Optimal approximation for functions prescribed at equally spaced points

Abstract

Expl icit upp er a nd lo\\"er bounds for the valuc F (1,) of a linear func t ional F applied Lo a funct ion u(x ) defined on the in terval 0 ::;1; ::; 1 arc given when u is prescribed at the N + l points i f N , i = O, .. . ,N, a nd a bound for the integral of U[k]2 is known. These bounds a re optimal in the sense that they a re attai ned for functions satisfyin g the prescri bed condit ion s. Their computation invol ves the in version of a mat rix of s ize k -l ra ther t han N , \\"hieh m eans that N is permi tted Lo be large.