Abstract Convergence properties of quadratic spline interpolation of continuous functions that does not necessarily take place at the midpoints of mesh intervals are investigated. A theorem giving lower bounds on the elements of the inverse of certain tridiagonal matrices is proved. This result is used to precisely relate the norm of certain interpolating projections to the points of interpolation and local mesh ratios. It is shown, for example, that for Lipschitz continuous functions, any choice of interpolation points, one in each mesh interval, uniformly bounded away from the mesh points, yields convergence at the best possible rate with no mesh ratio restriction.
[1]
S. Demko.
Local approximation properties of spline projections
,
1977
.
[2]
J. Gillis,et al.
Matrix Iterative Analysis
,
1961
.
[3]
H. Schwarz.
Numerical analysis of symmetric matrices
,
1974
.
[4]
M. Marsden,et al.
Cubic spline interpolation of continuous functions
,
1974
.
[5]
Richard S. Varga,et al.
Quandratic interpolatory splines
,
1974
.
[6]
M. J. Marsden.
Quadratic spline interpolation
,
1974
.
[7]
D. Kershaw.
Inequalities on the elements of the inverse of a certain tridiagonal matrix
,
1970
.
[8]
C. D. Boor,et al.
Quadratic Spline Interpolation and the Sharpness of Lebesgue's Inequality.
,
1976
.
[9]
R. Varga,et al.
L-Splines
,
1967
.