## Reproducing Kernel Hilbert Space Approach to Spline Problems with Applications in Estimation and ControlDepartment of Electrical Engineering, Stanford University., 1972 - Mathematical optimization - 68 pages The solutions to several optimization problems involve generalized spline functions. Existing algorithms for calculating splines cannot be easily updated with the addition of new data, and are therefore not suitable for real-time computation when data are acquired sequentially. In the thesis, recursive algorithms are developed for the problems of optimal interpolation and smoothing, and optimal approximation of linear functionals, when the underlying space of functions is a reproducing kernel Hilbert space. It is shown that these deterministic problems have equivalent stochastic least-squares estimation problems, and that the recursive solution of each deterministic problem corresponds to the recursive solution of the associated stochastic problem in which a discrete innovation sequence is computed. In addition, it is shown that the problem of computing the minimum-energy control of a linear time-varying system that yields an output satisfying certain functional constraints is a spline problem and can be solved recursively using the above methods. (Author). |

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### Contents

INTRODUCTION | x |

INTERPOLATION WITH LgSPLINES | 7 |

REFORMULATION AND SOLUTION OF THE INTERPOLATION PROBLEM | 15 |

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### Common terms and phrases

analog filter approximation of linear bounded linear operator cardinal spline coefficients compute the spline congruence relation covariance matrix data are acquired deBoor and Lynch definition Equation estimate of y(t estimation problem evaluation functionals Example 4.1 fact feU(r given by Eq Golomb and Weinberger Gram-Schmidt procedure Green,s function i,j element implies interpolating spline interpolation constraints Jerome and Schumaker kernel Hilbert space Kimeldorf and Wahba Lg-spline interpolating linear functionals linear least-squares estimate linear mapping linear operations linear subspace linearly independent Lyche and Schumaker minimizing Eq minimum-energy control problem n+1 n+1 null space optimal approximation orthonormal basis polynomial of degree projection theorem Proof proved by deBoor real-time computation recursive algorithm recursive solution reformulate representer reproducing kernel Hilbert reproducing property satisfy Eq Schumaker 14 side of Eq smoothing problem solution to Eq space H spline function spline interpolation spline problem spline that interpolates subspace Theorem 3.3 time-varying system unique