ISSN:
1432-0940
Keywords:
Walsh array
;
Best rational approximants
;
Entire functions
;
Smooth coefficients
;
Asymptotics
;
Padé approximants
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let $$f(z): = \sum\nolimits_{j = 0}^\infty {a_j z^J } $$ be entire, witha j≠0,j large enough, $$\lim _{J \to \infty } a_{j + 1} /a_J = 0$$ , and, for someq∈C, $$q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q$$ asj→∞. LetE mn(f; r) denote the error in best rational approximation off in the uniform norm on |z‖≤r, by rational functions of type (m, n). We study the behavior ofE mn(f; r) asm and/orn→∞. For example, whenq above is not a root of unity, or whenq is a root of unity, butq m has a certain asymptotic expansion asm→∞, then we show that, for each fixed positive integern, ,m→∞. In particular, this applies to the Mittag-Leffler functions $$f(z): = \sum\nolimits_{j = 0}^\infty {z^j /\Gamma (1 + j/\lambda )} $$ and to $$f(z): = \sum\nolimits_{j = 0}^\infty {z^j /(j!)^{I/\lambda } } $$ , λ〉0. When |q‖〈1, we also handle the diagonal case, showing, for example, that ,n→∞. Under mild additional conditions, we show that we can replace 1+0(1) n by 1+0(1). In all cases we show that the poles of the best approximants approach ∞ asm→∞.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01890411
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