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Approximating Continuous Functions on Persistence Diagrams Using Template Functions
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Computes lattice Green's functions in two dimensions for square lattices. The method is a set of recurrence relations implemented in high-precision arithmetic. The square lattice results were needed for a particular research project, which is why this was developed, but similar recursion relations are available for three-dimensional lattices and the same method should work.more » « less
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Abstract We establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g.,$$B({\mathcal H})$$. In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the$$k{\mathrm {th}}$$directional derivative of any NC function at a scalar point is ak-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball.more » « less
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10208-022-09567-7
