Uniform rational approximation of functions with first derivative in the real Hardy space Re \(H^ 1\) (Q809291)

The authors prove that if f is absolutely continuous with derivative in the real Hardy space \(H_ 1\) over the whole real line then individual functions can be approximated by rationals of degree n with uniform error o(1/n) while estimates for the whole class will be O(1/n). Together with interpolation theory this pretty much settles the problem of rational approximation on the whole line. The proofs use atomic decomposition, rational approximation of the atoms, and pasting arguments.