A new approach to Euler splines

AbstractStarting from the exponential Euler polynomials discussed by Euler in “Institutions Calculi Differentialis,” Vol. II, 1755, the author introduced in “Linear operators and approximation,” Vol. 20, 1972, the so-called exponential Euler splines. Here we describe a new approach to these splines. Let t be a constant such that t=|t|eiα, −π<α<π,t≠0,t≠1.. Let S1(x:t) be the cardinal linear spline such that S1(v:t) = tv for all v ϵ Z. Starting from S1(x:t) it is shown that we obtain all higher degree exponential Euler splines recursively by the averaging operation Sn(x;t)=∫x−12x+12 Sn−1(u;t) du|∫−1212 Sn−1(u;t)du (n=2,3,…,).. Here Sn(x:t) is a cardinal spline of degree n if n is odd, while Sn(x + 12:t) is a cardinal spline if n is even. It is shown that they have the properties Sn(v:t) = tv for v ϵ Z. Sn(v;t)=tvfor v ϵ Z, lim>n→∞ Sn(x;t)=tx=|t|x eiαx