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

• Autorzy: Tamás Erdélyi
• Języki publikacji: EN

Abstrakty

EN

Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by 𝒫ₙ (resp. $𝒫ₙ^{c}$) the set of all polynomials of degree at most~n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $Pₙ ∈ 𝒫ₙ^{c}$ 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
$∥Sₙ∥^{real}_{∞,∂D}:= sup_{Pₙ∈𝒫ₙ} (max_{z∈∂D} |Sₙ(Pₙ)(z)|)/(max_{z∈∂D}|Pₙ(z)|)$,
$∥Sₙ∥^{comp}_{∞,∂D}:= sup_{Pₙ∈𝒫ₙ^{c}} (max_{z∈∂D} |Sₙ(Pₙ)(z)|)/(max_{z∈∂D} |Pₙ(z)|$.
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.

Kategorie tematyczne

• 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol\cprime ski{\u\i}-type inequalities)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2001
• Tom: 90
• Numer: 2
• Strony: 287-293

Daty

wydano

2001

Twórcy

autor

Tamás Erdélyi

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

Identyfikatory

DOI

10.4064/cm90-2-8