Bernstein type inequality in monotone rational approximation
arXiv:1009.44301 citationsh-index: 14
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This provides a theoretical bound for monotone rational approximation, relevant to approximation theory researchers.
The paper establishes a Bernstein-type inequality for monotone rational functions, showing that the derivative is bounded by an exponential factor in degree n, with sharp estimates for odd rational functions.
The following analog of Bernstein inequality for monotone rational functions is established: if $R$ is an increasing on $[-1,1]$ rational function of degree $n$, then $$ R'(x)<\frac{9^n}{1-x^2}\|R\|,\quad x\in (-1,1). $$ The exponential dependence of constant factor on $n$ is shown, with sharp estimates for odd rational functions.