Vector-valued Lg-splines I. Interpolating splines

The Derivation of Structural Properties of LM-g Splines by Necessary Condition of Optimal Control

“…it is called L-g interpolating spline [4]. If { } =1 ( ≥ ) are EHB functionals defined by (10), then solving the minimization problem (33) is to find a control function ( ) which satisfies certain conditions of piecewise smoothness to minimize the functional…”

“…Splines defined in terms of one linear differential operator are called L-g splines. The structural and continuity properties of these splines were discussed in detail in [4,5]. They play an important role in the approximation of functions and optimal control.…”

A variational approach to splines

“…Other extensions have come from researchers in electrical engineering: Sidhu and Weinert [49] presented one such extension in 1979. They consider the problem of simultaneous interpolation, that is, a method by which one could interpolate several functions at once.…”

Detection of stochastic processes

“…where denotes the inner product and the norm in the RKHS , associated with , characterized by the following properties: and (108) Note that the second of these properties suggests the name "reproducing kernel," since the kernel reproduces the elements of when applied to them in the form of a linear operator. We shall show presently that (109) By definition, is obtained by applying to the same operations used to get from , so that (110) Substituting (109)- (110) into (105) gives the formula (111) This is a nice generalization of the white noise formula, which can be regarded as a "limiting" case in which and the inner product is the usual (L2) inner product. Note also that from (104) we can see that the singular case of perfect detection will arise if and only if , which by (109) means It can be checked that which connects with the earlier discussion (below (92)).…”

“…We shall show presently that (109) By definition, is obtained by applying to the same operations used to get from , so that (110) Substituting (109)- (110) into (105) gives the formula (111) This is a nice generalization of the white noise formula, which can be regarded as a "limiting" case in which and the inner product is the usual (L2) inner product. Note also that from (104) we can see that the singular case of perfect detection will arise if and only if , which by (109) means It can be checked that which connects with the earlier discussion (below (92)). However, to see what may have been gained by this new formulation, apart from an elegant interpretation, let us reconsider the example introduced earlier of exponentially correlated noise-see (98) and (99).…”