On Regularity and Mass Concentration Phenomena for the Sign Uncertainty Principle
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On Regularity and Mass Concentration Phenomena for the Sign Uncertainty Principle Felipe Gonçalves1 · Diogo Oliveira e Silva2 · João P. G. Ramos3 Received: 24 March 2020 / Accepted: 5 September 2020 © Mathematica Josephina, Inc. 2020
Abstract The sign uncertainty principle of Bourgain et al. asserts that if a function f : Rd → R and its Fourier transform f are nonpositive at the origin and not identically zero, then they cannot both be nonnegative outside an arbitrarily small neighborhood of the origin. In this article, we establish some equivalent formulations of the sign uncertainty principle, and in particular prove that minimizing sequences exist within the Schwartz class when d = 1. We further address a complementary sign uncertainty principle, and show that corresponding near-minimizers concentrate a universal proportion of their positive mass near the origin in all dimensions. Keywords Sign uncertainty principle · Fourier transform · Schwartz class · Bandlimited function · Mass concentration Mathematics Subject Classification 42A85 · 42B10 · 46A11
1 Introduction Motivated by a problem in the theory of zeta functions over algebraic number fields, Bourgain et al. [1] investigated the class of functions A+ (d), defined as follows. Given d ≥ 1, a function f : Rd → R is said to be eventually nonnegative if f (x) ≥ 0 for
B
João P. G. Ramos [email protected] Felipe Gonçalves [email protected] Diogo Oliveira e Silva [email protected]
1
Hausdorff Center for Mathematics, 53115 Bonn, Germany
2
School of Mathematics, University of Birmingham, B15 2TT Birmingham, England, UK
3
Instituto Nacional de Matemática Pura e Aplicada, 22460-320 Rio de Janeiro, Brazil
123
F. Gonçalves et al.
all sufficiently large |x|. Normalize the Fourier transform, f (ξ ) =
Rd
f (x)e−2πix,ξ dx,
(1.1)
where ·, · represents the usual inner product in Rd . Let A+ (d) denote the set of functions f : Rd → R which satisfy the following conditions: • • •
f ∈ L 1 (Rd ), f ∈ L 1 (Rd ), and f is real-valued (i.e., f is even); f is eventually nonnegative while f (0) ≤ 0; f is eventually nonnegative while f (0) ≤ 0.
Note that any function f ∈ A+ (d) is uniformly continuous. Consider the quantity r ( f ) := inf{r > 0 : f (x) ≥ 0 if |x| ≥ r }, which corresponds to the radius of the last sign change of f . The product r ( f )r ( f) is unchanged if we replace f with x → f (λx) for some λ > 0, and thus becomes a natural object to consider. One of the initial observations in [1] is that the quantity A+ (d) :=
inf
f ∈A+ (d)\{0}
r ( f )r ( f)
(1.2)
is uniformly bounded from below away from zero. In fact, the following two-sided inequality is established in [1, §3]: 1 A+ (d) A+ (d) 1 ≤ lim inf √ ≤√ . ≤ lim sup √ √ d→∞ 2π e 2π d d d→∞
(1.3)
In particular, the radii r ( f ), r ( f ) of the last sign change of f , f , respectively, cannot both be made arbitrarily small, unless f ∈ A+ (d) is identically zero. Consequently, the aforementioned results can be regarded as manifestations of a sign unc
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