SPECIAL QUADRATURE ERROR ESTIMATES AND THEIR APPLICATION IN THE HARDY-LITTLEWOOD MAJORANT PROBLEM
The Hardy-Littlewood majorant problem has a positive answer only for exponents p which are even integers, while there are counterexam- ples for all p = 2 2N. Montgomery conjectured that there exist counterex- amples even among idempotent polynomials. This was proved recently by Mockenhaupt and Schlag with some four-term idempotents. However, Mockenhaupt conjectured that even the classical 1 + e 2ix e 2i (k+2)x three-term character sums, should work for all 2k < p < 2k + 2 and for all k 2 N. In two previous papers we proved this conjecture for k = 0;1;2;3;4, i.e. in the range 0 < p < 10, p = 2 2N. Here we demonstrate that even the k = 5 case holds true. Renements in the technical features of our approach include use of total variation and integral mean estimates in error bounds for a certain fourth order quadrature. Our estimates make good use of the special forms of functions we encounter: linear combinations of powers and powers of log- arithms of absolute value squares of trigonometric polynomials of given degree. Thus the quadrature error estimates are less general, but we can nd better constants which are of practical use for us.
HOW TO CONCENTRATE IDEMPOTENTS
Call a sum of exponentials of the form f(x) = exp (2πiN1x) + exp (2πiN2x) + · · ·+ exp (2πiNmx), where the Nk are distinct integers, an idempotent. We have L interval concentration if there is a positive constant a, depending only on p, such that for each interval I ⊂ [0, 1] there is an idempotent f so that R I |f (x)| dx R 1 0 |f (x)| dx > a. We will explain how to produce such concentration for each p > 0. The origin of this question and the history of the development of its solution will be surveyed.
Lp norm local estimates for exponential sums
Abstract We prove that for every real number p>1 there is an explicitly calculated constant Kp>0 such that, for any arc J of the one-dimensional torus T=R/Z with |J|>0, no matter how small, one can find some 1-periodic exponential sum f(x)= ∑ m k=1 exp (2iπN k x) so that ( ∫ J |f(x)| p d x) 1/p ≥K p ·( ∫ J |f(x)| p d x) 1/p (Lp norm “local concentration” on J). For p≥2 the result remains true (with another Kp) if the arc J is replaced by any set E⊂ T with |E|>0. The special case p=2 of our results had been studied by many authors about twenty years ago, but the general case p>1 had remained open. The limit case p=1 remains unsettled to this date.