On rational approximation to ¦X¦

Best Uniform Rational Approximation of x on [0, 1]

For the uniform approximation of xα on [0, 1] by rational functions the following strong error estimate is proved: Let Enn(x, [0, 1]) denote the minimal approximation error in the uniform norm on [0, 1] of rational approximants to xα with numerator and denominator degree at most n, then the the limit lim n→∞ e √ Enn(x , [0, 1]) = 4| sinπα| holds for all α > 0.

Best Uniform Rational Approximation of x on [ 0

UNIFORM RATIONAL APPROXIMATION OF x α ON [ 0 , 1 ]

where ‖ · ‖K denotes the sup norm on K ⊆ R. It is well known that the best approximant r∗ mn exists and is unique within Rmn (cf. [Me, §§9.1, 9.2] or [Ri, §5.1]). The unique existence also holds in the special case (n = 0) of best polynomial approximants. Since fα(x) := |x|α is an even function on [−1, 1], the same is true for its unique approximant r∗ mn = r ∗ mn(fα, [−1, 1]; ·), and consequen...

Diophantine approximation on rational quadrics

We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays with a fixed maximal singular direction, which move away into one end of a locally symmetric space at linear depth, infinitely many times.

Rational Approximation

Recently several authors have investigated the question of approximating certain functions by reciprocals of polynomials under the uniform norm on the positive real axis . Perhaps these results have some applications in industry and elsewhere (cf. [1, 36]) . Our present motivation is to give a detailed list of all the known results with simplified proofs in some cases and many new results and f...