O. Kovrizhkin
We prove a new version of the Uncertainty Principle of the form $\int |f|^2 \lesssim \int_{E^c} |f|^2 + \int_{Σ^c}|\hat f|^2 $ where the sets $E$ and $Σ$ are $ε$-thin in the following sense: $|E \cap D(x, ρ_1(x))| \le ε|D(x, ρ_1(x))|$ and $|Σ\cap D(x, ρ_2(x))| \le ε|D(x, ρ_2(x))|$. This is an intermediate result between Logvinenko-Sereda's and Wolff's versions of the Uncertainty Principle.
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The windowed offset linear canonical transform (WOLCT) can be identified as a generalization of the windowed linear canonical transform (WLCT). In this paper, we generalize several different uncertainty principles for the WOLCT, including Heisenberg uncertainty principle, Hardy's uncertainty principle, Beurling's uncertainty principle, Lieb's uncertainty principle, Donoho-Stark's uncertainty principle, Amrein-Berthier-Benedicks's uncertainty principle, Nazarov's uncertainty principle and Logarithmic uncertainty principle.
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In this paper, we generalise Hardy's uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised.
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Breitenberger's uncertainty principle on the torus $\mathbb{T}$ and its higher-dimensional analogue on $\mathbb{S}^{d-1}$ are well understood. We give describe an entire family of uncertainty principles on compact manifolds $(M,g)$, which includes the classical Heisenberg-Weyl uncertainty principle (for $M=B(0,1) \subset \mathbb{R}^d$ the unit ball with the flat metric) and the Goh-Goodman uncertainty principle (for $M=\mathbb{S}^{d-1}$ with the canonical metric) as special cases. This raises a new geometric problem related to small-curvature low-distortion embeddings: given a function $f:M \r…
We construct $δ$-regular sets with $δ\ge \frac12$ for which the analog of the Bourgain--Dyatlov Fractal Uncertainty Principle fails for the Walsh--Fourier transform.
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In this paper, we generalize the continuous quaternion shearlet transform on $\mathbb{R}^{2}$ to $\mathbb{R}^{2d}$, called the multivariate two sided continuous quaternion shearlet transform. Using the two sided quaternion Fourier transform, we derive several important properties such as (reconstruction formula, reproducing kernel, plancherel's formula, etc.). We present several example of the multivariate two sided continuous quaternion shearlet transform. We apply the multivariate two sided continuous quaternion shearlet transform properties and the two sided quaternion Fourier transform to …
The aim of this paper is establish the Heisenberg-Pauli-Weyl uncertainty principle and Donho-Stark's uncertainty principle for the Weinstein $L^2$-multiplier operators.
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New Sign Uncertainty Principles
该研究提出并证明了可广泛推广已有成果的新型符号不确定性原理，将其应用于多种变换与函数空间，给出数值证据揭示离散与连续情形的关联，并探索其与球体堆积、球面设计的联系。
Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are ``well-distributed'' in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equi-distribution, as have Fourier analysts when working with the ``uncertainty principle''. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples.
The aim of this paper is to prove an uncertainty principle for the representation of a vector in two bases. Our result extends previously known qualitative uncertainty principles into quantitative estimates. We then show how to transfer this result to the discrete version of the Short Time Fourier Transform. An application to trigonometric polynomials is also given.
Fractal uncertainty principle states that no function can be localized in both position and frequency near a fractal set. This article provides a review of recent developments on the fractal uncertainty principle and of their applications to quantum chaos, including lower bounds on mass of eigenfunctions on negatively curved surfaces and spectral gaps on convex co-compact hyperbolic surfaces.
We extend Hardy's uncertainty principle for a square integrable function and its Fourier transform to the multidimensional case using a symplectic diagonalization. We use this extension to show that Hardy's uncertainty principle is equivalent to a statement on the Wigner distribution of the function. We give a geometric interpretation of our results in terms of the notion of symplectic capacity of an ellipsoid. Furthermore, we show that Hardy's uncertainty principle is valid for a general Lagrangian frame of the phase space. Finally, we discuss an extension of Hardy's theorem for the Wigner di…
We prove an explicit formula for the dependence of the exponent in the fractal uncertainty principle of Bourgain-Dyatlov on the dimension and on the regularity constant for the regular set. In particular, this implies an explicit essential spectral gap for convex co-compact hyperbolic surfaces when the Hausdorff dimension of the limit set is close to 1.
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