LM-g splines

Abstract

As an extension of the notion of an L-g spline, three mathematical structures called LM-g splines of types I, II, and III are introduced. Each is defined in terms of two differential operators L = D n + ∑ j=0 n−1 a j(t)D j and M = ∑ i=0 m b i(t)D i, where n ⩾ m ges; 0, n > 0, D = d dt the coefficients a j , j = 0,…, n − 1, and b i , i = 0,…, m, are sufficiently smooth; and b m is bounded away from zero on [0, T]. Each of the above types of splines is the solution of an optimization problem more general than the one used in the definition of the L-g spline and hence it is recognized as an entity which is distinct from and more general mathematically than the L-g spline. The LM-g splines introduced here reduce to an L-g spline in the special case in which m = 0 and b 0 = constant ≠ 0. After the existence and uniqueness conditions, characterization, and best approximation properties for the proposed splines are obtained in an appropriate reproducing kernel Hilbert space framework, their usefulness in extending the range of applicability of spline theory to problems in estimation, optimal control, and digital signal processing are indicated. Also, as an extension of recent results in the generalized spline literature, state variable models for the LM-g splines introduced here are exhibited, based on which existing least squares algorithms can be used for the recursive calculation of these splines from the data.