Particular cases:
Lagrange interpolation/control polygion
Taylor interpolation/control polygon
Hermite interpolation/control polygon
Fast change of another functional basis to a respective interpolation basis: a general
property.
Existence and uniqueness of the solution of the interpolation problem
nxn – exists unique (Vandermonde-type determinants);
mxn, m<n – exists, not unique, m>n generally does not exist.
Some deficiencies of polynomial interpolation
Numerically ill-conditioned for high degrees (exception: Bernstein basis for every
degree)
Non-local support of the basis functions =>
global propagation of local errors,
full matrices of systems of linear equations.
Improvement: spline interpolation (B-spline basis as an interpolation basis).
For cubic and higher-degree splines: local errors stay local (not for quadratic
splines!)
Local support of the basis functions: sparse (band-limited) matrices of systems of
linear equations.
Two ways of looking at the splines:
‘Bezier way’: B-spline basis
‘Hermite way’: polynomials pieces with Hermite interpolation conditions at the
boundary.
Important practical examples:
Reparametrization of Bezier curves with interpolation conditions at the
endpoints;
Piecewise cubic Hermite interpolation
Higher then cubic degree spline interpolation (Schoenberg interpolation)
Attention: the missing conditions at the endpoints of the interval (to balance the
number of equations and the number of unknowns)
Examples: generalized Vandermonde determinants.
 the above deficiencies of polynomial interpolation are overcome.
Other deficiencies of interpolation still unresolved:
Robustness to outliers in the data
Over-fitting data
Improvement: Interpolation -> less restrictive types of Approximation (least squares,
etc.).