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

Tamás Erdélyi. "The norm of the polynomial truncation operator on the unit disk and on [-1,1]." Colloquium Mathematicae 90.2 (2001): 287-293. <http://eudml.org/doc/283624>.

@article{TamásErdélyi2001,
abstract = {Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. $ₙ^\{c\}$) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $Pₙ ∈ ₙ^\{c\}$ of the form $Pₙ(z) := ∑_\{j=0\}^\{n\} a_\{j\}z_\{j\}$, $a_\{j\} ∈ C$, by $Sₙ(Pₙ)(z):= ∑_\{j=0\}^\{n\} ã_\{j\}z_\{j\}$, $ã_\{j\}:= a_\{j\}|a_\{j\}| min\{|a_\{j\}|,1\}$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $∥Sₙ∥^\{real\}_\{∞,∂D\}:= sup_\{Pₙ∈ₙ\} (max_\{z∈∂D\} |Sₙ(Pₙ)(z)|)/(max_\{z∈∂D\}|Pₙ(z)|)$, $∥Sₙ∥^\{comp\}_\{∞,∂D\}:= sup_\{Pₙ∈ₙ^\{c\}\} (max_\{z∈∂D\} |Sₙ(Pₙ)(z)|)/(max_\{z∈∂D\} |Pₙ(z)|$. Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁ > 0 such that $c₁√(2n+1) ≤ ∥Sₙ∥^\{real\}_\{∞,∂D\} ≤ ∥Sₙ∥^\{comp\}_\{∞,∂D\} ≤ √(2n+1)$ This settles a question asked by S. Kwapień. Moreover, an analogous result in $L_\{p\}(∂D)$ for p ∈ [2,∞] is established and the case when the unit circle ∂D is replaced by the interval [−1,1] is studied.},
author = {Tamás Erdélyi},
journal = {Colloquium Mathematicae},
keywords = {truncation of polynomials},
language = {eng},
number = {2},
pages = {287-293},
title = {The norm of the polynomial truncation operator on the unit disk and on [-1,1]},
url = {http://eudml.org/doc/283624},
volume = {90},
year = {2001},
}

TY - JOUR
AU - Tamás Erdélyi
TI - The norm of the polynomial truncation operator on the unit disk and on [-1,1]
JO - Colloquium Mathematicae
PY - 2001
VL - 90
IS - 2
SP - 287
EP - 293
AB - Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. $ₙ^{c}$) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $Pₙ ∈ ₙ^{c}$ of the form $Pₙ(z) := ∑_{j=0}^{n} a_{j}z_{j}$, $a_{j} ∈ C$, by $Sₙ(Pₙ)(z):= ∑_{j=0}^{n} ã_{j}z_{j}$, $ã_{j}:= a_{j}|a_{j}| min{|a_{j}|,1}$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $∥Sₙ∥^{real}_{∞,∂D}:= sup_{Pₙ∈ₙ} (max_{z∈∂D} |Sₙ(Pₙ)(z)|)/(max_{z∈∂D}|Pₙ(z)|)$, $∥Sₙ∥^{comp}_{∞,∂D}:= sup_{Pₙ∈ₙ^{c}} (max_{z∈∂D} |Sₙ(Pₙ)(z)|)/(max_{z∈∂D} |Pₙ(z)|$. Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁ > 0 such that $c₁√(2n+1) ≤ ∥Sₙ∥^{real}_{∞,∂D} ≤ ∥Sₙ∥^{comp}_{∞,∂D} ≤ √(2n+1)$ This settles a question asked by S. Kwapień. Moreover, an analogous result in $L_{p}(∂D)$ for p ∈ [2,∞] is established and the case when the unit circle ∂D is replaced by the interval [−1,1] is studied.
LA - eng
KW - truncation of polynomials
UR - http://eudml.org/doc/283624
ER -