§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)

24.17.1		$$\sum _{j=a}^{n-1}{(-1)}^{j}f(j+h)=\frac{1}{2}\sum _{k=0}^{m-1}\frac{E_k\left(h\right)}{k!}\left({(-1)}^{n-1}{f}^{(k)}(n)+{(-1)}^a{f}^{(k)}(a)\right)+R_m(n),$$
Symbols: $E_n\left(x\right)$: Euler polynomials, $!$: factorial (as in $n!$), $j$: integer, $k$: integer, $m$: integer, $h$: parameter, $a$: integer, $n$: integer, $f^{(m)}(x)$: integrable function and $R_m(n)$: remainder Permalink: http://dlmf.nist.gov/24.17.E1 Encodings: TeX, pMML, png See also: Annotations for 24.17(i)

where

24.17.2		$$R_m(n)=\frac{1}{2(m-1)!}{\int }_a^n{f}^{(m)}(x){\tilde{E}}_{m-1}\left(h-x\right)dx.$$
Symbols: $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, ${\tilde{E}}_n\left(x\right)$: periodic Euler functions, $m$: integer, $x$: real or complex, $h$: parameter, $a$: integer, $n$: integer, $f^{(m)}(x)$: integrable function and $R_m(n)$: remainder Permalink: http://dlmf.nist.gov/24.17.E2 Encodings: TeX, pMML, png See also: Annotations for 24.17(i)

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

24.17.3		$$S_n(x)=\frac{{\tilde{E}}_n\left(x+\frac{1}{2}n+\frac{1}{2}\right)}{{\tilde{E}}_n\left(\frac{1}{2}n+\frac{1}{2}\right)},$$
$n=0,1,\mathrm{\dots }$,
Symbols: ${\tilde{E}}_n\left(x\right)$: periodic Euler functions, $n$: integer and $x$: real or complex Permalink: http://dlmf.nist.gov/24.17.E3 Encodings: TeX, pMML, png See also: Annotations for 24.17(ii)

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

24.17.4		$$S_n(k)={(-1)}^k,$$
$k\in \mathbb{Z}$.
Symbols: $\in$: element of, $\mathbb{Z}$: set of all integers, $k$: integer and $n$: integer Permalink: http://dlmf.nist.gov/24.17.E4 Encodings: TeX, pMML, png See also: Annotations for 24.17(ii)

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

24.17.5
Symbols: $B_n$: Bernoulli numbers, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $n$: integer, $x$: real or complex and $M_n(x)$: function Permalink: http://dlmf.nist.gov/24.17.E5 Encodings: TeX, pMML, png See also: Annotations for 24.17(ii)

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

24.17.6		$$M_n(k)=0,$$
$k\in \mathbb{Z}$.
Symbols: $\in$: element of, $\mathbb{Z}$: set of all integers, $k$: integer, $n$: integer and $M_n(x)$: function Referenced by: §24.17(ii) Permalink: http://dlmf.nist.gov/24.17.E6 Encodings: TeX, pMML, png See also: Annotations for 24.17(ii)

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

24.17.7		$$M_n(x)=O\left({|x|}^{\gamma }\right),$$
$x\to \pm \mathrm{\infty }$,
Symbols: $O\left(x\right)$: order not exceeding, $n$: integer, $x$: real or complex and $M_n(x)$: function Permalink: http://dlmf.nist.gov/24.17.E7 Encodings: TeX, pMML, png See also: Annotations for 24.17(ii)

for some positive constant $\gamma$.

For any $n\ge 2$ the function

24.17.8		$$F(x)={\tilde{B}}_n\left(x\right)-2^{-n}{B}_n$$
Symbols: $B_n$: Bernoulli numbers, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $n$: integer and $x$: real or complex Permalink: http://dlmf.nist.gov/24.17.E8 Encodings: TeX, pMML, png See also: Annotations for 24.17(ii)

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