Plancherel Theorem Research Articles

An infinite–rank Lie algebra associated to SL(2,R) and SL(2,R)/U(1)

We construct a generalised notion of Kac-Moody algebras using smooth maps from the non-compact manifolds M= SL(2,R) and M= SL(2,R)/U(1) to a finite-dimensional simple Lie group G. This construction is achieved through two equivalent ways: by means of the Plancherel Theorem and by identifying a Hilbert basis within L2(M). We analyze the existence of central extensions and identify those in duality with Hermitean operators on M. By inspecting the Clebsch–Gordan coefficients of sl(2,R), we derive the Lie brackets characterising the corresponding generalised Kac-Moody algebras. The root structure of these algebras is identified, and it is shown that an infinite number of simultaneously commuting operators can be defined. Furthermore, we briefly touch upon applications of these algebras within the realm of supergravity, particularly in scenarios where the scalar fields coordinatize the non-compact manifold SL(2,R)/U(1).

The Kontorovich - Lebedev Transformation on Sobolev Type Spaces

The Kontorovich-Lebedev transformation $$(KLf)(x)=\int_0^\inftyK_{i\tau}(x)f(\tau)d\tau, \;\, x \in {\mathbf R}_+$$ is consideredas an operator, which maps the weighted space $L_p(\mathbf R_+;$$\omega(\tau)d\tau), \;\, 2 \le p \le \infty$ into the Sobolevtype space $S_p^{N, \alpha}({\mathbf R}_+)$ with the finite norm$$||u||_{S_p^{N,\alpha}({\mathbf R}_+)}= \biggl( \sum_{k= 0}^N\int_0^\infty |A_x^k u|^p x^{\alpha_k p -1} dx\biggr)^{1/p} <\infty,$$where $\alpha= (\alpha_0, \alpha_1, \dots, \alpha_N), \alpha_k \in{\mathbf R}, k=0, \dots, N$, and $ A_x$ is the differentialoperator of the form$$A_x u= x^2u(x) - x\frac{d}{dx}\biggl[x\frac{du}{dx }\biggr], $$and $A_x^k$ means $k$-th iterate of $A_x, \ A_x^0u= u$. Elementary properties for the space $S_p^{N, \alpha} ({\mathbf R}_+)$ are derived. Boundedness and inversion properties for the Kontorovich-Lebedev transform are studied. In the Hilbert case ($p=2$) the isomorphism between these spaces is established for the special type of weights and Plancherel's type theorem is proved. 2000 Mathematics Subject Classification. 44A15, 46E35, 26D10

Discrete Octonion Linear Canonical Transform: Definition and Properties

In this paper, the discrete octonion linear canonical transform (DOCLCT) is defined. According to the definition of the DOCLCT, some properties associated with the DOCLCT are explored, such as linearity, scaling, boundedness, Plancherel theorem, inversion transform and shift transform. Then, the relationship between the DOCLCT and the three-dimensional (3-D) discrete linear canonical transform (DLCT) is obtained. Moreover, based on a new convolution operator, we derive the convolution theorem of the DOCLCT. Finally, the correlation theorem of the DOCLCT is established.

Octonion Beurling theorem: Uncertainty principles for Hermite functions on ℝ3

We generalize Beurling's theorem to the octonion Fourier transform for octonion‐valued functions. Despite challenges such as the failure of the octonion Fourier transform to change differentiation into multiplication and the absence of the octonionic Plancherel's theorem, we establish the octonionic Beurling theorem for Hermite functions on by introducing an integral condition involving the octonion Fourier transform. This extension of uncertainty principles by Hardy, Gelfand–Shilov, and Cowling–Price to the octonionic setting demonstrates that the function and its Fourier transform cannot have arbitrary Gaussian decay simultaneously.

Localization operators and inversion formulas for the Dunkl–Weinstein–Stockwell transform

Abstract We define and study the Stockwell transform S g \mathscr{S}_{g} associated to the Dunkl–Weinstein operator Δ k , β \Delta_{k,\beta} and prove a Plancherel theorem and an inversion formula. Next, we define a reconstruction function f Δ f_{\Delta} and prove Calderón’s reproducing inversion formula for the Dunkl–Weinstein–Stockwell transform S g \mathscr{S}_{g} . Moreover, we define the localization operators L g ⁢ ( σ ) \mathcal{L}_{g}(\sigma) associated to this transform. We study the boundedness and compactness of these operators and establish a trace formula. Finally, we introduce and study the extremal function F η , k ∗ := ( η ⁢ I + S g ∗ ⁢ S g ) − 1 ⁢ S g ∗ ⁢ ( k ) F^{\ast}_{\eta,\smash{k}}:=(\eta I+\mathscr{S}^{\ast}_{g}\mathscr{S}_{g})^{-1}\mathscr{S}^{\ast}_{g}(k) , and we deduce best approximate inversion formulas for the Dunkl–Weinstein–Stockwell transform S g \mathscr{S}_{g} on the Sobolev space H k , β s ⁢ ( R + d + 1 ) \mathscr{H}^{s}_{k,\beta}(\mathbb{R}_{+}^{d+1}) .

An uncertainty principle on the Lorentz spaces

Via subtly combining the Plancherel theorem with various inequalities including Chebyshev’s inequality, generalized Hölder’s and Hausdorff–Young’s inequalities, we establish a new nonlinear uncertainty principle on the Lorentz spaces Lp,q(Rn) for whole ranges of the indexes p,q∈([1,∞)×(0,1])∪((1,∞)×(1,∞)).

The Poisson geometry of Plancherel formulas for triangular groups

In this paper we establish the existence of canonical coordinates for generic co-adjoint orbits on triangular groups. These orbits correspond to a set of full Plancherel measure on the associated dual groups. This generalizes a well-known coordinatization of co-adjoint orbits of a minimal (non-generic) type originally discovered by Flaschka. The latter had strong connections to the classical Toda lattice and its associated Poisson geometry. Our results develop connections with the Full Kostant–Toda lattice and its Poisson geometry. This leads to novel insights relating the details of Plancherel theorems for Borel Lie groups to the invariant theory for Borels and their subgroups. We also discuss some implications for the quantum integrability of the Full Kostant Toda lattice.

Hyperbolic linear canonical transforms of quaternion signals and uncertainty

This paper is concerned with Linear Canonical Transforms (LCTs) associated with two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure, which we call Quaternion Hyperbolic Linear Canonical Transforms (QHLCTs). These transforms are defined by replacing the Euclidean plane wave with a corresponding hyperbolic relativistic plane wave in one dimension multiplied by quadratic modulations in both the hyperbolic spatial and frequency domains, giving the hyperbolic counterpart of the Euclidean LCTs. We prove the fundamental properties of the partial QHLCTs and the right-sided QHLCT by employing hyperbolic geometry tools and establish main results such as the Riemann–Lebesgue Lemma, the Plancherel and Parseval Theorems, and inversion formulas. The analysis is carried out in terms of novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHLCTs. The results are applied to establish two quaternionic versions of the Heisenberg uncertainty principle for the right-sided QHLCT. These uncertainty principles prescribe a lower bound on the product of the effective widths of quaternion-valued signals in the hyperbolic spatial and frequency domains. It is shown that only hyperbolic Gaussian quaternion functions minimize the uncertainty relations.

Cauchy problems for the time-fractional degenerate diffusion equations

This paper is devoted to the Cauchy problems for the one-dimensional linear time-fractional diffusion equations with $\partial^{\alpha}_{t}$ the Caputo fractional derivative of order $\alpha\in(0,1)$ in the variable t and time-degenerate diffusive coefficients $t^{\beta}$ with $\beta >1-\alpha$. The solutions of  Cauchy problems for the one-dimensional time-fractional degenerate diffusion equations with the time-fractional derivative $\partial^{\alpha}_{t}$ of order $\alpha\in(0,1)$  in the variable $t$, are shown. In the "Problem statement and main results" section of the paper, the solution of the time-fractional degenerate diffusion equation in a variable coefficient with two different initial conditions are considered. In this work, a solution is found by using the Kilbas-Saigo function $E_{\alpha,m,l}(z)$ and applying the Fourier transform $F$ and inverse Fourier transform $\mathcal{F}^{-1}$. Convergence of solution of problem 1 and problem 2 are proven using Plancherel theorem. The existence and uniqueness of the solution of the problem are confirmed.&nbsp

Uncertainty Principles for the Two-Sided Quaternion Windowed Quadratic-Phase Fourier Transform

A recent addition to the class of integral transforms is the quaternion quadratic-phase Fourier transform (Q-QPFT), which generalizes various signal and image processing tools. However, this transform is insufficient for addressing the quadratic-phase spectrum of non-stationary signals in the quaternion domain. To address this problem, we, in this paper, study the (two sided) quaternion windowed quadratic-phase Fourier transform (QWQPFT) and investigate the uncertainty principles associated with the QWQPFT. We first propose the definition of QWQPFT and establish its relation with quaternion Fourier transform (QFT); then, we investigate several properties of QWQPFT which includes inversion and the Plancherel theorem. Moreover, we study different kinds of uncertainty principles for QWQPFT such as Hardy’s uncertainty principle, Beurling’s uncertainty principle, Donoho–Stark’s uncertainty principle, the logarithmic uncertainty principle, the local uncertainty principle, and Pitt’s inequality.

Superhigh charge density and direct-current output in triboelectric nanogenerators via peak shifting modified charge pumping

Due to the great potentials in self-powered sensors and micro-nano power sources, triboelectric nanogenerator (TENG) has attracted numerous attentions. However, enhancing charge density and realizing direct-current (DC) are long-term significant issues for TENGs. In this work, we investigated a DC TENG with outstanding electricity generation characteristics based on quasi- synchronous double-stage self-charge pumping. The TENG works in a contact-separation mode with a four-layered parallel structure. The TENG possesses a high open-circuit voltage of 2300 V, a high short-circuit current of 3 mA, a high power of 150 mW and a superhigh transferred charge of 20 μC at 4 Hz, which are significantly enhanced from those in single-stage pumping and non pumped counterpart TENGs. The property-improving mechanism lies in that through the unidirectional self-charge pump with a slight time difference among TENGs, negative halves are moved to the adjacent positive halves, leads to the widening, higher and DC outputs at the final terminal, which employs the Plancherel theorem for peak shifting in framework of energy conservation law. The TENG can power light emitting diodes, electronic devices and an electrophoretic deposition of pigment coatings. This work provides a method for developing high-performance DC TENGs with superhigh transferred charge for various applications.

A Plancherel Theorem On a Noncommutative Hypergroup

Let G be a locally compact hypergroup and let K be a compact sub-hypergroup of G. (G, K) is a Gelfand pair if Mc(G//K), the algebra of measures with compact support on the double coset G//K, is commutative for the convolution. In this paper, assuming that (G, K) is a Gelfand pair, we define and study a Fourier transform on G and then establish a Plancherel theorem for the pair (G, K).

Positivity and Hankel transforms

In this work we prove that some integrals of special functions are positive by applying the Plancherel theorem for Hankel transforms and positivity of the modified Bessel functions. We also prove that, except an extra elementary factor, Hankel transforms map subsets of completely monotonic functions into complete monotonic functions.

Nonparametric estimation of the expected discounted penalty function in the compound Poisson model

We propose a nonparametric estimator of the expected discounted penalty function in the compound Poisson risk model. We use a projection estimator on the Laguerre basis and we compute the coefficients using Plancherel theorem. We provide an upper bound on the MISE of our estimator, and we show it achieves parametric rates of convergence on Sobolev–Laguerre spaces without needing a bias-variance compromise. Moreover, we compare our estimator with the Laguerre deconvolution method. We compute an upper bound of the MISE of the Laguerre deconvolution estimator and we compare it on Sobolev–Laguerre spaces with our estimator. Finally, we compare these estimators on simulated data.

The Fourier Transform and its application in solving the equation

Fourier transform is a linear transform that applicants in the field of engineering and physics. In this research study, we summarize the fundamental properties of the Fourier transform and its application in solving the wave equation. First, we review the definition of Fourier transform and its inverse form. Then, we show Schwartz space is the invariant subspace of the Fourier transform. We also prove the Plancherel theorem. At last, we apply the Fourier transform to solve the wave equation.

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.

Electrooculogram signal identification for elderly disabled using Elman network

This study proposes an algorithm for classifying the different eye movements gathered from 20 subjects through Electrooculogram (EOG) by means of neural networks. Reference Power and Plancherel theorem were used to excavate outstanding attributes from raw signals to recognize the eye movement tasks performed by individuals. Elman Recurrent Neural Network (ERNN) was projected for categorization of filtered EOG signals. Classification performance of Reference Power using the ERNN was observed to be better with a range of 90.28 to 91.95% in comparison with Plancherel features. Recognition performance was identified using single trial analysis. The single trial analysis shows that subject S20 and S14 has the maximum recognition and identification rate compared to the other subjects involved in this study for reference power and plancherel theorem, and also this study proves that even though the classification accuracy was the maximum the average recognition rate was minimum for some of the subjects at the same time minimum classification accuracy subject shows the maximum recognition and identification rate for some subjects. Result depicted that average recognition and identification rate was maximum for plancherel features compared to reference features using ERNN and also single trial analysis evident that depends upon the subject performance and involvement the identification and recognition rate was vary for all the subjects.

Ψec: A local spectral exterior calculus

We introduce $\Psi \mathrm{ec}$, a discretization of Cartan's exterior calculus of differential forms using wavelets. Our construction consists of differential $r$-form wavelets with flexible directional localization that provide tight frames for the spaces $\Omega^r(\mathbb{R}^n)$ of forms in $\mathbb{R}^2$ and $\mathbb{R}^3$. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the $k$-dimensional integral of an $r$-form is an $(r-k)$-form. They also verify Stokes' theorem for differential forms, with the most efficient finite dimensional approximation attained using directionally localized, curvelet- or ridgelet-like forms. The construction of $\Psi \mathrm{ec}$ builds on the geometric simplicity of the exterior calculus in the Fourier domain. We establish this structure by extending existing results on the Fourier transform of differential forms to a frequency description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators.

Harmonic analysis associated to the canonical Fourier Bessel transform

ABSTRACT The aim of this paper is to develop a new harmonic analysis related to a Bessel type operator on the real line: We define the canonical Fourier Bessel transform and study some of its important properties. We prove a Riemann–Lebesgue lemma, inversion formula and operational formulas for this transformation. We derive Plancherel theorem and Babenko inequality for In the present paper, several uncertainty inequalities and theorems for the canonical Fourier Bessel transform are given, including the Heisenberg inequality, Hardy theorem, Nash-type inequality, Carlson-type inequality, global uncertainty principle, local uncertainty principle, logarithmic uncertainty principle in terms of entropy and Miyachi uncertainty principle.

Harmonic manifolds of hypergeometric type and spherical Fourier transform

The spherical Fourier transform on a harmonic Hadamard manifold (Xn,g) of positive volume entropy is studied. If (Xn,g) is of hypergeometric type, namely spherical functions of X are represented by the Gauss hypergeometric functions, the inversion formula, the convolution rule together with the Plancherel theorem are shown by the representation of the spherical functions in terms of the Gauss hypergeometric functions. A geometric characterization of hypergeometric type is derived in terms of volume density of geodesic spheres. Geometric properties of (Xn,g) are also discussed.