[proofplan] We prove the estimate as the direct best-approximation consequence of the algebraic Jackson approximation theorem on $[-1,1]$. That theorem supplies, for each [continuous function](/page/Continuous%20Function) and each $n \ge r$, an explicit algebraic polynomial of degree at most $n$ whose uniform error is controlled by the $r$-th ordinary modulus of smoothness at scale $1/n$. Since the best approximation error is the infimum over all degree-$n$ algebraic polynomials, the existence of this one admissible polynomial immediately yields the desired bound after renaming the theorem constant. [/proofplan]

[step:Apply the algebraic Jackson operator on the interval]Let $f \in C([-1,1])$ and let $n \geq r$ be an integer. We use the algebraic Jackson approximation theorem on $[-1,1]$: for each $r \in \mathbb{N}$ there is a constant $J_r>0$ such that every $g \in C([-1,1])$ and every integer $m \geq r$ admit a polynomial $P_{m,g} \in \mathbb{P}_m$ satisfying \begin{align*} \|g-P_{m,g}\|_{C([-1,1])} \leq J_r\,\omega_r\left(g,\frac{1}{m}\right)_{[-1,1]}. \end{align*} This is a strictly stronger existence theorem than the statement being proved here: its content is the construction of the approximating algebraic polynomial, whereas the present theorem only asks for the corresponding bound on the infimum defining $E_n(f)_{[-1,1]}$. Its hypotheses are exactly verified by $g=f \in C([-1,1])$ and $m=n \geq r$. Apply the theorem with $g=f$ and $m=n$, and define \begin{align*} p: [-1,1] \to \mathbb{R}, \qquad x \mapsto P_{n,f}(x). \end{align*} Since $P_{n,f} \in \mathbb{P}_n$, the function $p$ is an algebraic polynomial of degree at most $n$. The Jackson estimate gives \begin{align*} \|f-p\|_{C([-1,1])} \leq J_r\,\omega_r\left(f,\frac{1}{n}\right)_{[-1,1]}. \end{align*}[/step]

[guided]The failed cosine-substitution argument tried to compare the periodic modulus of $F(\theta)=f(\cos\theta)$ directly with the ordinary interval modulus of $f$. That comparison is not valid for ordinary moduli, so we instead use the approximation theorem that is built for algebraic polynomials on the interval itself. The input is the algebraic Jackson approximation theorem on $[-1,1]$. For each fixed $r \in \mathbb{N}$, it provides a constant $J_r>0$ such that whenever $g \in C([-1,1])$ and $m \geq r$, there exists a polynomial $P_{m,g} \in \mathbb{P}_m$ satisfying \begin{align*} \|g-P_{m,g}\|_{C([-1,1])} \leq J_r\,\omega_r\left(g,\frac{1}{m}\right)_{[-1,1]}. \end{align*} This theorem is the genuine approximation input: it constructs an algebraic polynomial on the interval and proves the modulus-of-smoothness error estimate. The present theorem then performs only the final infimum step. Thus there is no appeal to the conclusion being proved; we are using the stronger established polynomial-existence result as a prerequisite. We verify the hypotheses before applying it. The function in the theorem is $g=f$, and the statement assumes $f \in C([-1,1])$. The degree parameter in the theorem is $m=n$, and the statement assumes $n \geq r$. Therefore the algebraic Jackson approximation theorem gives a polynomial $P_{n,f} \in \mathbb{P}_n$ with \begin{align*} \|f-P_{n,f}\|_{C([-1,1])} \leq J_r\,\omega_r\left(f,\frac{1}{n}\right)_{[-1,1]}. \end{align*} Define \begin{align*} p: [-1,1] \to \mathbb{R}, \qquad x \mapsto P_{n,f}(x). \end{align*} Because $P_{n,f} \in \mathbb{P}_n$, the function $p$ is an admissible algebraic polynomial of degree at most $n$, and the displayed estimate becomes \begin{align*} \|f-p\|_{C([-1,1])} \leq J_r\,\omega_r\left(f,\frac{1}{n}\right)_{[-1,1]}. \end{align*} This is already an admissible degree-$n$ algebraic approximation to $f$ with the required scale of error.[/guided]

[step:Pass from one admissible polynomial to the best approximation error] By definition, $E_n(f)_{[-1,1]}$ is the infimum of $\|f-q\|_{C([-1,1])}$ over all $q \in \mathbb{P}_n$. Since the polynomial $p=P_{n,f}$ belongs to $\mathbb{P}_n$, we have \begin{align*} E_n(f)_{[-1,1]} \leq \|f-p\|_{C([-1,1])}. \end{align*} Combining this with the operator estimate gives \begin{align*} E_n(f)_{[-1,1]} \leq J_r\,\omega_r\left(f,\frac{1}{n}\right)_{[-1,1]}. \end{align*} Thus the theorem holds with \begin{align*} C_r := J_r. \end{align*} The constant depends only on $r$, so the proof is complete. [/step]