$H^{r,} ^{\infty }(R)$- and $W^{r,\infty }(R)$-splines

Abstract:

Let E be a subset of R the real line and $f:E \to R$. Necessary and sufficient conditions are derived for $\inf (\left \|{D^r}x\right \|_{{L^\infty }}:x{|_E} = f)$ to have a solution. When restricted to quasi-uniform partitions E, necessary and sufficient conditions are derived for the solution to be in ${L^\infty }$. For finite partitions E it is shown that a solution to the ${L^\infty }$ infimum problem can be obtained by solving $\inf (\left \|{D^r}x\right \|_{{L^p}}:x{|_E} = f)$ and letting p go to infinity. In this way it was discovered that solutions to the ${L^\infty }$ problem could be chosen to be piecewise polynomial (of degree r or less). The solutions to the ${L^p}$ problem are called ${H^{r,p}}$-splines and were studied extensively by Golomb in [3].