Let and , and be integers such that , , and is absolutely integrable over . Then with the notation of §24.2(iii)
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).
A function of the form , with is called a cardinal monospline of degree . Again with the notation of §24.2(iii) define
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).
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)).