Abstract
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.
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Acknowledgements
Kasso A. Okoudjou was partially supported by the U. S. Army Research Office grant W911NF1910366, and an MLK visiting professorship at MIT. Jiang, Lan, Sule, and Venkat would like to acknowledge the REU at Cornell University.
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Jiang, Q., Lan, T., Okoudjou, K.A. et al. Sobolev Orthogonal Polynomials on the Sierpinski Gasket. J Fourier Anal Appl 27, 38 (2021). https://doi.org/10.1007/s00041-021-09819-0
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DOI: https://doi.org/10.1007/s00041-021-09819-0