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A fractal is an infinitely self-similar geometric shape. The graph Laplacian can be used to emulate differential equations over graph approximations of fractals. It is possible to define a Green’s function to provide solutions to Laplacian-based differential equations, which can be used to generate polynomials over fractals. These polynomials can be orthogonalized with the Gram-Schmidt process. ... read moreThis thesis begins with a Jacobi matrix generated from orthogonal polynomials over the Sierpinski Gasket, a fractal that has been popularly studied. Then, using the three-term recurrence relation of classical orthogonal polynomials, this paper derives new families of orthogonal polynomials over the real line. This thesis concludes that the distribution of the zeros of these families of orthogonal polynomials conforms to the interlacing property and the Christoffel-Darboux identities. Furthermore, this thesis attempts to identify the probability measure generating the underlying Jacobi matrix.
Thesis (B.S.C.S.)--Tufts University, 2023.
Submitted to the Dept. of Mathematics.
Advisor: Kasso Okoudjou.read less
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