The convergence properties of a type II complete deficient quartic spline

A novel deficient complete quartic spline S having type II complete endpoint conditions S′′a=f′′a,S′′b=f′′b is constructed, with the expression given in the terms of the second derivative on the mesh nodes, and taking prescribed values on these grid nodes and at midpoints. The existence and uniqueness of this spline is proved, and the error estimates are provided. For less smooth interpolated functions, the interpolation error estimate is given in terms of the uniform modulus of continuity considering S′′a=S′′b=0 as endpoint conditions. Two numerical examples illustrate the geometric properties of this deficient quartic spline interpolation operator. The error estimates are obtained for S,S′,S′′, and S′′′ showing an optimal order of convergence O(h5).