# Refinements of the Peano kernel theorem

## Abstract:

It is shown that by starting with a general form of the Peano kernel theorem which makes no reference to the interchange of linear functionals and integrals, the most general results can be obtained in an elementary manner. In particular, we classify how the Peano kernels become increasingly smooth and satisfy boundary (or equivalently moment) conditions as the corresponding linear functionals become continuous on wider classes of functions. These results are then used to give new representations of the continuous duals of $C^r[a,b]$ and $W_p^r[a,b]$, $1\le p<\infty$.

Keywords: Peano kernel theorem, mass theorem, quotient theorem, Sard's factorisation theorem, open mapping theorem, representation of linear functionals, functions of (normalised) bounded variation, Riemann-Stieltjes measure,integration by parts, moment (orthogonality) condition, Sobolev space, interpolation, quadrature, B-spline

Math Review Classification: 41A05, 41A10, 41A65, 41A80 (primary), 41A55, 46E99, 65J05 (secondary)

Length: 18 pages

Comment: Written in TeX

Last updated: 22 October 1998

Status: Submitted