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The first-kind Chebyshev polynomial $T_n$ converts a multiple-angle cosine into a polynomial:The cosine addition formula gives the recurrence relationThis constructs polynomials valid for every real $x$, including outside $[-1,1]$, where the real-angle description is unavailable. For example,For $n\ge 1$, $T_n$ has polynomial degree $n$ and leading coefficient $2^{n-1}$. On $[-1,1]$ its uniform norm is one, even as its oscillations become more numerous.^[ref][ref]

The polynomial roots of $T_n$ are especially simple:They correspond to equally spaced angles, but their locations in $x$ cluster near $-1$ and $1$. For $T_3$, the polynomial roots are $\sqrt{3}/2,0,-\sqrt{3}/2$. For $T_8$, the positive polynomial roots are approximately $0.980785,0.831470,0.555570,0.195090$, with their negatives completing the set. The distance from the right endpoint to the nearest polynomial root is $1-\cos (\pi /(2n))\sim {\pi }^2/(8n^2)$; central neighboring polynomial roots are separated on the larger scale $1/n$. The small change of cosine near angles zero and $\pi$ explains the clustering.

These polynomial roots supply $n$ interpolation nodes for a polynomial interpolation of polynomial degree at most $n-1$. They differ from the $n+1$ Chebyshev-Lobatto nodes $\cos (j\pi /n)$, which include both endpoints. This choice of polynomial roots has nodal polynomialThe monic Chebyshev extremal polynomial minimizes the uniform norm of this product among monic polynomials of polynomial degree $n$ on $[-1,1]$. Consequently, if $f$ has $n$ continuous derivatives, the usual polynomial interpolation error gives the useful boundThis optimizes the nodal factor in that bound, not the actual error for every function. Endpoint clustering helps control the large endpoint oscillations of equally spaced interpolation associated with Runge's phenomenon. The two node conventions and their geometric construction are discussed in Trefethen's authored text.^[ref]

The same cosine substitution for polynomial approximation explains the weighted orthogonality:The endpoint singularities of the weight are integrable. Thus the Chebyshev polynomials are orthogonal polynomials for this weight, making coefficients in a least-squares polynomial in an orthogonal-polynomial basis separate cleanly. Their polynomial roots also give Chebyshev–Gauss quadrature, connecting polynomial interpolation to Gaussian quadrature.^[ref]

Accuracy still depends on the target function. The absolute value function $|x|$ has a corner at zero, so the hypothesis requiring continuous derivatives in the displayed polynomial interpolation error bound fails. Endpoint clustering cannot remove that corner. A jump poses a stronger obstruction: polynomials are continuous functions, so they cannot converge in the uniform norm to a discontinuous function on the whole interval. Chebfun's documented examples show persistent overshoot for such jumps; these are a limitation of global polynomial interpolation, not a universal promise of rapid accuracy from Chebyshev polynomials.^[ref]

Take $n\ge 1$, as intended by the stated normalization. The Chebyshev polynomial recurrence $T_{r+1}=2x{T}_r-T_{r-1}$, with $T_0=1$ and $T_1=x$, shows by mathematical induction that $T_n$ has polynomial degree $n$ and leading coefficient $2^{n-1}$. Consequently $f_n=2^{1-n}{T}_n$ is a monic polynomial. At the ordered Chebyshev extreme points $t_j=-\cos (j\pi /n)$,and $\Vert f_n{\Vert }_{\infty }=2^{1-n}$.

Suppose a monic polynomial $q$ of the same polynomial degree had $\Vert q{\Vert }_{\infty }<2^{1-n}$. The difference $f_n-q$ has degree at most $n-1$ because its leading coefficients cancel. Its signs at the $n+1$ ordered extreme points strictly alternate. The intermediate value theorem supplies a distinct root of a polynomial between each consecutive pair, giving at least $n$ roots. This contradicts its degree bound. Hence the monic Chebyshev extremal polynomial satisfiesAt $n=0$ the printed formula instead gives $f_0=2$, which is not monic; the degree-zero minimizer is the constant one. This endpoint case requires the separate normalization $f_0=1$.

For $n\ge 1$, set $p=2^{1-n}{T}_n$. It is monic by part (a), and its supremum norm on $[-1,1]$ is $E=2^{1-n}$. At the $n+1$ successive extreme points $x_j=\cos (j\pi /n)$ it alternates between $E$ and $-E$.

Suppose a monic degree-$n$ polynomial $q$ had $\Vert q{\Vert }_{\infty }<E$. At each $x_j$, the difference $p-q$ would have the sign of $p$. These signs alternate, so the intermediate value theorem forces a root in each of the $n$ successive gaps. But the leading monic terms cancel, making $\mathrm{deg}(p-q)\le n-1$, a contradiction. ThereforeThis is the monic monic Chebyshev extremal polynomial. The positive-degree restriction matters: at $n=0$ the printed scaling gives the constant two, while the only monic constant is one, so that version is false. The customary theorem uses $n\ge 1$.

Put $x=\cos \theta$. The trigonometric identitygivesStarting with $T_0=1$ and $T_1=x$, this recurrence proves inductively that each Chebyshev polynomial is a polynomial. For $n\ge 1$, the highest-degree term in $2x{T}_n$ cannot cancel against $T_{n-1}$, so the degree increases by one and its leading coefficient doubles. ThusThe constant polynomial $T_0$ has leading coefficient one.

For $n\ge 1$, the points $x_j=\cos (j\pi /n)$, $j=0,\dots ,n$, give $T_n(x_j)=(-1{)}^j$. They are all the points where $|T_n|=1$, since $|\cos (n\theta )|=1$ exactly when $n\theta$ is an integer multiple of $\pi$. Therefore there are $n+1$ equioscillation points, including the two endpoints. Reversing their order makes an increasing alternating sequence. For $n=0$ the polynomial is constant: every point has unit value, but an alternating sequence has at most one point.

Now take $n\ge 1$ and $c=2^{1-n}$. Since $c{T}_n$ is monic, $p_0(x)=x^n-c{T}_n(x)$ has degree at most $n-1$, and $\Vert x^n-p_0{\Vert }_{\infty }=c$. This proves the upper bound for the best error.

For the lower bound, suppose a polynomial $p$ of degree at most $n-1$ had $\Vert x^n-p{\Vert }_{\infty }<c$. At each of the $n+1$ ordered extrema, $h=p-p_0$ must have the same strictly nonzero sign as $T_n$: if $c{T}_n=c$, then $|c-h|<c$ gives $h>0$, and if $c{T}_n=-c$, then $|-c-h|<c$ gives $h<0$. Between every consecutive pair, the intermediate value theorem gives a zero of $h$. These are $n$ distinct zeros, impossible for a nonzero polynomial of degree at most $n-1$; $h=0$ is incompatible with the strict error improvement. This first-principles sign argument proves the monic Chebyshev extremal polynomial bound without using the alternation theorem:The displayed $p_0$ supplies an actual best approximating polynomial.