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The centre manifold theorem states that near an equilibrium whose linearization has centre, stable, and unstable spectral subspaces $E^c,E^s,E^u$, there is a local invariant manifold $W^c$ tangent to $E^c$ at the equilibrium. It can be written locally as a graph over $E^c$, and the dynamics on it govern the local nonhyperbolic behaviour. The manifold need not be unique, but its finite Taylor expansion is determined to the required order by the invariance equation.

For a parameter-dependent system, the key step in forming the extended centre manifold is to promote the parameter to a dynamical variable:The parameter direction then has zero eigenvalue and is included in the extended centre subspace.

Chebyshev's equal-ripple criterion says that a polynomial $q$ of degree at most $m$ is a best uniform approximation to a continuous function $F$ if and only if the error $F-q$ attains alternating extrema of maximum magnitude at $m+2$ ordered points.

Since $T_n$ has leading coefficient $2^{n-1}$, the polynomial $2^{1-n}{T}_n$ is monic and alternates between $\pm 2^{1-n}$ at $t_j=\cos (j\pi /n)$. Therefore

At each $t_j$, both $T_n-f$ and $T_n+f$ have sign $(-1{)}^j$, because $|f(t_j)|<1$. Each has a root in every interval between consecutive extrema, hence all its at most $n$ roots lie in $(-1,1)$. For $t>1$ both polynomials remain positive, giving $|f(t)|<T_n(t)$. For $t<-1$ both have sign $(-1{)}^n$, giving $|f(t)|<|T_n(t)|$. On $[-1,1]$ the hypothesis gives $|f|<1$. Thus