COLLOQUIUM MATHEMATICUM VOL. 90 2001 NO. 2 THE NORM OF THE POLYNOMIAL TRUNCATION OPERATOR ON THE UNIT DISK AND ON [−1, 1] BY TAM ´ AS ERD ´ ELYI (College Station, TX) Abstract. Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by P n (resp. P c n ) 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 ∈P c n of the form P n (z) := ∑ n j=0 a j z j , a j ∈ C, by S n (P n )(z) := n  j=0  a j z j ,  a 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 ‖S n ‖ real ∞,∂D := sup P n ∈P n max z∈∂D |S n (P n )(z)| max z∈∂D |P n (z)| , ‖S n ‖ comp ∞,∂D := sup P n ∈P c n max z∈∂D |S n (P n )(z)| max z∈∂D |P n (z)| . 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 √ 2n +1 ≤‖S n ‖ real ∞,∂D ≤‖S n ‖ 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. 1. New result. Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by P n (resp. P c n ) 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 ∈P c n of the form P n (z ) := n  j=0 a j z j , a j ∈ C, 2000 Mathematics Subject Classification : Primary 41A17. Key words and phrases : truncation of polynomials, norm of the polynomial truncation operator, Lov´asz–Spencer–Vesztergombi theorem. Research supported, in part, by NSF under Grant No. DMS-0070826. [287]