Discrete cubic \({\mathcal X}\)-splines (Q1122073)

Generalizing the concept of cubic \({\mathcal X}-splines\) introduced by \textit{C. W. Clenshaw} and \textit{B. Negus} [J. Inst. Math. Appl. 22, 109-119 (1978; Zbl 0392.65005)] the author defines discrete cubic \({\mathcal X}\)- splines in the following way: For a given partition \(a=x_ 0<x_ 1...<x_ n=b\) (i) the spline s(x,h) is continuous over the interval [a,b], (ii) its restriction \(s_ i\) to each subinterval \([x_{i-1},x_ i]\) is a polynomial of degree 3, (iii) s(x,h) interpolates given values of a continuous function \(f:s(x_ i,h)=f_ i\), \(i=0(1)n\), (iv) the first central difference quotient is continuous at the inner gridpoints \(x_ i\), \(i=1(1)n-1\), \[ jump x_ iD^ 1_ hs(x,h):=jump x_ i((s(x+h)-s(x-h)/2h)=0, \] (v) the second and third central difference quotients fulfill the relation \(jump x_ iDh^ 2s(x,h)=\alpha_ ijump x_ iD_ h^ 3s(x,h),\) where the \(\alpha_ i\) denote given constants, and the jump-function is defined by \(jump_ xf:=f(x+0)-f(x-0).\) Under certain conditions on h and \(\alpha_ i\) the author proves an existence and uniqueness theorem and an error estimation depending on the modulus of continuity of the first difference quotient.