The norm of the polynomial truncation operator on the unit disk and on [-1,1]
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© 2001, Instytut Matematycznys. All rights reserved. Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by Pn (resp. Pcn) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sn for polynomials Pn ∈ Pcn of the form Pn(z) := Σnj=0, ajzj, by aj ϵ ℂ, by (Formula presented.) (here 0/0 is interpreted as 1). We define the norms of the truncation operators by (Formula presented.) Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c1 > 0 such that (Formula presented.) This settles a question asked by S. Kwapień. Moreover, an analogous result in Lp(∂D) for p ϵ [2,∞] is established and the case when the unit circle ∂D is replaced by the interval [−1, 1] is studied.
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