On optimal periodic spline interpolation

On periodic M-Splines

“…we obtain for the Green function of L k For a similar construction of other types of splines we refer, to the papers [3,4,5,9] and the monograph of Karlin I-8].…”

Multivariate Bernoulli splines and the periodic interpolation problem

“…Periodic spline interpolation is well understood in one variable, see, for example, [11 [3], [5], [6], [8]-[101 [20], and [24]. The univariate Bernoulli spline e m i.e.…”

“…The univariate Bernoulli spline e m i.e. the periodic extension of the ruth Bernoulli polynomial, is a useful tool for describing the spline interpolation operators (see [5], [6], and [8]- [10]). The minimum norm properties, well known for interpolating spline functions of odd degree, can be developed in a very compact manner using the notion of a "reproducing kernel" of a Hilbert space (see [1], [4J, and [11]).…”

“…Given an arbitrary finite set X c ~ the corresponding interpolation operator is well defined, and the function s s possesses minimal norm in H among all functions g ~ H which interpolate f with respect to X (see Chapter 6 of 1,17]). The Bernoulli spline B~, given in (1) can be used in order to derive an explicit representation of the reproducing kernel of certain Sobolev spaces (see [8], and [10]). This is due to the fact that Bm is Green's function of the differential operator (see [27]).…”

“…Here Hm denotes the usual Sobolev space of(m -1)-times continuously differentiable functions of period 2n with f~'~-t) absolutely continuous and ft'~ square integrable. Suitable choices of inner products on Hm for which the reproducing.kernel is a piecewise polynomial function are studied in I 1], 1,4], [6], and [10]. Extensions of the spline theory concerning Hilbert-space methods to the multivariate setting deal mostly with tensor product spaces [ 1], [2], [7], [13], [26]; there are some more general developments for nonperiodic functions in [ 15], [16], [19], and [21].…”