On Variation Diminishing Spline Approximation Methods

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


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
$$S_i \left( x \right) = \sum\limits_{j = 0}^l {f\left( {\frac{j} {l}} \right)N_j \left( x \right)} .$$


Spline Function Arithmetic Progression Bernstein Polynomial Divided Difference Shape Preserve Property 
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Copyright information

© Springer Science+Business Media New York 1988

Authors and Affiliations

  • Martin Marsden
    • 1
  • I. J. Schoenberg
    • 1
  1. 1.University of WisconsinMadisonUSA

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