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Let $\mathcal{C}$ be the class of continuous functions that— map $[0, 1]$ to $[0, 1]$, and equal neither 0 nor 1 on the open interval $(0, 1)$. A function $f(x)$ is algebraic over the rational numbe …

Let $f:[0,1]\to [0,1]$ be continuous. Let— $$B_n(f)(x)=\sum_{k=0}^n f(k/n) {n \choose k} x^k (1-x)^{n-k},$$ be the Bernstein polynomial of $f$ of degree $n$. This question relates to the difference b …

After analyzing the proof of Güntürk and Li (2021), Theorem 2.4, I believe I found explicit error bounds for the Micchelli–Felbecker polynomials (iterated Bernstein polynomials) when $f(x)$ has a giv …

Questions Let $f(\lambda):[0,1]\to (0,1)$ have a $\beta-\lfloor\beta\rfloor$)-Hölder continuous $\lfloor\beta\rfloor$-th derivative, where $\beta>0$. Find explicit bounds, with no hidden constants, o …

Main Question This question is about finding explicit, calculable, and fast error bounds when approximating continuous functions with polynomials to a user-specified error tolerance. EDIT (Apr. 23): …

Main Question This question is about finding explicit, calculable, and fast error bounds (no hidden constants) when approximating continuous functions with polynomials or simpler functions to a user-s …

Here are some results on certain polynomials. Tachev (2022)[^4] has published an error bound that relates to the polynomial— $$L_{2,n/2} = 2B_n(f) - B_{n/2}(f).$$ Their Theorem 2 describes the bound p …

Main Question Suppose $f:[0,1]\to[0,1]$ is continuous, polynomially bounded, and belongs to a large class of functions (for example, the $k$-th derivative, $k\ge 0$, is continuous, Lipschitz continuou …