The L1 norm of the approximation error for splines with equidistant knots

Abstract

This paper is concerned with the estimation of the L 1 norm of the difference between a function f of bounded variation in [0, 1] and the associated variation-diminishing spline function with equidistant knots, S m,n f, with see Schoenberg [4]. For f bounded on [0, 1] and for integers m, n such that $$ n \geqslant m \geqslant 2, $$ (1.1) the function S m,n f is defined by $$ S_{m.n} f\left( x \right) = \sum\limits_{j = 0}^\iota {f\left( {n^{ - 1} \xi \left( {m,j} \right)} \right)\bar N_{m.j} \left( {nx} \right),} $$ (1.2) where $$ l = m + n - 2, $$ (1.3) $$\xi \left( {m,j} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{\left( {j + 1} \right)j}}{{2\left( {m - 1} \right)}},} \hfill & {j = 0, \ldots ,m - 2} \hfill {j + 1 - \frac{m}{2},} \hfill & {j = m - 1, \ldots ,n - 1,} \hfill {n - \xi \left( {m,l - j} \right),} \hfill & {j = n, \ldots ,l,} \hfill \end{array} } \right.$$ (1.4) $${{\tilde{N}}_{{m,j\left( x \right)}}} = \left\{ {\begin{array}{*{20}{c}} {\frac{{j + 1}}{m}{{h}_{{m.j + 1}}}\left( x \right),} \hfill & {j = 0, \ldots ,m - 2,} \hfill {{{h}_{m}}\left( {x + m - j - 1} \right),} \hfill & {j = m - 1, \ldots ,n - 1,} \hfill {{{{\bar{N}}}_{{m.l - j}}}\left( {n - x} \right),} \hfill & {j = n, \ldots ,l,} \hfill \end{array} } \right.$$ (1.5) $${{h}_{{m,k}}}\left( x \right) = \frac{m}{{k!}}\sum\limits_{{i = 1}}^{k} {{{{( - 1)}}^{{k - i}}}{{i}^{{k - m}}}} \left( {_{i}^{k}} \right)\left( {i - x} \right)_{ + }^{{m - 1}}{\mkern 1mu} {\text{for }}x0,$$ (1.6) $$ h_m \left( x \right) = h_{m.m} \left( x \right).$$ (1.7) .