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

Tamás Erdélyi

Colloquium Mathematicae, Tome 89 (2001), p. 287-293 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

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Publié le : 2001-01-01

Zbl 0995.41002

EUDML-ID : urn:eudml:doc:283624