Some qualitative uncertainty principles for the Fractional Dunkl Transform

In this paper, we prove various mathematical aspects of the qualitative and quantitative uncertainty principle, including Hardy’s, Cowling–Price and its variants, Beurling and its variants, Gelfand–Shilov, Donoho–Stark’s uncertainty principle, and variants of Heisenberg’s inequalities for the generalized Fourier transform associated to a Dunkl-type operator on the real line.

Uncertainty principle for the quaternion Fourier transform

With due apologies to physicists for any shortcomings, we begin with a little bit of physics ! In quantum mechanics a “wave function” f(x) associated with a particle is a complex valued function on, say, ℝ3 such that |f(x)|2 is the probability density function of the position of the particle i.e. \(\int\limits_{{\mathbb{R}^3}} {{{\left| {f\left( x \right)} \right|}^2}dx = 1}\) and the probability of finding the particle in a subset E of ℝ3 is given by \(\int\limits_E {{{\left| {f\left( x \right)} \right|}^2}dx}\). The expected value of position is given by a ∈ ℝ3, where \({a_j} = {\int\limits_{{\mathbb{R}^3}} {{x_j}\left| {f\left( x \right)} \right|} ^2}dx,a = \left( {{a_1},{a_2},{a_3}} \right),x = \left( {{x_1},{x_2},{x_3}} \right)\). Ignoring physical constants (!) like the famous Planck’s constant, the probability density function of momentum is (more or less) given by \({\left| {\hat f\left( y \right)} \right|^2}\), where \(\hat f\) denotes the 3-dimensional Fourier transform of f. If b is the expected value of momentum, the celebrated Heisenberg uncertainty principle asserts that: $$\left( {\int\limits_{{\mathbb{R}^3}} {{{\left\| {x - a} \right\|}^2}{{\left| {f\left( x \right)} \right|}^2}dx} } \right)\left( {\int\limits_{{\mathbb{R}^3}} {{{\left\| {y - b} \right\|}^2}{{\left| {\hat f\left( y \right)} \right|}^2}dy} } \right) \geqslant C$$ (1.1) where C is a positive universal constant and ‖ · ‖ denotes the length of a vector in ℝ3. What the above says is that if the probability of finding the particle near its mean position is high, then the probability of its momentum being near its mean momentum cannot be high, and we can make an identical statement with position and momentum interchanged. For a very readable account of the Heisenberg uncertainty principle, see [20].

We survey various mathematical aspects of the uncertainty principle, including Heisenberg’s inequality and its variants, local uncertainty inequalities, logarithmic uncertainty inequalities, results relating to Wigner distributions, qualitative uncertainty principles, theorems on approximate concentration, and decompositions of phase space.

It has long been known that if f ∈ L 2 ( R n ) f \in {L^2}({{\mathbf {R}}^n}) and the supports of and its Fourier transform f ^ \hat f are bounded then f = 0 f = 0 almost everywhere. More recently it has been shown that the same conclusion can be reached under the weaker condition that the supports of f f and f ^ \hat f have finite measure. These results may be thought of as qualitative uncertainty principles since they limit the "concentration" of the Fourier transform pair ( f , f ^ ) (f,\hat f) . Little is known, however, of analogous results for functions on locally compact groups. A qualitative uncertainty principle is proved here for unimodular groups of type I.

Qualitative uncertainty principles for groups with finite dimensional irreducible representations

A qualitative uncertainty principle for functions generating a Gabor frame on LCA groups

In this paper, we prove various mathematical aspects of the qualitative uncertainty principle, including Hardy’s, Cowling–Price’s theorem, Morgan’s theorem, Beurling, Gelfand–Shilov, Miyachi theorems.

For locally compact abelian groups it is known that if the product of the measures of the support of an $L^1$-function $f$ and its Fourier transform is less than $1$, then $f = 0$ almost everywhere. This is a weak version of the classical qualitative uncertainty principle. In this paper we focus on compact groups. We obtain conditions on the structure of a compact group under which there exists a lower bound for all products of the measures of the support of an integrable function and its Fourier transform, and conditions under which this bound equals $1$. For several types of compact groups, we determine the exact set of values which the product can attain.

We define lacunary Fourier series on a compact connected semisimple Lie group G. If f ∈ L 1 (G) has lacunary Fourier series and f vanishes on a non empty open subset of G, then we prove that f vanishes identically. This result can be viewed as a qualitative uncertainty principle.

The ‐Hankel wavelet transform ( ‐HWT) is a novel addition to the class of wavelet transforms, which has gained a respectable status in the realm of time‐frequency signal analysis within a short span of time. Knowing the fact that the study of uncertainty principles is both theoretically interesting and practically useful, we formulate several qualitative uncertainty principles for the ‐Hankel wavelet transform. First, we formulate the Heisenberg's uncertainty principle governing the simultaneous localization of a signal and the corresponding ‐HWT via three approaches: ‐type, ‐type, and ‐entropy based. Second, we derive some weighted uncertainty inequalities such as the Pitt's and Beckner's uncertainty inequalities for the ‐HWT. We culminate our study by formulating several concentration‐based uncertainty principles, including the Amrein–Berthier–Benedicks's and local inequalities for the ‐Hankel wavelet transform.

ABSTRACTThe nonisotropic angular Stockwell transform (NAST) employs an angular, scalable and localized window function, to effectively capture directional characteristics and nonisotropic features presented in the signal. In this paper, our main contributions are to derive three distinct qualitative uncertainty principles (UPs) for the NAST: the Benedicks–Amrein–Berther UP, the logarithmic Sobolev‐type UP, and the Donoho–Stark UP. These uncertainty principles provide rather valuable insights into the behavior and limitations of the NAST regarding localization and concentration properties.

In this paper, we introduce the definitions of the Fourier transform and spherical Fourier transform on quaternionic Heisenberg group, then we present some of their properties, in particular, we will determine the spherical functions and the heat kernel of quaternionic Heisenberg group. Finally, we prove a qualitative uncertainty principle ‘Donoho–Stark's uncertainty principle’.

In this paper, we introduce the canonical Jacobi–Dunkl transform and study some applications of this transform. We study the generalized translation operator associated with the square of the Jacobi–Dunkl operator and present some basic properties. Then, we define the generalized convolution product associated with the canonical Jacobi–Dunkl transform and establish Young's inequality. As the applications, we investigate some qualitative uncertainty principles and provide the solutions of the generalized heat and Schrödinger's equations associated with this transform. In particular, we establish Hardy's, Beurling's and Miyachi's uncertainty principles for the canonical Jacobi–Dunkl transform.

The present paper reframes what we call the Hinton Hypothesis, which states that everything of human nature can be duplicated in artificial intelligence. We consider competition to be a significant aspect of human nature. We assume as our working hypothesis that a perfectly competitive AI-agent society is duplicated from human society, like a free financial market. Human agents hesitate between being non-cooperative and cooperative, a hesitation governed by the invisible hand. AI agents observe each other to gain an advantage through more accurate information. By the economic rationality, every agent tends to be the final observer. Thus, the order between observations satisfies the noncommutative law. This is called the qualitative artificial uncertainty principle, which serves as a model of artificial invisible hand.