ISSN:
1432-0940
Keywords:
Algebraic polynomials
;
Entire functions
;
Asymptotic expansions
;
Chebyshev approximation
;
41A10
;
41A25
;
41A50
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract LetE n (f) denote the sup-norm-distance (with respect to the interval [−1, 1]) betweenf and the set of real polynomials of degree not exceedingn. For functions likee x , cosx, etc., the order ofE n (f) asn→∞ is well known. A typical result is $$2^{n - 1} n!E_{n - 1} (e^x ) = 1 + 1/4n + O(n^{ - 2} ).$$ It is shown in this paper that 2 n−1 n!E n−1(e x ) possesses a complete asymptotic expansion. This result is contained in the more general result that for a wide class of entire functions (containing, for example, exp(cx), coscx, and the Bessel functionsJ k (x)) the quantity $$2^{n - 1} n!E_{n - 1} \left( f \right)/f^{(n)} \left( 0 \right)$$ possesses a complete asymptotic expansion (providedn is always even (resp. always odd) iff is even (resp. odd)).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01890576
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