§24.17 Mathematical Applications

§24.17(i) Summation

¶ Euler–Maclaurin Summation Formula

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

¶ Boole Summation Formula

Let and , and be integers such that , , and is absolutely integrable over . Then with the notation of §24.2(iii)

where

¶ Calculus of Finite Differences

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

§24.17(ii) Spline Functions

¶ Euler Splines

Let denote the class of functions that have continuous derivatives on and are polynomials of degree at most in each interval , . The members of are called cardinal spline functions. The functions

are called Euler splines of degree . For each , is the unique bounded function such that and

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

¶ Bernoulli Monosplines

A function of the form , with is called a cardinal monospline of degree . Again with the notation of §24.2(iii) define

is a monospline of degree , and it follows from (24.4.25) and (24.4.27) that

For each the function is also the unique cardinal monospline of degree satisfying (24.17.6), provided that

for some positive constant .

For any the function

is the unique cardinal monospline of degree having the least supremum norm on (minimality property).

§24.17(iii) Number Theory

Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and -series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); -adic analysis (Koblitz (1984, Chapter 2)).