Quadrature, interpolation and observability

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Quadrature, interpolation and observability

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dc.creator Hodges, Lucille McDaniel
dc.date.available 2011-02-18T19:40:25Z
dc.date.issued 1997-12
dc.identifier.uri http://hdl.handle.net/2346/11588 en_US
dc.description.abstract Methods of interpolation and quadrature have been used for over 300 years. Improvements in the techniques have been made by many, most notably by Gauss, whose technique applied to polynomials is referred to as Gaussian Quadrature. Stieltjes extended Gauss's method to certain non-polynomial functions as early as 1884. Conditions that guarantee the existence of quadrature formulas for certain collections of functions were studied by TchebychefF, and his work was extended by others. Today, a class of functions which satisfies these conditions is called a Tchebycheff System. This thesis contains the definition of a TchebychefF System, along with the theorems, proofs, and definitions necessary to guarantee the existence of quadrature formulas for such systems. Solutions of discretely observable linear control systems are of particular interest, and observability with respect to a given output function is defined. The output function is written as a linear combination of a collection of orthonormal functions. Orthonormal functions are defined, and their properties are discussed. The technique for evaluating the coefficients in the output function involves evaluating the definite integral of functions which can be shown to form a Tchebycheff system. Therefore, quadrature formulas for these integrals exist, and in many cases are known. The technique given is useful in cases where the method of direct calculation is unstable. The condition number of a matrix is defined and shown to be an indication of the the degree to which perturbations in data affect the accuracy of the solution. In special cases, the number of data points required for direct calculation is the same as the number required by the method presented in this thesis. But the method is shown to require more data points in other cases. A lower bound for the number of data points required is given.
dc.language.iso en_US en_US
dc.publisher Texas Tech University en_US
dc.subject Potential theory en_US
dc.subject Asymptotic expansions en_US
dc.subject Chebyshev systems en_US
dc.title Quadrature, interpolation and observability en_US
dc.type Electronic Dissertation en_US
thesis.degree.name Ph.D.
thesis.degree.level Doctoral
thesis.degree.discipline Mathematics
thesis.degree.grantor Texas Tech University
thesis.degree.department Mathematics
thesis.degree.department Mathematics and Statistics
dc.degree.department Mathematics en_US
dc.rights.availability unrestricted en_US

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