An Improved Constant for the Muntz-Jackson Theorem
Publications de l'Institut Mathématique, _N_S_45 (1989) no. 59, p. 103
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We improve a Newman result [2,3] from 1974 concerning
approximation of a continuous function by generalized polynomials. He
proved that every $f\in C[0,1]$ there exists a generalized polynomial
$P(x)=\sum_{k=0}^N c_kx^{\lambda k}$ such that
$
|f(x)-P(x)| łeq Aw_f(\varepsilon),\qquad x\in [0,1]\tag 1
$
holds. Here $0=\lambda_0\lambda_1\cdots\lambda_N$ are given numbers
$w_f$ is the modulus of continuity of $f$, $\varepsilon=\max\{|B(z)/z|:
Re\, z=1\}$, $B(z)$ is the Blaschke product corresponding to the above
set of $\lambda_k$'s and $A$ is a constant. Newman [2] proved that (1)
holds with $A=368$. We show that (1) is valid with $A=66$. We prove
this by slightly modifying Newman's proof and choosing the size of an
interval, to which a suitable contradiction is extended, optimally.
Classification :
41A30 41A25
@article{PIM_1989_N_S_45_59_a15,
author = {H. N. Odogwu},
title = {An {Improved} {Constant} for the {Muntz-Jackson} {Theorem}},
journal = {Publications de l'Institut Math\'ematique},
pages = {103 },
publisher = {mathdoc},
volume = {_N_S_45},
number = {59},
year = {1989},
language = {en},
url = {http://geodesic.mathdoc.fr/item/PIM_1989_N_S_45_59_a15/}
}
H. N. Odogwu. An Improved Constant for the Muntz-Jackson Theorem. Publications de l'Institut Mathématique, _N_S_45 (1989) no. 59, p. 103 . http://geodesic.mathdoc.fr/item/PIM_1989_N_S_45_59_a15/