Morsche, ter, H.G., Boehmer, K., Meinardus, G., Schempp, W., Mathematics and Computer Science, and Research 1957/58 - 1978/79
Let m be a natural number and let Sm denote the class of cardinal spline functions of degree m. The object of this note is to establish a linear relationship between the 2m+2 quantities s(i+x), s(i+1+x),...,s(i+m+x), s(k)(i+y), s(k)(i+1+y),...,s(k)(i+m+y), where x,y ∈ [0,1], i=0,±1,±2,... s ∈ Sm and where s(k) denotes the k-th derivative of s (k=0,1,2,...,m−1). Using the shift operator E, we represent this relation in a simple form, involving the exponential Euler polynomials. The results are applied to cardinal spline interpolation.