### Information

- March 30, 2007
- 10:15 am - 10:40 am

# A best approximation problem with application to parallel computing

##### Martin Gander (Université de Genève)

The classical best approximation problem is the following: given a
real-valued continuous function on a compact interval and a class of
functions defined on the same interval, find an element in the class
that realizes the distance of the function to the class. If the
class is the linear space of polynomials of degree less than or equal
to *n*, and the distance is measured in the *L-∞* norm, then the
approximation problem is called a Chebyshev best approximation
problem.

We are interested in a best approximation problem in a more general
setting: we search for a given function *f: C → C* the
polynomial *s _{n}^{*}* of degree less than or equal to

*n*that minimizes over all

*s*of degree less than or equal to

*n*the quantity

sup z ∈ K |(s(z) - f(z)) / (s(z) + f(z)) exp(-l f(z))|

where *K* is a compact set in *C*, and *l* is a non-negative real
parameter. The solution of this best approximation problem is
important in parallel computing: it leads to the fastest iterative
domain decomposition methods.