A note on rational approximation on [0, ∞)

A Note on Approximation by Rational Functions

The theory of the approximation by rational functions on point sets E of the js-plane (z = x+iy) has been summarized by J. L. Walsh who himself has proved a great number of important theorems some of which are fundamental. The results concern both the case when E is bounded and when E extends to infinity. In the present note a Z^-theory (0<p< oo) will be given for the following point sets exten...

A Note on the Rational Points of X+0(N)

Let C be the image of a canonical embedding φ of the Atkin-Lehner quotient X 0 (N) associated to the Fricke involution wN . In this note we exhibit some relations among the rational points of C. For each g = 3 (resp. the first g = 4) curve C we found that there are one or more lines (resp. planes) in P whose intersection with C consists entirely of rational Heegner points or the cusp point, whe...

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...