On Asymptotics of the~Jump of~Highest Derivative for a~Polynomial Spline

B. S. Kindalev

Matematičeskie trudy, Tome 5 (2002) no. 1, pp. 66-73

Résumé

In this article, we obtain $2[n/2]+2$ terms ($[\boldsymbol{\cdot}]$ stands for the integer part) of the asymptotic expansion of the error $$ \bigl(S^{(n)}({}\,\overline{\kern-.3mm x}_i+0)-S^{(n)}({}\,\overline{\kern-.3mm x}_i-0)\bigr)\big/h-f^{(n+1)}({}\,\overline{\kern-.3mm x}_i), $$ where $S(x)$ is a periodic spline of degree $n\ge 0$ and deficiency 1 that interpolates a periodic sufficiently smooth function $f(x)$ at the nodes $x_i$ ($i=0,\pm1,\dots$) of a uniform mesh of width $h$. The nodes of the spline are the points ${}\,\overline{\kern-.3mm x}_i=x_i+h\bigl(1+(-1)^n\bigr)/4$. The expansion coefficients are represented explicitly in terms of the values of the Bernoulli polynomials at 0 for $n$ odd and 1/2 for $n$ even.