The norm of the polynomial truncation operator on the unit disk and on $[-1,1]$
Tom 90 / 2001
Streszczenie
Let $D$ and $\partial D$ denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ${\cal P}_n$ (resp. ${\cal P}_n^{\rm c}$) the set of all polynomials of degree at most~$n$ with real (resp. complex) coefficients. We define the truncation operators $S_n$ for polynomials $P_n \in {\cal P}_n^{\rm c}$ of the form $P_n(z) := \sum_{j=0}^n a_jz^j$, $a_j \in {\Bbb C}$, by $$ S_n(P_n)(z) := \sum_{j=0}^n \widetilde{a}_jz^j, \quad\ \widetilde{a}_j := {a_j\over |a_j|}\min\{|a_j|,1\} $$ (here $0/0$ is interpreted as $1$). We define the norms of the truncation operators by $$\eqalign{ \|S_n\|_{\infty, \partial D}^{\rm real} :={}& \sup_{P_n \in {\cal P}_n} {\frac{\max_{z \in \partial D}{|S_n(P_n)(z)|}} {\max_{z \in \partial D}{|P_n(z)|}}},\cr \|S_n\|_{\infty, \partial D}^{\rm comp} :={}& \sup_{P_n \in {\cal P}_n^{\rm c}} {\frac{\max_{z \in \partial D}{|S_n(P_n)(z)|}} {\max_{z \in \partial D}{|P_n(z)|}}}.\cr} $$ Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant $c_1 > 0$ such that $$ c_1 \sqrt {2n + 1} \leq \|S_n\|_{\infty, \partial D}^{\rm real} \leq \|S_n\|_{\infty, \partial D}^{\rm comp} \leq \sqrt {2n+1}. $$ This settles a question asked by S.~Kwapie/n. Moreover, an analogous result in $L_p(\partial D)$ for $p \in [2,\infty]$ is established and the case when the unit circle $\partial D$ is replaced by the interval $[-1,1]$ is studied.
