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Volume 27, Issue 3
Asymptotic Property of Approximation to $x^α$sgn$x$ by Newman Type Operators

Qian Zhan & Shusheng Xu

Commun. Math. Res., 27 (2011), pp. 193-199.

Published online: 2021-05

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  • Abstract

The approximation of $|x|$ by rational functions is a classical rational problem. This paper deals with the rational approximation of the function $x^α$sgn$x$, which equals $|x|$ if $α = 1$. We construct a Newman type operator $r_n(x)$ and show $$\mathop{\rm min}\limits_{|x|≤1} \{|x^α{\rm sgn}x − r_n(x)| \} ∼ Cn^{−\frac{α}{2}}e^{−\sqrt{2nα}},$$ where $C$ is a constant depending on $α$.

  • Keywords

rational approximation, asymptotic property, Newman type operator.

  • AMS Subject Headings

41A05, 41A25

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{CMR-27-193, author = {Zhan , Qian and Xu , Shusheng}, title = {Asymptotic Property of Approximation to $x^α$sgn$x$ by Newman Type Operators}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {27}, number = {3}, pages = {193--199}, abstract = {

The approximation of $|x|$ by rational functions is a classical rational problem. This paper deals with the rational approximation of the function $x^α$sgn$x$, which equals $|x|$ if $α = 1$. We construct a Newman type operator $r_n(x)$ and show $$\mathop{\rm min}\limits_{|x|≤1} \{|x^α{\rm sgn}x − r_n(x)| \} ∼ Cn^{−\frac{α}{2}}e^{−\sqrt{2nα}},$$ where $C$ is a constant depending on $α$.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19082.html} }
TY - JOUR T1 - Asymptotic Property of Approximation to $x^α$sgn$x$ by Newman Type Operators AU - Zhan , Qian AU - Xu , Shusheng JO - Communications in Mathematical Research VL - 3 SP - 193 EP - 199 PY - 2021 DA - 2021/05 SN - 27 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19082.html KW - rational approximation, asymptotic property, Newman type operator. AB -

The approximation of $|x|$ by rational functions is a classical rational problem. This paper deals with the rational approximation of the function $x^α$sgn$x$, which equals $|x|$ if $α = 1$. We construct a Newman type operator $r_n(x)$ and show $$\mathop{\rm min}\limits_{|x|≤1} \{|x^α{\rm sgn}x − r_n(x)| \} ∼ Cn^{−\frac{α}{2}}e^{−\sqrt{2nα}},$$ where $C$ is a constant depending on $α$.

Zhan , Qian and Xu , Shusheng. (2021). Asymptotic Property of Approximation to $x^α$sgn$x$ by Newman Type Operators. Communications in Mathematical Research . 27 (3). 193-199. doi:
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