Concentration of the Integral Norm of Idempotents

Bonami, Aline; Révész, Szilárd Gy.

doi:10.1007/978-0-8176-4888-6_8

Cited by 3 publications

(18 citation statements)

References 11 publications

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“…But in a sense this is due to a "cheating" in the extent that we can simulate powers of Dirichlet kernels by products of their scaled versions. In a forthcoming note [10] we show, however, that on the finite groups Z/qZ uniform in q L 1 concentration does really fail.…”

Section: Remark 14mentioning

confidence: 80%

Integral concentration of idempotent trigonometric polynomials with gaps

Bonami¹,

Révész²

2009

ajm

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Abstract. We prove that for all p > 1/2 there exists a constant γ p > 0 such that, for any symmetric measurable set of positive measure E ⊂ T and for any γ < γ p , there is an idempotent trigonometrical polynomial f satisfying E |f | p > γ T |f | p . This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of γ p > 0 for p > 1 and conjectured that it does not exists for p = 1.Furthermore, we prove that one can take γ p = 1 when p > 1 is not an even integer, and that polynomials f can be chosen with arbitrarily large gaps when p = 2. This shows striking differences with the case p = 2, for which the best constant is strictly smaller than 1/2, as it has been known for twenty years, and for which having arbitrarily large gaps with such concentration of the integral is not possible, according to a classical theorem of Wiener.We find sharper results for 0 < p ≤ 1 when we restrict to open sets, or when we enlarge the class of idempotent trigonometric polynomials to all positive definite ones. (2000): Primary 42A05. Secondary 42A16, 42A61, 42A55, 42A82, 42B05. Mathematics Subject Classification

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Section: Remark 14mentioning

confidence: 80%

Integral concentration of idempotent trigonometric polynomials with gaps

Bonami¹,

Révész²

2009

ajm

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“…Now if ρ ≤ k + 1, we also have 0 ≤ τ ≤ k + 1, so the number τ of choosing e 1 s is exactly λ, the mod k + 2 reduced residue of ν, and consequently σ = (ν − λ)/(k + 2) = µ. That results in formula (4) for a ± (ν), while (5) follows by Parseval's formula. Whence the assertion is proved.…”

Section: Notations and A Few General Formula For The Numerical Analysismentioning

confidence: 92%

“…As the error is at most δ, it suffices to show that p(t) := P 10 (t) + δ < 0 in [3,4]. Now From the explicit formula of p(t) we consecutively compute also p ′′ (3) = −23.12291565 < 0, p ′′′ (3) = −93.80789264 < 0 and p (4) (3) = −324.0046433, p (5) (3) = −978.7532737..., p (6) (3) = −3144.062078..., p (7) (3) = −5587.909055..., all < 0. Thus p (j) (3) < 0 for j = 0, .…”

Section: Consequently We Havementioning

confidence: 99%

“…In case of t 0 = 4.75 we have for all ξ ∈ [4. 5,5] (51) max m e · ξ m , 9 ξ log m 9 ≤ max m 4.5e m , 9 5 2 m log m 3 = 9 5 log m 9 (∀m < 37).…”

Section: Consequently We Havementioning

confidence: 99%

“…For details we refer to [4]. For the history and relevance of this closely related problem of idempotent polynomial concentration in L p see [4,5], the detailed introduction of [8], the survey paper [1], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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On mockenhoupt’s conjecture in the Hardy-Littlewood majorant problem

Krenedits¹

2013

J. Contemp. Mathemat. Anal.

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The Hardy-Littlewood majorant problem has a positive answer only for exponents p which are even integers, while there are counterexamples for all p / ∈ 2N. Montgomery conjectured that even among the idempotent polynomials there must exist some counterexamples, i.e. there exist some finite set of characters and some ± signs with which the signed character sum has larger p th norm than the idempotent obtained with all the signs chosen + in the character sum. That conjecture was proved recently by Mockenhaupt and Schlag.However, Mockenhaupt conjectured that even the classical 1 + e 2πix ± e 2πi(k+2)x threeterm character sums, used for p = 3 and k = 1 already by Hardy and Littlewood, should work in this respect. That remained unproved, as the construction of Mockenhaupt and Schlag works with four-term idempotents. In our previous work we proved this conjecture for k = 0, 1, 2, i.e. in the range 0 < p < 6, p / ∈ 2N. Continuing this work here we demonstrate that even the k = 3, 4 cases hold true. Several refinement in the technical features of our approach include improved fourth order quadrature formulae, finite estimation of G ′2 /G (with G being the absolute value square function of an idempotent), valid even at a zero of G, and detailed error estimates of approximations of various derivatives in subintervals, chosen to have accelerated convergence due to smaller radius of the Taylor approximation. (2000): Primary 42A05. Mathematics Subject Classification

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Special quadrature error estimates and their application in the hardy-littlewood majorant problem

Krenedits¹

2012

Preprint

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The Hardy-Littlewood majorant problem has a positive answer only for exponents p which are even integers, while there are counterexamples for all p / ∈ 2N. Montgomery conjectured that there exist counterexamples 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 2πix ± e 2πi(k+2)x threeterm character sums, should work for all 2k < p < 2k + 2 and for all k ∈ 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 / ∈ 2N. Here we demonstrate that even the k = 5 case holds true.Refinements 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 logarithms of absolute value squares of trigonometric polynomials of given degree. Thus the quadrature error estimates are less general, but we can find better constants which are of practical use for us.

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Concentration of the Integral Norm of Idempotents

Cited by 3 publications

References 11 publications

Integral concentration of idempotent trigonometric polynomials with gaps

Integral concentration of idempotent trigonometric polynomials with gaps

On mockenhoupt’s conjecture in the Hardy-Littlewood majorant problem

Special quadrature error estimates and their application in the hardy-littlewood majorant problem

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