We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket (\(SG\)), which is a fractal set that can be viewed as a limit of a sequence of finite graphs. These orthogonal polynomials arise through the Gram–Schmidt orthogonalisation process applied on the set of monomials on \(SG\) using several notions of a Sobolev inner products. After establishing some recurrence relations for these orthogonal polynomials, we give estimates for their \(L^2\), \(L^\infty \), and Sobolev norms, and study their asymptotic behavior. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation.

我们在Sierpiński垫片（\（SG \））上开发了Sobolev正交多项式的理论，这是一个分形集，可以看作是有限图序列的极限。这些正交多项式是通过使用Sobolev内积的几个概念应用于\（SG \）上的单项式集合的Gram–Schmidt正交化过程产生的。在为这些正交多项式建立一些递归关系之后，我们给出它们的\（L ^ 2 \），\（L ^ \ infty \）和Sobolev范数的估计，并研究它们的渐近行为。最后，我们研究多项式零集的性质，并开发快速的计算工具来探索正交和插值的应用。