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• For iterated Boolean sums of Bernstein polynomials ($U_{n,k}$ in Micchelli 1973; see also Günturk and Li), find an explicit bound, with no hidden constants, on the approximation error for functions with continuous $r$-th derivative, or verify my proof of those error bounds.
• For linear combinations of Bernstein polynomials (Butzer 1953, Tachev 2022), verify my proof of those error bounds in my Proposition B10.
• For the "Lorentz operator", find an explicit bound, with no hidden constants, on the approximation error for the operator $Q_{n_r}$ and for the polynomials $(f_n)$ and $(g_n)$ formed with it, and find the hidden constants $\theta_\alpha$, $s$, and $D$ as well as those in Lemmas 15, 17 to 22, and 24 in the paper. Or verify my proof of the order-2 operator's error bounds in my Proposition B10A.
• Find other polynomial operators meeting the requirements above and having explicit error bounds, with no hidden constants, especially operators that preserve polynomials of a higher degree than linear functions.

Thus, people have developed alternatives, including linear combinations and iterated Boolean sums of Bernstein polynomials, to improve the convergence rate. These include Micchelli (1973), Guan (2009), Güntürk and Li (2021a, 2021b), the "Lorentz operator" in Holtz et al. (2011), Draganov (2014), and Tachev (2022).

These alternative polynomials usually include results where the error bound is the desired $O(1/n^{k/2})$, but nearly all those results (e.g., Theorem 4.4 in Micchelli; Theorem 5 in Güntürk and Li) have hidden constants with no upper bounds given, making them unimplementable (that is, it can't be known beforehand whether a given polynomial will come close to the target function within a user-specified error tolerance).

(**) My implementation experience shows that Chebyshev interpolants are far from being readily convertible to Bernstein form without using transcendental functions or paying attention to the difference between first vs. second kind, Chebyshev points vs. coefficients, and the interval [-1, 1] vs. [0, 1]. For purposes of this question, Chebyshev interpolants are impractical, and so are other approximating functions that introduce transcendental functions. By contrast, other schemes (which are of greater interest to me) involve polynomials that are already in Bernstein form or that use only rational arithmetic to transform to Bernstein form (these include linear combinations and iterated Boolean sums of Bernstein polynomials). Indeed, unlike with rational arithmetic (where arbitrary precision is trivial), transcendental functions require special measures to support arbitrary accuracy, such as constructive/recursive reals — floating-point numbers won't do for purposes of this question.

• For iterated Boolean sums (linear combinations of iterates) of Bernstein polynomials ($U_{n,k}$ in Micchelli 1973; see also Güntürk and Li), find an explicit bound, with no hidden constants, on the approximation error for functions with continuous $r$-th derivative, or verify my proofs of these bounds in Propositions B10C and B10D.
• For linear combinations of Bernstein polynomials (Butzer 1953, Tachev 2022), verify my proof of those error bounds in my Proposition B10.
• For the "Lorentz operator", find an explicit bound, with no hidden constants, on the approximation error for the operator $Q_{n,r}$ and for the polynomials $(f_n)$ and $(g_n)$ formed with it, and find the hidden constants $\theta_\alpha$, $s$, and $D$ as well as those in Lemmas 15, 17 to 22, 24, and 25 in the paper. Or verify my proof of the order-2 operator's error bounds in my Proposition B10A.
• Let $f:[-1,1]\to [0,1]$ be continuous. Find explicit bounds, with no hidden constants, on the error in approximating $f$ with the following polynomials: The polynomials are similar to Chebyshev interpolants, but evaluate $f$ at rational values of $\lambda$ that converge to Chebyshev points (that is, converging to $\cos(j\pi/n)$ with increasing $n$). The error bounds must be close to those of Chebyshev interpolants (see, e.g., chapters 7, 8, and 12 of Trefethen, Approximation Theory and Approximation Practice, 2013).
• Find other polynomial operators meeting the requirements of the main question (see "Main Question", above) and having explicit error bounds, with no hidden constants, especially operators that preserve polynomials of a higher degree than linear functions.

Thus, people have developed alternatives, including linear combinations and iterated Boolean sums of Bernstein polynomials, to improve the convergence rate. These include Micchelli (1973), Guan (2009), Güntürk and Li (2021a, 2021b), the "Lorentz operator" in Holtz et al. (2011) (see also "New coins from old, smoothly"), Draganov (2014), and Tachev (2022).

These alternative polynomials usually include results where the error bound is the desired $O(1/n^{k/2})$, but most of those results (e.g., Theorem 4.4 in Micchelli; Theorem 5 in Güntürk and Li) have hidden constants with no upper bounds given, making them unimplementable (that is, it can't be known beforehand whether a given polynomial will come close to the target function within a user-specified error tolerance).

• For iterated Boolean sums of Bernstein polynomials ($U_{n,k}$ in Micchelli 1973; see also Günturk and Li), find an explicit bound, with no hidden constants, on the approximation error for functions with continuous $r$-th derivative, or verify my proof of those error bounds.
• For linear combinations of Bernstein polynomials (Butzer 1953, Tachev 2022), verify my proof of those error bounds in my Proposition B10.
• For the "Lorentz operator", find an explicit bound, with no hidden constants, on the approximation error for the operator $Q_{n_r}$ and for the polynomials $(f_n)$ and $(g_n)$ formed with it, and find the hidden constants $\theta_\alpha$, $s$, and $D$ as well as those in Lemmas 15, 17 to 22, and 24 in the paper.
• Find other polynomial operators meeting the requirements above and having explicit error bounds, with no hidden constants, especially operators that preserve polynomials of a higher degree than linear functions.

• For iterated Boolean sums of Bernstein polynomials ($U_{n,k}$ in Micchelli 1973; see also Günturk and Li), find an explicit bound, with no hidden constants, on the approximation error for functions with continuous $r$-th derivative, or verify my proof of those error bounds.
• For linear combinations of Bernstein polynomials (Butzer 1953, Tachev 2022), verify my proof of those error bounds in my Proposition B10.
• For the "Lorentz operator", find an explicit bound, with no hidden constants, on the approximation error for the operator $Q_{n_r}$ and for the polynomials $(f_n)$ and $(g_n)$ formed with it, and find the hidden constants $\theta_\alpha$, $s$, and $D$ as well as those in Lemmas 15, 17 to 22, and 24 in the paper. Or verify my proof of the order-2 operator's error bounds in my Proposition B10A.
• Find other polynomial operators meeting the requirements above and having explicit error bounds, with no hidden constants, especially operators that preserve polynomials of a higher degree than linear functions.

• For iterated Boolean sums of Bernstein polynomials ($U_{n,k}$ in Micchelli 1973; see also Günturk and Li), find an explicit bound, with no hidden constants, on the approximation error for functions with continuous $r$-th derivative, or verify my proof of those error bounds.
• For linear combinations of Bernstein polynomials (Butzer 1953, Tachev 2022), verify my proof of those error bounds in Proposition B10.
• For the "Lorentz operator", find an explicit bound, with no hidden constants, on the approximation error for the operator $Q_{n_r}$ and for the polynomials $(f_n)$ and $(g_n)$ formed with it, and find the hidden constants $\theta_\alpha$, $s$, and $D$ as well as those in Lemmas 15, 17 to 22, and 24 in the paper.
• Find other polynomial operators meeting the requirements above and having explicit error bounds, with no hidden constants, especially those that preserve polynomials of a higher degree than linear functions.

• For iterated Boolean sums of Bernstein polynomials ($U_{n,k}$ in Micchelli 1973; see also Günturk and Li), find an explicit bound, with no hidden constants, on the approximation error for functions with continuous $r$-th derivative, or verify my proof of those error bounds.
• For linear combinations of Bernstein polynomials (Butzer 1953, Tachev 2022), verify my proof of those error bounds in my Proposition B10.
• For the "Lorentz operator", find an explicit bound, with no hidden constants, on the approximation error for the operator $Q_{n_r}$ and for the polynomials $(f_n)$ and $(g_n)$ formed with it, and find the hidden constants $\theta_\alpha$, $s$, and $D$ as well as those in Lemmas 15, 17 to 22, and 24 in the paper.
• Find other polynomial operators meeting the requirements above and having explicit error bounds, with no hidden constants, especially operators that preserve polynomials of a higher degree than linear functions.