Some new Chebyshev spaces (Q1048992)

In this nice, short note, the authors show that \[ span\{1,x,x^2,\dots,x^{m-1}, \frac {T_m(x)}{\sqrt{(1-x^2}}\}, span\{1,x,x^2,\dots,x^{m-1}, \frac {T_m(x)arccos x}{\sqrt{(1-x^2}}\} \] are extended Chebyshev spaces over \((-1,1)\), where \(m>0\) and \(T_m(x)\) is the \(m\)th degree Chebyshev polynomial of the first kind. This is achieved by using induction to obtain a formula for the \(m\)th derivative operator applied to the last functions in the span, there by showing the derivatives to be strictly positive. This is then used to deduce that \(span\{\sin y,\sin 2y,\dots,\sin my,y\cos my\}\) is an extended Chebyshev space on \((n\pi,(n+1)\pi)\) for all \(n>0\).