I. J. Schoenberg

Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions

Introduction. Let there be given a sequence of ordinates

On Hermite-Birkhoff Interpolation

\left\{ {{y_n}} \right\}\quad \left( {n = 0, \pm 1 \pm 2, \ldots } \right),

Über variationsvermindernde lineare Transformationen

corresponding to all integral values of the variable x = n. If these ordinates are the values of a known analytic function F(x), then the problem of interpolation between these ordinates has an obvious and precise meaning: we are required to compute intermediate values F(x) to the same accuracy to which the ordinates are known. Undoubtedly, the most convenient tool for the solution of this problem is the polynomial central interpolation method. It uses the polynomial of degree k — 1, interpolating k successive ordinates, as an approximation to F(x) only within a unit interval in x, centrally located with respect to its k defining ordinates. Assuming k fixed, successive approximating arcs for F(x) are thus obtained which present discontinuities on passing from one arc to the next if k is odd, or discontinuities in their first derivatives if k is even (see section 2.121). Actually these discontinuities are irrelevant in our present case of an analytic function F(x). Indeed, if the interpolated values obtained are sufficiently accurate, these discontinuities will be apparent only if we force the computation beyond the intrinsic accuracy of the y n.

Cardinal interpolation and spline functions: II interpolation of data of power growth

We denote by T 1 the class of entire functions which are limits, uniform in every finite domain, of real polynomials with only real non-positive zeros. Likewise we denote by T 2 the wider class of entire functions obtained if in the previous definition we only require that the approximating polynomials be real and have only real zeros. From the classical work of Laguerre and Polya(2) we know that ϕ(s) ∈ T 1 if and only if ϕ(s) admits a representation of the form

The Finite Fourier Series and Elementary Geometry

\begin{array}{*{20}{c}} {\Phi \left( s \right) = C{e^{\gamma s}}{s^m}\prod\limits_{v = 1}^\infty {\left( {1 + {\delta _v}s} \right)} ,} \\ {\left( {C\;real,\;\lambda \geqq 0,\quad {\delta _v} \geqq 0,\quad \sum {{\delta _v} < \infty } } \right),} \end{array}

A note on the cyclotomic polynomial

(1) and also that the elements ψ(s) of the class T 2 are characterized by the representation