal{G}(p_{n+1})\), and \(\mathcal{G}(g)(x):=-\int_{SG}G(x,y)g_{n+1}(y)dy\),
\[a_n = -\frac{\langle f_{n+2}S_{n+1}\rangle_H}{\|S_{n+1}\|_H^2}\] \[b_n = -\frac{\langle f_{n+2},S_{n}\rangle_H}{\|S_{n}\|_H^2}\]

Remark:/ When \(k = 2\) or \(3\), the same recursive relation is still valid if we replace the Sobolev inner product by \(\langle f,g\rangle_H:=\) \[\langle f,g\rangle_{L^2}+\lambda_1 \langle \Delta f,\Delta g\rangle_{L^2}+\lambda_2\,\varepsilon(f,g)+\] \[[f(q_0)\,f(q_1)\,f(q_2)] M [g(q_0)\,g(q_1)\,g(q_2)]^T\] for nonnegative constants \(\lambda_1\) and \(\lambda_2\), positive semidefinite \(3\times3\) matrix \(M\)

Corollary:/ When \(k =2\) or \(3\), \((a_n, b_n)\) is the unique solution to the system \(a_n S_{n+1}(q_1)+b_nS_n(q_1)=S_{n+2}(q_1)\) and \(a_n \partial_n S_{n+1}(q_1)+b_n\partial_n S_n(q_1)=\partial_n S_{n+2}(q_1)\). In particular, the matrix \(\left[\begin{matrix} S_{n+1}(q_1)&S_n(q_1)\\ \partial_n S_{n+1}(q_1) &\partial_n S_n(q_1) \end{matrix}\right]\) is non-singular for any integer \(n\geq 0\).

Asymptotics (Case 1: \(k = 2\) or \(3\))

  • First, We are interested in the case when \(\lambda\rightarrow\infty\).

  • Estimates:/ \(\|S_n\|_{H}^2=\Theta (\lambda)\), \(|a_n|=O(\lambda^{-1})\), \(|b_n|=\Theta(\lambda^{-1})\) \[\|\Delta S_n\|_{L^2}^2\le \lambda^{-1}\|G\|_{L^2}^2\|p_{n-1}\|_{L^2}^2+\|p_{n-1}\|_{L^2}^2\] \[\|S_n\|_{L^{\infty}}\le C(1+\lambda^{-\frac12})\|p_{n-1}\|_{L^2}\] (C is independent of \(n\) and \(\lambda\))

Theorem:/ Suppose \(k=2\) or \(3\). Then for any \(n\ge3\), \(S_n(x;\lambda)\) converges to \(f_n\) uniformly in \(x\) as \(\lambda\rightarrow\infty\). Consequently \(\Delta S_n\rightarrow p_{n-1}\) uniformly as \(\lambda\rightarrow\infty\). Also,

\[\lambda(S_n(\lambda)-f_n)\rightarrow-\frac{\langle f_n,f_{n-1}\rangle_{L^2}}{\|p_{n-2}\|_{L^2}^2}f_{n-1}-\frac{\|p_{n-1}\|_{L^2}^2}{\|p_{n-3}\|_{L^2}^2}f_{n-2}\] uniformly in \(x\) as \(\lambda\rightarrow\infty\)

Recurrence Relations (Case 2: \(k = 1\))

  • More complicated!

  • Requires a conjecture: \(\partial_n f_t(q_0)\neq 0\), where \(f_t:=\mathcal{G} (p_{t-1})\).

  • Theorem:/ Let \(S_{-1}:=0\), k=1, \(f_{n+2}= \mathcal{G}(p_{n+1})\) and suppose that \(\partial_n f_{n+2}(q_0)\neq 0\), then \(S_{n+3}-a_nS_{n+2} - b_nS_{n+1}-c_nS_n = f_{n+3}+d_nf_{n+2}\), The matrix\[\begin{aligned} \begin{bmatrix} S_{n+2}(q_1)&S_{n+1}(q_1)&S_{n}(q_1)\\ \partial_n S_{n+2}(q_1) &\partial_n S_{n+1}(q_1)&\partial_n S_{n}(q_1)\\ S_{n+2}(q_0)&S_{n+1}(q_0)&S_{n}(q_0) \end{bmatrix} \end{aligned}\] is non-singular.

Asymptotics (Case 2: \(k = 1\))

Theorem:/ Assume the normal derivative conjecture is true. Then there exists a sequence of monic polynomials \(\{g_n\}_{n=0}^{\infty}\) independent of \(\lambda\) such that for any \(n\ge0\), \(deg\, g_n=n\), \(S_n\) converges uniformly in \(x\) to \(g_n\). And \(g_{n+3}+d_ng_{n+2}=f_{n+3}+d_nf_{n+2}\) for any \(n\ge 1\). For the basic cases, \(g_0=p_0\), \(g_1=p_1\), \(g_{2}+d_{-1}g_{1}=f_{2}+d_{-1}f_{1}-\frac{\langle f_2+d_{-1}f_1,{g_0}\rangle_{L^2}}{{\|g_0\|_{L^2}^2}}g_{0}\), and \(g_{3}+d_{0}g_{2}=f_{3}+d_{0}f_{2}-\frac{\langle f_3+d_{0}f_2,{g_0}\rangle_{L^2}}{{\|g_0\|_{L^2}^2}}g_{0}\). Moreover, for any \(\alpha<1\), \(n\ge0\), \(\lim\limits_{\lambda \rightarrow\infty}\lambda^\alpha(S_n(\lambda)-g_n)=0\) uniformly in \(x\).

Generalized Recurrence (\(k = 2\) or \(3\))

  • Consider the inner product: \(\langle f,g\rangle_{H^m} = \sum\limits_{l = 0}^m \lambda_l\int_{SG}\Delta^lf\Delta^lg\,d\mu\)

  • Theorem:/ \(S_{n+m+1} - \mathcal{F}_{n+m+1} - \sum\limits_{l = 0}^{2m-1}a_{n, l}S_{n+m-l} = 0\), where \(\mathcal{F}_{n+m+1} := \mathcal{G}^mp_{n+1}\)

  • Remark:/ It is still true if we consider \(\langle f,g\rangle_{H^m}= \sum\limits_{l = 0}^m \lambda_l\int_{SG}\Delta^lf\Delta^lg\,d\mu+\sum\limits_{l=0}^{m-1}\beta_l\,\varepsilon(\Delta^l f,\Delta^l g)+\nonumber \sum\limits_{l=0}^{m-1}[\Delta ^lf(q_0)\,\Delta ^lf(q_1)\,\Delta ^lf(q_2)] M_l [\Delta ^lg(q_0)\,\Delta ^lg(q_1)\,\Delta ^lg(q_2)]^T\), where \(M_l\) are positive definite \(3\times 3\) matrices.

Generalized Asymptotics (\(k = 2\) or \(3\))

Asymptotic: we consider the case \(\lambda_m\rightarrow\infty\), and the other parameters are bounded.
Theorem:/ Suppose \(k=2\) or \(3\), and there exists \(M>0\) such that \(\lambda_l\le M\) for any \(l<m\). Then for any \(n\ge 2m+1\), we have\[\|S_n-\mathcal{F}_n\|_{L^2}\le C(n,M,m,\mu)\lambda_m^{-1}\] Consequently, \(\lim\limits_{\lambda_m\rightarrow\infty}\|\Delta^i S_n-\mathcal{G}^{m-i}p_{n-m}\|_{L^\infty}\rightarrow 0\) for any \(0\le i\le m\).

Zeros

Theorem:/ (Topological result): Let \(f\) be a continuous function defined on \(SG\). Suppose \(f\) has finitely many zeros. Let \(Z_0\) be the intersection of zero set \(Z\) of \(f\) and \(V^*\). Then for any connected component \(D\) in \(SG\setminus Z_0\), either \(f\ge0\) on \(D\) or \(f\le0\) on \(D\).

Theorem:/ Suppose \(f(x)=\sum\limits^\infty_{j=t_1}c_{j1}P_{j1}^{(0)}(x)+\sum\limits^\infty_{j=t_2}c_{j2}P_{j2}^{(0)}(x)+\sum\limits^\infty_{j=t_3}c_{j3}P_{j3}^{(0)}(x)\) where \(c_{t_1,1}\), \(c_{t_2,2}\) and \(c_{t_3,3}\) are nonzero and has zero set \(Z\). Then

  1. \(Z\) is compact and nowhere dense in \(SG\).

  2. If \(t_3<t_1-1\) and \(t_3<t_2\), then \(f\) has infinitely many zeros that has limit point \(q_0\). Moreover, suppose the conjecture \(P_{j1}>0\) is true, \(t_1\le t_2\) and \(t_1\le t_3\). Then \(q_0\) has a neighborhood \(U\) such that \(Z\cap U\subset\{q_0\}.\)