Browsing by Author "Bailey, Benjamin Aaron"
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Item Area of polygons in hyperbolic geometry(Texas Tech University, 2004-08) Bailey, Benjamin AaronConsider the Poincare model for hyperbolic geometry on the unit disc and an arbitrary n-gon in this geometry. Chapter I gives a brief introduction to the conformal metric, which generates hyperbolic geometry. In Chapter H, there is an analytic derivation of a convenient computational formula for the hyperbolic area of a hyperbolic n-gon in terms of the coordinates of its vertices, as well as an insightful geometric interpretation of this formula in terms of naturally occurring angles of the n-gon. Chapter III extends the formulas in Chapter II to the closed unit disc with an alternative, more geometrically motivated proof. Chapter IV uses the results of Chapter I to establish identities between hyperbolic area and perimeter of an n-gon. A proof of the existence of a solution for the following isoperimetric problem also is given: Maximize (if such a maximum exists) the area of an n-gon with fixed perimeter.Item Studies in Interpolation and Approximation of Multivariate Bandlimited Functions(2012-10-19) Bailey, Benjamin AaronThe focus of this dissertation is the interpolation and approximation of multivariate bandlimited functions via sampled (function) values. The first set of results investigates polynomial interpolation in connection with multivariate bandlimited functions. To this end, the concept of a uniformly invertible Riesz basis is developed (with examples), and is used to construct Lagrangian polynomial interpolants for particular classes of sampled square-summable data. These interpolants are used to derive two asymptotic recovery and approximation formulas. The first recovery formula is theoretically straightforward, with global convergence in the appropriate metrics; however, it becomes computationally complicated in the limit. This complexity is sidestepped in the second recovery formula, at the cost of requiring a more local form of convergence. The second set of results uses oversampling of data to establish a multivariate recovery formula. Under additional restrictions on the sampling sites and the frequency band, this formula demonstrates a certain stability with respect to sampling errors. Computational simplifications of this formula are also given.