![]() We shed new light on their approach by considering a different set of unknowns and by using the theory of formal orthogonal polynomials. ![]() It can be seen as a continuation of the pioneering work done by Delves and Lyness. Our approach could therefore be called a logarithmic residue based quadrature method. This form involves the logarithmic derivative $f'/f$ of $f$. Our principal means of obtaining information about the location of the zeros is a certain symmetric bilinear form that can be evaluated via numerical integration along $\gamma$. We start by studying the problem of computing all the zeros of an analytic function $f$ that lie inside a positively oriented Jordan curve $\gamma$. This thesis is a blend of computational complex analysis and numerical linear algebra. The possibilities are literally infinite. The 100,000 new subfamilies generated by combining classical interpolation problems with classical integral transforms is only a lower bound. For example, the Geddes-Newton-Laplace subfamily of multivariate series performs Newton interpolation in the Laplace transform domain. The explosion results from combining the abstract interpolation properties which characterize classical series expansions (Newton, Taylor, Hermite, Fourier, etc.) with classical linear integral transforms (Laplace, Fourier, etc.). This opens the door for a combinatorial explosion of fundamentally new types of infinite series expansions for multivariate functions. It is based on the simple observation that the family of abstract splitting operators I invented (to generate Geddes series expansions) is an invariant family under conjugation by invertible linear transformations. The 100,000 subfamilies of Geddes series expansions resulted from a new idea I developed after I submitted my thesis. When implemented as a hybrid symbolic-numeric method, Geddes series expansions provides the computational means to solve high-dimensional multivariate problems which were once intractable. ![]() This is achieved in a natural way by specifying an abstract interpolation problem for the given multivariate function and generating the unique multivariate series expansion which solves it. ![]() The key is to replace the fixed basis functions of the last three centuries with an infinite variety of dynamically generated basis functions. This is a major paradigm shift, and it unlocks a world of new possibilities. I have invented over 100,000 different types of Geddes series and have named each type after the mathematicians whose historic contributions made this work possible (e.g., Geddes-Newton-Laplace series). In my PhD thesis, I introduced a vast new family of infinite series expansions for multivariate functions, which I named "Geddes series" in honor of my thesis supervisor. ![]()
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