Quadrature, interpolation and observability

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Title: Quadrature, interpolation and observability
Author: Hodges, Lucille McDaniel
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 .
URI: http : / /hdl .handle .net /2346 /11588
Date: 1997-12


Hodges, Lucille McDaniel Quadrature, interpolation and observability. Doctoral dissertation, Texas Tech University. Available electronically from http : / /hdl .handle .net /2346 /11588 .

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