= Even-polynomial approximation with exact endpoint interpolation
{title2=$b(t)=p(t^2),\quad b(0)=a(0),\ b(1)=a(1)$}
On $[0,1]$, polynomials in $t^2$ form a unital algebra closed under <complex conjugation> separating points, so the <Stone-Weierstrass theorem> makes them dense in the continuous complex functions. Their approximation can preserve both endpoint values exactly. Given $\|a-p(t^2)\|_\infty<\eta$, add $[a(0)-p(0)](1-t^2)+[a(1)-p(1)]t^2$. This is still an even <polynomial>, fixes both endpoints, and changes the approximation by at most $\eta$, since its correction is a <convex combination> of the endpoint errors. Taking $\eta<\varepsilon/2$ yields any prescribed uniform error $\varepsilon$.
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