The norm of the polynomial truncation operator on the unit disk and on $[-1,1]$

Tamás Erdélyi¹

¹ Department of Mathematics Texas A&M University College Station, TX 77843, U.S.A.

Colloquium Mathematicum, Tome 90 (2001) no. 2, pp. 287-293.

Résumé

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.

Détail

DOI : 10.4064/cm90-2-8

Keywords: partial denote unit disk unit circle complex plane respectively denote cal resp cal set polynomials degree real resp complex coefficients define truncation operators polynomials cal form sum bbb n sum widetilde quad widetilde min here interpreted define norms truncation operators eqalign infty partial real sup cal frac max partial n max partial infty partial comp sup cal frac max partial n max partial main theorem establishes right order magnitude above norms there absolute constant sqrt leq infty partial real leq infty partial comp leq sqrt settles question asked kwapie moreover analogous result partial infty established unit circle partial replaced interval studied