Best uniform rational approximation of xα on [0, 1]

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 xa ON [0, 1]

A strong error estimate for the uniform rational approximation of xa on [0, 1] is given, and its proof is sketched. Let E„n(xa, [0, 1]) denote the minimal approximation error in the uniform norm. Then it is shown that lim e2*^"Enn(xa , [0, 1]) = 41+a| sin7ta| n—»oo holds true for each a > 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...

The best uniform polynomial approximation of two classes of rational functions

In this paper we obtain the explicit form of the best uniform polynomial approximations out of Pn of two classes of rational functions using properties of Chebyshev polynomials. In this way we present some new theorems and lemmas. Some examples will be given to support the results.