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Let $g : [-1, 1] \to R$ be a $1$-Lipschitz function and $f_{k,d} : [-1, 1] \to R$ a $1$-Lipschitz function whose restriction to any subinterval $[h_i, h_{i+1}] \subset [-1, 1]$, $i = 0 ... (k-1)$ with nodes $h_i = 2(i / k) - 1$ is a polynomial of degree $d$.

I want to upper bound the quantity $\| f_{k,d} - g \|_\infty$ depending on $k$ and $d$ for some best approximating function $f_{k,d}$. In case $k = 1$, and general Polynomials, such a bound is given by Jackson's theorem (see, e.g. Corollary 7.5 in http://fourier.math.uoc.gr/~mk/approx1011/carothers.pdf). But here, applying Jackson's bound does not work, since one requires the function $f_{k,d}$ to be Lipschitz.

I'm glad for any help or pointers to the literature!

Edit: Setting $f_{k,d}(h_i) = g(h_i)$ on the node values and affine in-between should give a $O(1/k)$ bound. My conjecture is, that the rate is $O(1 / (kd))$ but it seems not straightforward to prove.

Edit2: Approximating $g$ with Bernstein Polynomials on each Interval should give a $O(1 / (k\sqrt{d}))$ rate, since the resulting function is Lipschitz.

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