Cardinal interpolation and spline fucntions V. The B-splines for cardinal Hermite interpolation

AbstractIn the third paper of this series on cardinal spline interpolation [4] Lipow and Schoenberg study the problem of Hermite interpolation S(v) = Yv, S′(v) = Yv′,…,S(r−1)(v) = Yv(r−1) for allv. The B-splines are there conspicuous by their absence, although they were found very useful for the case γ = 1 of ordinary (or Lagrange) interpolation (see [5–10]). The purpose of the present paper is to investigate the B-splines for the case of Hermite interpolation (γ > 1). In this sense the present paper is a supplement to [4] and is based on its results. This is done in Part I. Part II is devoted to the special case when we want to solve the problem S(v) = Yv, S′(v) = Yv′ for all v by quintic spline functions of the class C‴(– ∞, ∞). This is the simplest nontrivial example for the general theory. In Part II we derive an explicit solution for the problem (1), where v = 0, 1,…, n

Eulerian polynomials and B-splines

AbstractAn interrelationship between Eulerian polynomials, Eulerian fractions and Euler–Frobe nius polynomials, Euler–Frobenius fractions, and B-splines is presented. The properties of Eulerian polynomials and Eulerian fractions and their applications in B-spline interpolation and evaluation of Riemann zeta function values at odd integers are given. The relation between Eulerian numbers and B-spline values at knot points are also discussed

Cardinal interpolation with polysplines on annuli

AbstractCardinal polysplines of order p on annuli are functions in C2p-2Rn⧹0 which are piecewise polyharmonic of order p such that Δp-1S may have discontinuities on spheres in Rn, centered at the origin and having radii of the form ej, j∈Z. The main result is an interpolation theorem for cardinal polysplines where the data are given by sufficiently smooth functions on the spheres of radius ej and center 0 obeying a certain growth condition in j. This result can be considered as an analogue of the famous interpolation theorem of Schoenberg for cardinal splines

Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough ($L^\infty$) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution $H$) minimizing the $L^2$ norm of the source terms; its (pre-)computation involves minimizing $\mathcal{O}(H^{-d})$ quadratic (cell) problems on (super-)localized sub-domains of size $\mathcal{O}(H \ln (1/ H))$. The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for $d\leq 3$, and polyharmonic for $d\geq 4$, for the operator -\diiv(a\nabla \cdot) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method ($\mathcal{O}(H)$ in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincar\'{e} inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.Comment: ESAIM: Mathematical Modelling and Numerical Analysis. Special issue (2013