I am trying to approximate the following function, $$f(x)=\frac{\sum^{N}_{c=1}\gamma_c x^{c+k-2}-\sum^{N}_{c=1}\beta_c x^{c-1}}{\sum^{N}_{c=1}\alpha_c x^{c}}$$ where $\alpha_c$'s, $\beta_c$'s and $\gamma_c$'s are some positive constants. $k$ is a positive integer. I wonder if such function can be tamed and approximated?

  • $\begingroup$ Approximate where and in what metric? This is a rational function $f(x)=\frac{\sum\limits_{i=1}^{N+k-2}a_i x^i}{\sum\limits_{i=1}^{N}b_i x^i}=P(x)+\frac{Q(x)}{R(x)}$ where $deg(P(x))=k-2$, $deg(Q)<deg(R)$ $\endgroup$ – AO1992 May 20 at 15:12
  • $\begingroup$ I was thinking of something along the line with Padé approximant en.wikipedia.org/wiki/Padé_approximant $\endgroup$ – William May 20 at 16:10

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