Linear extrapolation by rational functions, exponentials and logarithmic functions

Integration of Rational Functions: Rational Computation of the Logarithmic Part

This formula is not satisfactory under a computing point of view because it introduces more algebraic quantities than necessary. The number P(a)/Q'(a) is called the residue of the root a of Q. We introduce the notion of multiplicity of a residue b that is the number of roots a having b as a residue. Trager (1976) introduced a new formula involving less algebraic numbers: let S(y) be the resulta...

Interpolation and Extrapolation of Smooth Functions by Linear Operators

Let C(R) be the space of functions on R whose m derivatives are Lipschitz 1. For E ⊂ R, let C(E) be the space of all restrictions to E of functions in C(R). We show that there exists a bounded linear operator T : C(E)→ C(R) such that, for any f ∈ C(E), we have Tf = f on E.

Rational Functions with Linear Relations

We find all polynomials f, g, h over a field K such that g and h are linear and f(g(x)) = h(f(x)). We also solve the same problem for rational functions f, g, h, in case the field K is algebraically closed.

Approximation by Rational Functions

Making use of the Hardy-Littlewood maximal function, we give a new proof of the following theorem of Pekarski: If f' is in L log L on a finite interval, then f can be approximated in the uniform norm by rational functions of degree n to an error 0(1/n) on that interval. It is well known that approximation by rational functions of degree n can produce a dramatically smaller error than that for p...

Approximation by Rational Functions