# On Variation Diminishing Spline Approximation Methods

• Martin Marsden
• I. J. Schoenberg
Chapter
Part of the Contemporary Mathematicians book series (CM)

## Abstract

Let the function f(x) be defined in [0,1] and let us approximate it by a piece-wise linear function S l (x) obtained as follows: l being a natural number, we divide [0, 1] into l equal parts and denote by S l (x) the continuous function which is linear in each subinterval ((i − 1)/l, i/il) while interpolating f(x) at its end points. Denoting by N j (x) the broken linear functions such that
$$N_j \left( {\frac{i} {l}} \right) = \delta _{ij} ,\,\left( {i,\,j = 0,\, \ldots ,\,l} \right),$$
we may represent the approximation in the form
$$S_i \left( x \right) = \sum\limits_{j = 0}^l {f\left( {\frac{j} {l}} \right)N_j \left( x \right)} .$$
(1)

## Keywords

Spline Function Arithmetic Progression Bernstein Polynomial Divided Difference Shape Preserve Property
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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