§24.17 Mathematical Applications

See §2.10(i). For a generalization see Olver (1997b, p. 284).

Let $0\le h\le 1$ and $a,m$, and $n$ be integers such that $n>a$, $m>0$, and $f^{(m)}(x)$ is absolutely integrable over $[a,n]$. Then with the notation of §24.2(iii)

where

See Milne-Thomson (1933), Nörlund (1924), or Jordan (1965). For a more modern perspective see Graham et al. (1994).

Let ${𝒮}_n$ denote the class of functions that have $n-1$ continuous derivatives on $ℝ$ and are polynomials of degree at most $n$ in each interval $(k,k+1)$, $k\in \mathbb{Z}$. The members of ${𝒮}_n$ are called cardinal spline functions. The functions

are called Euler splines of degree $n$. For each $n$, $S_n(x)$ is the unique bounded function such that $S_n(x)\in {𝒮}_n$ and

The function $S_n(x)$ is also optimal in a certain sense; see Schoenberg (1971).

A function of the form $x^n-S(x)$, with $S(x)\in {𝒮}_{n-1}$ is called a cardinal monospline of degree $n$. Again with the notation of §24.2(iii) define

$M_n(x)$ is a monospline of degree $n$, and it follows from (24.4.25) and (24.4.27) that

For each $n=1,2,\mathrm{\dots }$ the function $M_n(x)$ is also the unique cardinal monospline of degree $n$ satisfying (24.17.6), provided that

for some positive constant $\gamma$.

For any $n\ge 2$ the function

is the unique cardinal monospline of degree $n$ having the least supremum norm ${\Vert F\Vert }_{\mathrm{\infty }}$ on $ℝ$ (minimality property).