Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 127, Number 1, January 1999, Pages 211-216
AN EXTREMAL PROBLEM
FOR TRIGONOMETRIC POLYNOMIALS
J. MARSHALL ASH AND MICHAEL GANZBURG
(Communicated by Christopher D. Sogge)
ABSTRACT. Let T, (x) = Zk=0 (ak cos kx + bksin kx) be a trigonometric poly-
nomial of degree n. The problem of finding Cnp, the largest value for C in
the inequality maxflaol, lall,..., lan? lbil,..., 1bn1} < (1/C) 11Tn11Pis studied.
We find Cnp exactly provided p is the conjugate of an even integer 2s and
n > 2s - 1, s = 1, 2,.... For general p, 1 < p < oo,we get an interval estimate
for Cnp, where the interval length tends to 0 as n tends to oo.
Let Lp be the Banach space of all 27r-periodicreal-valued functions f with finite
norm IIf defined to be (f If(x)l Pdx) when 1 < p < oc, and esssup If(x)IXs[-7r,7r]
when p = ox. Let pi = P be the conjugate exponent and for n > 1 let Tn be the
class of all trigonometric polynomials with real coefficients of degree n or less.