The norm of the polynomial truncation operator on the unit disk and on \([-1,1]\) (Q2773269)

The truncation polynomial operator \(S_n:P_n^{complex}\rightarrow P^{complex}_n\) is defined by NEWLINE\[NEWLINE S_n(P(z))=\sum_{j=0}^n \widetilde{a}_j z^j, NEWLINE\]NEWLINE where \(P(z)=\sum_{j=0}^na_jz^j\) is a complex polynomial of degree \(\leq n\) and \(\widetilde{a}_j:=\frac{a_j}{|a_j|}\min(|a_j|,1)\). Associated to this operator the author defines the norms (here, \(C\) denotes the unit circle or the interval [-1,1]): NEWLINE\[NEWLINE \|S_n\|_{\infty,C}^{real}=\sup{p\in P^{real}_n}\frac{\max_{z\in C}|S_n(p)(z)|}{\max_{z\in C}|p(z)|} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \|S_n\|_{\infty,C}^{complex}=\sup{p\in P^{complex}_n}\frac{\max_{z\in C}|S_n(p)(z)|}{\max_{z\in C}|p(z)|} NEWLINE\]NEWLINE and studies the asymptotic behaviour of these norms (for \(n\rightarrow\infty\)).