Bounds on Moments of Weighted Sums of Finite Riesz Products
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(2020) 26:84
Bounds on Moments of Weighted Sums of Finite Riesz Products Aline Bonami1 · Rafał Latała2 · Piotr Nayar2 · Tomasz Tkocz3 Received: 11 March 2020 / Revised: 14 October 2020 / Accepted: 26 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let n j be a lacunary sequence of integers, such that n j+1 /n j ≥ r . We are interested in linear combinations of the sequence of finite Riesz products Nj=1 (1 + cos(n j t)). We p prove that, whenever the Riesz products are normalized in L norm ( p ≥ 1) and when r is large enough, the L p norm of such a linear combination is equivalent to the p norm of the sequence of coefficients. In other words, one can describe many ways of embedding p into L p based on Fourier coefficients. This generalizes to vector valued L p spaces. Keywords Riesz products · Moment estimates · Lacunary sequences · Trigonometric polynomials Mathematics Subject Classification Primary: 42A55 · Secondary: 26D05, 42A05
Communicated by Krzysztof Stempak.
B
Tomasz Tkocz [email protected] Aline Bonami [email protected] Rafał Latała [email protected] Piotr Nayar [email protected]
1
Institut Denis Poisson, CNRS-UMR 2013, Université d’Orléans, Orléans, France
2
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
3
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA 0123456789().: V,-vol
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Journal of Fourier Analysis and Applications
(2020) 26:84
1 introduction Let T = R/2π Z be the one dimensional torus and m be the normalized Haar measure on T. Let (n j ) j≥1 be an increasing sequence of positive integers. Riesz products are defined on T by R0 ≡ 1 and R N (t) :=
N
(1 + cos(n j t)) for N = 1, 2, . . .
(1)
j=1
To simplify the notation we also put X 0 ≡ 1 and X j (t) := 1 + cos(n j t),
j = 1, 2, . . .
It was Frigyes Riesz who first realized the usefulness of these objects treated as probability measures. When n j+1 /n j ≥ 2 for j ≥ 1, the numbers Nj=1 ε j n j are all nonzero for nonzero vectors (ε j ) Nj=1 ∈ {−1, 0, 1} N , due to the fact that for every l, l k=1 n k < n l+1 . In particular, the zero mode of R N has Fourier weight 1 and thus R N are densities of probability measures μ N . The weak-∗ limit of (μ N ) is a singular measure which admits a number of remarkable Fourier-analytic properties. The reader is referred for instance to [12] for more information on properties of Riesz products and general trigonometric polynomials as well as to the short survey [6] of some applications of Riesz products. We will always assume that n j+1 /n j ≥ 3 for j ≥ 1, so that every integer n can be written at most once as Nj=1 ε j n j for nonzero vectors (ε j ) Nj=1 ∈ {−1, 0, 1} N . N vk Rk where vk are vectors in a normed In this article we shall study the sum k=0 space (E, · ). By the triangle inequality, we trivially have N N vk . vk Rk dm ≤ T k=0
(2)
k=0
We are interested in the reverse inequality and in L p
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