On Variation Diminishing Spline Approximation Methods

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)