Applications of parabolic $B$-splines for solving interpolation problems

The use of local basis functions ($B$-splines) in interpolation by parabolic splines is studied. Constraints on the mesh of interpolation nodes whereby diagonal dominance is ensured are obtained, and the limits of good stipulation of the matrix of the system of defining relations of the interpolation problem, are indicated. An expression is obtained for passing from one representation of a spline to another. Explicit expressions are obtained for constructing local approximation splines. It is shown that the latter have approximation properties similar to those of interpolation splines.