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Consider the unit $D$-simplex $S^D=\left\lbrace (x_0, x_1, \ldots, x_D) \in \mathbb{R}^{D+1} \mid \sum\limits_{i=0}^{D}x_i = 1, x_i \geq 0 \right\rbrace$. I have a bounded, convex function $f:S^D\to\mathbb{R}$ that I would like to approximate by a linear combination of basis functions $g_j:S^D\to\mathbb{R}$, i.e. \begin{equation} f(x) \approx \hat{f}(x) = \sum\limits_{j=1}^N \alpha_j g_j(x), \end{equation} where $\alpha_j \in\mathbb{R}$ are the weights of the basis functions. I am interested in an approximation that minimizes the supremum norm $\sup\limits_{x\in S^D}|f(x)-\hat{f}(x)|$.

A concrete example: $$ f(x_0,x_1,\ldots,x_D) = \sum\limits_{i}x_i\ln x_i $$ i.e. the negative Shannon information entropy of a random variable having probability mass function (pmf) $x\in S^D$. However, generally I am interested in negative uncertainty functions that are convex on $S^D$: informally speaking, greater values at the center of the simplex ($x$ is uniform pmf), smaller values at the corners of the simplex ($x$ is pmf with probability mass concentrated on a single value).

What would be an appropriate choice for the basis functions $g_j$? Is there a choice for which the supremum norm tends to zero as $N\to\infty$?

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    $\begingroup$ So a case of special interest is $$f(x_0,x_1,\ldots,x_D)=\sum_i x_i\ln x_i?$$ But you would like to see a general method for selecting a good sequence of functions $g_j, j\ge1$. $\endgroup$ Oct 29, 2015 at 19:10
  • $\begingroup$ Correct. To give some more background, I'm interested in uncertainty functions which are non-negative concave functions on the simplex (c.f. DeGroot: Optimal statistical decisions). I thought Shannon entropy might be an example familiar to most people. $\endgroup$
    – mikkola
    Oct 29, 2015 at 19:46
  • $\begingroup$ Do you allow the convex function to take infinite values? $\endgroup$
    – user71040
    Nov 4, 2015 at 9:59
  • $\begingroup$ @user71040 We can assume the function to be bounded. I have added clarifications to the main question based on the comments above. $\endgroup$
    – mikkola
    Nov 4, 2015 at 11:23

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