Hausdorff-Young Inequality Research Articles

Biquaternion ambiguity function and associated uncertainty principles

A mathematical tool used to analyze signals in high-dimensional areas is the biquaternion ambiguity function. While typical ambiguity functions deal with time delay and Doppler frequency, biquaternion ambiguity functions extend the concept to rotations and scaling factors. Biquaternion ambiguity functions offer a more thorough representation of signal ambiguity in these high-dimensional spaces, which can help develop better signal processing algorithms for a range of uses. In this paper, we introduce the notion of biquaternion ambiguity function which is formed from conventional ambiguity function by invoking biquaternion algebra. We first establish the various properties, namely dilation, translation, shift, nonlinearity, boundedness, reconstruction formula, Moyal’s formula, and inversion formula. An interesting relationship between the biquaternion Fourier transform and the biquaternion ambiguity function is also established. Furthermore, we establish the various versions of uncertainty inequalities namely, Pitt’s inequality, sharp Hausdorff–Young inequality, sharp Local uncertainty principle and logarithmic uncertainty principle for biquaternion ambiguity function. Some potential applications are also presented at the end.

Titchmarsh’s-type theorem for two-sided quaternion Fourier transform and sharp Hausdorff–Young inequality for quaternion linear canonical transform

In this work, we first introduce the two-sided quaternion Fourier transform and demonstrate its essential properties. We generalize Titchmarsh’s-type theorem in the framework of the two-sided quaternion Fourier transform. Based on the interaction between the quaternion Fourier transform and quaternion linear canonical transform we explore sharp Hausdorff–Young inequality for the quaternion linear canonical transform. The obtained result can be considered as a generalized version of sharp Hausdorff–Young inequality for the two-dimensional quaternion Fourier transformation in the literature.

The Hausdorff–Young Inequality and Freud weights

The Hausdorff–Young Inequality and Freud weights

Octonion Special Affine Fourier Transform: Pitt’s Inequality and the Uncertainty Principles

The special affine Fourier transform (SAFT) is an extended version of the classical Fourier transform and incorporates various signal processing tools which include the Fourier transforms, the fractional Fourier transform, the linear canonical transform, and other related transforms. This paper aims to introduce a novel octonion special affine Fourier transform (O−SAFT) and establish several classes of uncertainty inequalities for the proposed transform. We begin by studying the norm split and energy conservation properties of the proposed (O−SAFT). Afterwards, we generalize several uncertainty relations for the (O−SAFT) which include Pitt’s inequality, Heisenberg–Weyl inequality, logarithmic uncertainty inequality, Hausdorff–Young inequality, and local uncertainty inequalities. Finally, we provide an illustrative example and some possible applications of the proposed transform.

Fractional Fourier Transform: Main Properties and Inequalities

The fractional Fourier transform is a natural generalization of the Fourier transform. In this work, we recall the definition of the fractional Fourier transform and its relation to the conventional Fourier transform. We exhibit that this relation permits one to obtain easily the main properties of the fractional Fourier transform. We investigate the sharp Hausdorff-Young inequality for the fractional Fourier transform and utilize it to build Matolcsi-Szücs inequality related to this transform. The other versions of the inequalities concerning the fractional Fourier transform is also discussed in detail. The results obtained in this paper are very significant, especially in the field of fractional differential equations.

An Inequality for the Convolutions on Unimodular Locally Compact Groups and the Optimal Constant of Young’s Inequality

Let mu be the Haar measure of a unimodular locally compact group G and m(G) as the infimum of the volumes of all open subgroups of G. The main result of this paper is that ∫Gf∘ϕ1∗ϕ2gdg≤∫Rf∘ϕ1∗∗ϕ2∗xdx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _{G}^{} f \\circ \\left( \\phi _1 * \\phi _2 \\right) \\left( g \\right) dg \\le \\int _{\\mathbb {R}}^{} f \\circ \\left( \\phi _1^* * \\phi _2^* \\right) \\left( x \\right) dx \\end{aligned}$$\\end{document}holds for any measurable functions phi _1, phi _2 :G rightarrow mathbb {R}_{ge 0} with mu ( textrm{supp} ; phi _1 ) + mu ( textrm{supp} ; phi _2 ) le m(G) and any convex function f :mathbb {R}_{ge 0} rightarrow mathbb {R} with f(0) = 0. Here phi ^* is the rearrangement of phi . Let Y_O(P,G) and Y_R(P,G) denote the optimal constants of Young’s and the reverse Young’s inequality, respectively, under the assumption mu ( textrm{supp} ; phi _1 ) + mu ( textrm{supp} ; phi _2 ) le m(G). Then we have Y_O(P,G) le Y_O(P,mathbb {R}) and Y_R(P,G) ge Y_R(P,mathbb {R}) as a corollary. Thus, we obtain that m (G) = infty if and only if H (p,G) le H (p, mathbb {R}) in the case of p' := p/(p-1) in 2 mathbb {Z}, where H(p, G) is the optimal constant of the Hausdorff–Young inequality.

2D Linear Canonical Transforms on Lp and Applications

As Fourier transformations of Lp functions are the mathematical basis of various applications, it is necessary to develop Lp theory for 2D-LCT before any further rigorous mathematical investigation of such transformations. In this paper, we study this Lp theory for 1≤p<∞. By defining an appropriate convolution, we obtain a result about the inverse of 2D-LCT on L1(R2). Together with the Plancherel identity and Hausdorff–Young inequality, we establish Lp(R2) multiplier theory and Littlewood–Paley theorems associated with the 2D-LCT. As applications, we demonstrate the recovery of the L1(R2) signal function by simulation. Moreover, we present a real-life application of such a theory of 2D-LCT by encrypting and decrypting real images.

Short-time free metaplectic transform: Its relation to short-time Fourier transform in $ L^2(\mathbb R^n) $ and uncertainty principles

<abstract><p>The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various applications in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inadequate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique localized transform coined as the short-time free metaplectic transform (ST-FMT). In this paper, we investigate the ST-FMT. First we propose the definition of the ST-FMT and provide the time-frequency analysis of the proposed transform in the FMT domain. Second we establish the relationship between the ST-FMT and short-time Fourier transform (STFT) in $ L^2(\mathbb R^n) $ and investigate the basic properties of the proposed transform including the reconstruction formula, Moyal's formula. The emergence of the ST-FMT definition and its properties broadens the development of time-frequency representation of higher-dimensional signals theory to a certain extent. We extend some different uncertainty principles (UPs) from quantum mechanics including Lieb's inequality, Pitt's inequality, Hausdorff-Young inequality, Heisenberg's UP, Hardy's UP, Beurling's UP, Logarithmic UP and Nazarov's UP. Finally, we give a numerical example and a possible applications of the proposed ST-FMT.</p></abstract>

Wigner distribution and associated uncertainty principles in the framework of octonion linear canonical transform

The most recent generalization of octonion Fourier transform (OFT) is the octonion linear canonical transform (OLCT) that has become popular in present era due to its applications in color image and signal processing. On the other hand the applications of Wigner distribution (WD) in signal and image analysis cannot be excluded. In this paper, we introduce novel integral transform coined as the Wigner distribution in the octonion linear canonical transform domain (WDOL). We first propose the definition of the one dimensional WDOL (1D-WDOL), we extend its relationship with 1D-OLCT and 1D-OFT. Then explore several important properties of 1D-WDOL, such as reconstruction formula, Rayleigh’s theorem. Second, we introduce the definition of three dimensional WDOL (3D-WDOL) and establish its relationships with the WD associated with quaternion LCT (WD-QLCT) and 3D-WD in LCT domain (3D-WDLCT). Then we study properties like reconstruction formula, Rayleigh’s theorem and Riemann–Lebesgue Lemma associated with 3D-WDOL. The crux of this paper lies in developing well known uncertainty principles (UPs) including Heisenberg’s UP, Logarithmic UP and Hausdorff–Young inequality associated with WDOL.

Equality in Hausdorff–Young for hypergroups

It was shown in [Colloq. Math. 131(2), 219--231 (2013)] that one can extend the domain of Fourier transform of a commutative hypergroup K to \(L^p(K)\) for \(1\le p \le 2\), and the Hausdorff–Young inequality holds true for these cases. In this article, we examine the structure of non-zero functions in \(L^p(K)\) for which equality is attained in the Hausdorff–Young inequality, for \(1<p<2\), and further provide a characterization for the basic uncertainty principle for commutative hypergroups with non-trivial center.

$$L^{p}$$–$$L^{q}$$ Boundedness of Fourier Multipliers on Fundamental Domains of Lattices in $$\mathbb {R}^d$$

In this paper we study the \(L^{p}\)–\(L^{q}\) boundedness of Fourier multipliers on the fundamental domain of a lattice in \(\mathbb {R}^{d}\) for \(1< p,q < \infty \) under the classical Hörmander condition. First, we introduce Fourier analysis on lattices and have a look at possible generalisations. We then prove the Hausdorff–Young inequality, Paley’s inequality and the Hausdorff–Young–Paley inequality in the context of lattices. This amounts to a quantitative version of the \(L^{p}\)–\(L^{q}\) boundedness of Fourier multipliers. We will show that this delivers also some \(L^{p}\)-estimates by using some standard Lebesgue space embeddings, which come from the finite measure of the fundamental domain. Moreover, the Paley inequality allows us to prove the Hardy–Littlewood inequality. As an application we treat some Sobolev embedding results, which also indicate the sharpness of our main inequalities.

Asymptotically sharp discrete nonlinear Hausdorff–Young inequalities for the SU(1,1)-valued Fourier products

Abstract We work in a discrete model of the nonlinear Fourier transform (following the terminology of Tao and Thiele), which appears in the study of orthogonal polynomials on the unit circle. The corresponding nonlinear variant of the Hausdorff–Young inequality can be deduced by adapting the ideas of Christ and Kiselev to the present discrete setting. However, the behavior of sharp constants remains largely unresolved. In this short note we give two results on these constants, after restricting our attention to either sufficiently small sequences or sequences that are far from being the extremizers of the linear Hausdorff–Young inequality.

Fusion bialgebras and Fourier analysis: Analytic obstructions for unitary categorification

We introduce fusion bialgebras and their duals and systematically study their Fourier analysis. As an application, we discover new efficient analytic obstructions on the unitary categorification of fusion rings. We prove the Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and their duals. We show that the Schur product property, Young's inequality and the sum-set estimate hold for fusion bialgebras, but not always on their duals. If the fusion ring is the Grothendieck ring of a unitary fusion category, then these inequalities hold on the duals. Therefore, these inequalities are analytic obstructions of categorification. We classify simple integral fusion rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be eliminated by applying the Schur product property on the dual. In general, these inequalities are obstructions to subfactorize fusion bialgebras.

On the monofractality of many stationary continuous Gaussian fields

In this article we focus on a general real-valued continuous stationary Gaussian field X characterized by its spectral density |g|2, where g is any even real-valued deterministic square integrable function. Our starting point consists in drawing a close connection between critical Besov regularity of the inverse Fourier transform of g and αX the random pointwise Hölder exponent function of X, which measures local roughness/smoothness of its sample paths at each point. Then, thanks to Littlewood-Paley methods and Hausdorff-Young inequalities, under weak conditions on g, we show that the random function αX is actually a deterministic constant which does not change from point to point. This result means that the field X is of monofractal nature. Also, it is worth mentioning that such a result can easily be extended to the case where X is no longer stationary but has stationary increments.

On the Multilinear Fractional Transforms

In this paper we first introduce multilinear fractional wavelet transform on Rn×R+n using Schwartz functions, i.e., infinitely differentiable complex-valued functions, rapidly decreasing at infinity. We also give multilinear fractional Fourier transform and prove the Hausdorff–Young inequality and Paley-type inequality. We then study boundedness of the multilinear fractional wavelet transform on Lebesgue spaces and Lorentz spaces.

Uncertainty Principle for Space–Time Algebra-Valued Functions

In this paper, we present a set of important properties of the special relativistic Fourier transformation (SFT) on the complex space–time algebra $${\mathcal {G}}{(3,1)}$$ , such as inversion property, the Plancherel theorem, and the Hausdorff–Young inequality. The main objective of this article is to introduce the concept of the vector derivative in geometric algebra and using it together with the notion of the space–time split to derive the Heisenberg–Pauli–Weyl inequality. Finally, we apply the SFT properties for proving the Donoho–Stark uncertainty principle for $${\mathcal {G}}{(3,1)}$$ multi-vector functions.

Extensions of the vector-valued Hausdorff–Young inequalities

In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff–Young, Hardy–Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff–Young and Hardy–Littlewood inequalities and state that the Fourier transform is bounded from Lp(Rd, | · | βp) into Lq(Rd, | · | -γq) under certain condition on p, q, β and γ. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and related geometric properties. Similar results are derived for Td and Zd by a transference argument. We prove sharpness of our results by providing elementary examples on lp-spaces. Moreover, connections with Rademacher (co)type are discussed as well.

On optimal autocorrelation inequalities on the real line

<p style='text-indent:20px;'>We study autocorrelation inequalities, in the spirit of Barnard and Steinerberger's work [<xref ref-type="bibr" rid="b1">1</xref>]. In particular, we obtain improvements on the sharp constants in some of the inequalities previously considered by these authors, and also prove existence of extremizers to these inequalities in certain specific settings. Our methods consist of relating the inequalities in question to other classical sharp inequalities in Fourier analysis, such as the sharp Hausdorff–Young inequality, and employing functional analysis as well as measure theory tools in connection to a suitable dual version of the problem to identify and impose conditions on extremizers.

Heisenberg’s and Hardy’s Uncertainty Principles for Special Relativistic Space-Time Fourier Transformation

The special relativistic (space-time) Fourier transform (SFT) in Clifford algebra $$Cl_{(3,1)}$$ of space-time, first introduced from a mathematical point of view in Hitzer (Adv Appl Clifford Algebras 17:497–517, 2007), extends the quaternionic Fourier transform to functions, fields and signals in space-time. The purpose of this paper is to advance the study of the SFT and investigate important properties such as continuity, Plancherel identity, Riemann–Lebesgue lemma, and to establish the associated Hausdorff–Young inequality. Moreover, using the observer related space-time split several uncertainty inequalities are established, including Heisenberg’s uncertainty principle and Hardy’s theorem.

On Hausdorff-Young inequalities in generalized Lebesgue spaces

On Hausdorff-Young inequalities in generalized Lebesgue spaces