Even-polynomial approximation with exact endpoint interpolation ($b(t)=p(t^2),b(0)=a(0),b(1)=a(1)$)

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 $\Vert a-p(t^2){\Vert }_{\infty }<\eta$, 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 $\eta <\epsilon /2$ yields any prescribed uniform error $\epsilon$.