User johan richter - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T01:33:23Z http://mathoverflow.net/feeds/user/9838 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41257/relative-error-approximation-by-polynomials/41276#41276 Answer by Johan Richter for Relative error approximation by polynomials Johan Richter 2010-10-06T14:23:29Z 2010-10-06T16:56:29Z <p>Take f to be non-zero but have infinitely many zeros in [-1,1]. A polynomial can only have a finite number of zeros so clearly the D(f,p) you defined will be infinite for every polynomial. </p> <p>On the other hand if f is bounded <strike>from below</strike> away from zero there will at least exist a sequence of polynomials such that the relative error converges to zero, by just applying the traditional absolute theory. </p> <p>Beyond these special cases I do not know anything but presumably relative approximation will have been studied before. I believe that when one chooses polynomials to approximate functions in practice (in calculators for example) one uses the relative error as the objective function. </p>