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Suppose we have a finite set $I \subseteq \mathbb{N}$, and $\alpha_i\in [0, 1]$ are fixed numbers for all $i\in I$.

Is there a way to approximate $\prod\limits_{(i, j)\in I^2} (1 - \alpha_i x_{i, j})$, where $0\leq x_{i, j}\leq 1 \; \forall (i, j)\in I^2$, as a linear function in terms of the $x_{i, j}$? If so, how accurate is the approximation to the actual product?

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  • $\begingroup$ For any differentiable function (not just this one), the first order approximation $f(\vec x_0) + \nabla f(\vec x_0)\cdot (\vec x-\vec x_0)$ is the best linear approximation near the point $\vec x_0$. For example, near $\vec 0$ the best linear approximation to your function is $1-\sum_{i=1}^k x_i$. $\endgroup$ Aug 5 at 20:26
  • $\begingroup$ @GregMartin Thanks for the response. I realized my question was too reduced though. I updated the question to capture the full nuance of my problem. $\endgroup$ Aug 5 at 20:39
  • $\begingroup$ OR sorry, I think your answer would still work for this updated problem $\endgroup$ Aug 5 at 20:40
  • $\begingroup$ Is there a norm you have in mind? A uniform approximation will differ from an $L^2$ approximation, and both from a Taylor series. $\endgroup$ Aug 5 at 22:23
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    $\begingroup$ OP, best to state your application. $\endgroup$ Aug 6 at 0:37

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