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- Resumen es exacto "The aim of this thesis is to study an extension to Krein spaces of the abstract interpolating and smoothing spline problems in Hilbert spaces, which we call the indefinite abstract interpolating and smoothing spline problems.We analyze contexts for possible applications of these problems in areas such as control theory and signal processing. The indefinite abstract interpolating spline problem is a quadratic programming problem with a quadratic constraint, where both the cost function and the restriction are sign indefinite, and thus, not convex. By studying the characteristics of the range of Tikhonov’s regularization operator, we develop tools and techniques to analyze this non-convex optimization problem. We find sufficient conditions for this problem to admit a solution for every initial data point, and we show that under these conditions the set of solutions is a single affine manifold in a generic case, in the sense that it is the case when the initial data point belongs to an open dense subset of the data space. Moreover, we show that these conditions are also necessary in the finite dimensional setting. The indefinite abstract smoothing spline problem is obtained by applying Tikhonov’s regularization procedure to the indefinite interpolating spline problem, resulting in an indefinite least squares problem. We prove that the indefinite smoothing spline problem admits a solution for every initial data point and every value of the regularization parameter if and only if the indefinite interpolating spline problem admits a solution for every initial data point. Under these conditions, for every initial data point belonging to an open dense subset of the data space, the indefinite interpolating spline problem can be posed as an indefinite smoothing spline problem.
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