Spherical Means Research Articles

A Novel Approach to the Fractional Laplacian via Generalized Spherical Means

Although at least ten equivalent definitions of the fractional Laplacian exist in unbounded domains, we introduce an additional equivalent definition based on the generalized spherical mean-value operator—a Fourier multiplier operator involving the normalized Bessel function. Specifically, we demonstrate that this new definition allows us to reduce any n-dimensional fractional Laplacian to a one-dimensional operator, which simplifies computation and enhances efficiency. We propose two methods for computing the generalized spherical means of a given function: one by solving standard wave equations and the other by solving Darboux’s equations.

On the injectivity of the shifted Funk–Radon transform and related harmonic analysis

AbstractNecessary and sufficient conditions are obtained for injectivity of the shifted Funk–Radon transform associated with k-dimensional totally geodesic submanifolds of the unit sphere Sn in ℝn+1. This result generalizes the well known statement for the spherical means on Sn and is formulated in terms of zeros of Jacobi polynomials. The relevant harmonic analysis is developed, including a new concept of induced Stiefel (or Grassmannian) harmonics, the Funk–Hecke type theorems, addition formula, and multipliers. Some perspectives and conjectures are discussed.

Characterization of Holomorphic Functions by Zero Spherical Means

Characterization of Holomorphic Functions by Zero Spherical Means

Injectivity of Spherical Means on H-Type Groups

Injectivity of Spherical Means on H-Type Groups

The Shifted Wave Equation on Non-flat Harmonic Manifolds

We solve the shifted wave equation ∂2∂t2φ(x,t)=(Δx+ρ2)φ(x,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\partial ^2}{\\partial t^2}\\varphi (x,t)=(\\Delta _x+\\rho ^2)\\varphi (x,t) \\end{aligned}$$\\end{document}on a non-compact simply connected harmonic manifold with mean curvature of the horospheres 2ρ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\rho >0$$\\end{document}. We give an explicit representation of the solution as the inverse dual Abel transform of the spherical means of their initial conditions using the local injectivity of the Abel transform and symmetry properties of the spherical mean value operator. Furthermore, we investigate the shifted wave equation using the Fourier transform on harmonic manifolds of rank one. Additionally, we obtain a result analogous to the classical Paley–Wiener theorem and use it to show an asymptotic Huygens principle as well as asymptotic equidistribution of the energy of a solution of the shifted wave equation under assumptions on the Harish–Chandra type c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{c}$$\\end{document}-function.

Обращение оператора свертки, ассоциированного со сферическими средними

Очевидным свойством произвольной ненулевой гладкой антипериодической функции является отсутствие соответствующего периода у ее производной. Другими словами, если $r$ - фиксированное положительное число и на вещественной оси $f(x+r)+f(x-r)=0$ и $f'(x+r)-f'(x-r)=0$, то $f=0$. Этот факт допускает нетривиальные обобщения на многомерные пространства. Одним из общих методов для таких обобщений является следующая теорема Брауна - Шрейбера - Тейлора о спектральном анализе: любое ненулевое подпространство $\mathcal{U}$ в $C(\mathbb{R}^n)$, инвариантное относительно всех движений $\mathbb{R}^n$, содержит радиальную функцию вида $(\lambda|x|)^{1-\frac{n}{2}}J_{\frac{n}{2}-1}(\lambda|x|)$, где $\lambda$ - некоторое комплексное число, $J_\nu$ -- функция Бесселя первого рода порядка $\nu$. В частности, если функция $f\in C^1(\mathbb{R}^n)$ и ее нормальная производная имеют нулевые интегралы по всем сферам фиксированного радиуса $r$ в $\mathbb{R}^n$, то $f=0$. В терминах сверток это означает инъективность оператора $\mathcal{P}f =(f\ast \Delta \chi_r, f\ast \sigma_r)$, $f\in C(\mathbb{R}^n)$, где $\Delta$ - оператор Лапласа, $\chi_{r}$ - индикатор шара $B_r=\{x\in\mathbb{R}^n: |x|<r\}$, $\sigma_{r}$ - поверхностная дельта-функция, сосредоточенная на сфере $S_r= \{x\in\mathbb{R}^n: |x|=r\}$. В данной работе изучается задача об обращении оператора $\mathcal{P}$ на классе распределений. Получена новая формула восстановления распределения $f\in \mathcal{D}'(\mathbb{R}^n)$ по известным сверткам $f\ast \Delta \chi_r$ и $f\ast \sigma_r$. В работе используются методы гармонического анализа, а также теории целых и специальных функций. Ключевым шагом в доказательстве основного результата является разложение дельта-функции Дирака по системе радиальных распределений с носителями в $\overline{B}_r$, биортогональной к некоторой системе сферических функций. Подобный подход можно использовать для обращения других операторов свертки с радиальными распределениями из $\mathcal{E}'(\mathbb{R}^n)$.

Spherical means-based free-water volume fraction from diffusion MRI increases non-linearly with age in the white matter of the healthy human brain

The term free-water volume fraction (FWVF) refers to the signal fraction that could be found as the cerebrospinal fluid of the brain, which has been demonstrated as a sensitive measure that correlates with cognitive performance and various neuropathological processes. It can be quantified by properly fitting the isotropic component of the magnetic resonance (MR) signal in diffusion-sensitized sequences. Using N=287 healthy subjects (178F/109M) aged 25-94, this study examines in detail the evolution of the FWVF obtained with the spherical means technique from multi-shell acquisitions in the human brain white matter across the adult lifespan, which has been previously reported to exhibit a positive trend when estimated from single-shell data using the bi-tensor signal representation. We found evidence of a noticeably non-linear gain after the sixth decade of life, with a region-specific variate and varying change rate of the spherical means-based multi-shell FWVF parameter with age, at the same time, a heteroskedastic pattern across the adult lifespan is suggested. On the other hand, the FW corrected diffusion tensor imaging (DTI) leads to a region-dependent flattened age-related evolution of the mean diffusivity (MD) and fractional anisotropy (FA), along with a considerable reduction in their variability, as compared to the studies conducted over the standard (single-component) DTI. This way, our study provides a new perspective on the trajectory-based assessment of the brain and explains the conceivable reason for the variations observed in FA and MD parameters across the lifespan with previous studies under the standard diffusion tensor imaging.

Gelfand pairs and spherical means on H-type groups

Gelfand pairs and spherical means on H-type groups

On the exactness of the universal backprojection formula for the spherical means Radon transform

The spherical means Radon transform is defined by the integral of a function f in over the sphere of radius r centered at a x, normalized by the area of the sphere. The problem of reconstructing f from the data where x belongs to a hypersurface and has important applications in modern imaging modalities, such as photo- and thermo- acoustic tomography. When Γ coincides with the boundary of a bounded (convex) domain , a function supported within Ω can be uniquely recovered from its spherical means known on Γ. We are interested in explicit inversion formulas for such a reconstruction. If , such formulas are only known for the case when Γ is an ellipsoid (or one of its partial cases). This gives rise to a question: can explicit inversion formulas be found for other closed hypersurfaces Γ? In this article we prove, for the so-called ‘universal backprojection inversion formulas’, that their extension to non-ellipsoidal domains Ω is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.

Renormalization of one-pion exchange in higher partial waves in chiral effective field theory for antinucleon-nucleon system* *Supported by the Doctoral Fund Project of Nanfang College, Guangzhou (2020BQ03)

The renormalization of the iterated one-pion exchange (OPE) has been studied in chiral effective field theory (χEFT) for the antinucleon-nucleon () scattering in some partial waves (Phys. Rev. C 105, 054005 (2022)). In this paper, we go further for the other higher partial waves but with total angular momenta . Contact interactions are represented by a complex spherical well in the coordinate space. Changing the radius of the spherical well means changing the cutoff. We check the cutoff dependence of the phase shifts, inelasticities, and mixing angles for the partial waves and show that contact interactions are needed at leading order in channels where the singular tensor potentials of OPE are attractive. The results are compared with the energy-dependent partial-wave analysis of scattering data. Comparisons between our conclusions and applications of χEFT to the nucleon-nucleon system are also discussed.

Inverse load identification in vibrating nanoplates

In this paper, we consider the uniqueness issue for the inverse problem of load identification in a nanoplate by dynamic measurements. Working in the framework of the strain gradient linear elasticity theory, we first deduce a Kirchhoff‐Love nanoplate model, and we analyze the well‐posedness of the equilibrium problem, clarifying the correct Neumann conditions on curved portions of the boundary. Our uniqueness result states that, given a transverse dynamic load , where and are known time‐dependent functions, if the transverse displacement of the nanoplate is known in an open subset of its domain for any interval of time, then the spatial components can be determined uniquely from the data. The proof is based on the spherical means method. The uniqueness result suggests a reconstruction technique to approximate the loads, as confirmed by a series of numerical simulations performed on a rectangular clamped nanoplate.

General Couplings Conditions of Four-Dimensional Maxwell-Klein-Gordon System

In this paper, we investigate the conditions for the Maxwell Klein Gordon system’s general gauge couplings in (3 + 1)-dimensional Minkowski spacetime. For the multi-interaction of photons and charged spin-0 particle, we begin with the Lagrangian with the general coupling turned on. The equation of motion is then derived in the form of a nonlinear wave equation. We obtained the general solution for the vector potential using the spherical means method. By considering the Coulomb gauge condition, we get the constraint for the couplings, which are critical in proving the global well-posedness of the solution.

Spherical means on the Heisenberg group: Stability of a maximal function estimate

Consider the surface measure μ on a sphere in a nonvertical hyperplane on the Heisenberg group ℍn, n ≥ 2, and the convolution f * μ. Form the associated maximal function Mf = supt>0 ∣f * μt∣ generated by the automorphic dilations. We use decoupling inequalities due to Wolff and Bourgain—Demeter to prove Lp-boundedness of M in an optimal range.

Boundary growth of Sobolev functions of monotone type for double phase functionals

Our aim in this paper is to deal with boundary growth of spherical means of Sobolev functions of monotone type for the double phase functional \(\Phi_{p,q}(x,t) = t^{p} + (b(x) t)^{q}\) in the unit ball B of \(\mathbb{R}^n\), where \(1 < p < q < \infty\) and \(b(\cdot)\) is a non-negative bounded function on B which is Hölder continuous of order \(\theta \in (0,1]\).

Accurate free-water estimation in white matter from fast diffusion MRI acquisitions using the spherical means technique.

To accurately estimate the partial volume fraction of free water in the white matter from diffusion MRI acquisitions not demanding strong sensitizing gradients and/or large collections of different b-values. Data sets considered comprise 32-64 gradients near plus 6 gradients near . The spherical means of each diffusion MRI set with the same b-value are computed. These means are related to the inherent diffusion parameters within the voxel (free- and cellular-water fractions; cellular-water diffusivity), which are solved by constrained nonlinear least squares regression. The proposed method outperforms those based on mixtures of two Gaussians for the kind of data sets considered. W.r.t. the accuracy, the former does not introduce significant biases in the scenarios of interest, while the latter can reach a bias of 5%-7% if fiber crossings are present. W.r.t. the precision, a variance near , compared to 15%, can be attained for usual configurations. It is possible to compute reliable estimates of the free-water fraction inside the white matter by complementing typical DTI acquisitions with few gradients at a lowb-value. It can be done voxel-by-voxel, without imposing spatial regularity constraints.

Half Way There: Theoretical Considerations for Power Laws and Sticks in Diffusion MRI for Tissue Microstructure

In this article, we consider how differing approaches that characterize biological microstructure with diffusion weighted magnetic resonance imaging intersect. Without geometrical boundary assumptions, there are techniques that make use of power law behavior which can be derived from a generalized diffusion equation or intuited heuristically as a time dependent diffusion process. Alternatively, by treating biological microstructure (e.g., myelinated axons) as an amalgam of stick-like geometrical entities, there are approaches that can be derived utilizing convolution-based methods, such as the spherical means technique. Since data acquisition requires that multiple diffusion weighting sensitization conditions or b-values are sampled, this suggests that implicit mutual information may be contained within each technique. The information intersection becomes most apparent when the power law exponent approaches a value of 12, whereby the functional form of the power law converges with the explicit stick-like geometric structure by way of confluent hypergeometric functions. While a value of 12 is useful for the case of solely impermeable fibers, values that diverge from 12 may also reveal deep connections between approaches, and potentially provide insight into the presence of compartmentation, exchange, and permeability within heterogeneous biological microstructures. All together, these disparate approaches provide a unique opportunity to more completely characterize the biological origins of observed changes to the diffusion attenuated signal.

Integral representation formulae for the solution of a wave equation with time‐dependent damping and mass in the scale‐invariant case

This paper is devoted to deriving integral representation formulae for the solution of an inhomogeneous linear wave equation with time‐dependent damping and mass terms that are scale invariant with respect to the so‐called hyperbolic scaling. Yagdjian's integral transform approach is employed for this purpose. The main step in our argument consists in determining the kernel functions for the different integral terms, which are related to the source term and to initial data. We will start with the one‐dimensional case (in space). We point out that we may not apply in a straightforward way Duhamel's principle to deal with the source term since the coefficients of lower order terms make our model not invariant by time translation. On the contrary, we shall begin with the representation formula for the inhomogeneous equation with vanishing data by using a revised Duhamel's principle. Then, we will derive the representation of the solution in the homogeneous case with nontrivial data. After deriving the formula in the one‐dimensional case, the classical approach by spherical means is used in order to deal with the odd‐dimensional case. Finally, using the method of descent, the representation formula in the even‐dimensional case is proved.

Maxwell Klein-Gordon System with General Gauge Coupling on 4-Dimensional Flat Spacetime

In this paper, we study the Maxwell Klein Gordon system (MKG) with the addition of general gauge couplings on four-dimensional Minkowski space. This system analyzes the interaction of the electromagnetic field (photon) coupled with the complex scalar field (spin-0 particles). This research considers the multi-field interactions by adding general coupling. Our method begins from the Lagrangian of Maxwell Klein Gordon equation with potential turn on. We derive the energy of the system, then write the equation of motion in the form of a non-linear partial differential equation. By using the spherical means method, we express the solutions in an integral form of its curvature. Then, by using temporal gauge condition and conservation of energy, we prove the inequality for some general coupling. The result shows that the first and the second derivative of it bounded by the energy and IL norm. This result will become the main key to proving the global existence of Maxwell Klein-Gordon’s theory.

On the Convergence of Bochner-Riesz’s Spherical Means of Fourier Double Integrals

In this paper we consider the convergence by measure of Fourier integral spherical means of Riesz at a critical exponent δ = 1/2 after changing the values of the integrable function on the given set of a small measure.

Limiting probability measures

The coordinates along any fixed direction(s), of points on the sphere $S^{n-1}(\sqrt{n})$, roughly follow a standard Gaussian distribution as $n$ approaches infinity. We revisit this classical result from a nonstandard analysis perspective, providing a new proof by working with hyperfinite dimensional spheres. We also set up a nonstandard theory for the asymptotic behavior of integrals over varying domains in general. We obtain a new proof of the Riemann--Lebesgue lemma as a by-product of this theory. We finally show that for any function $f \co \mathbb{R}^k \to \mathbb{R}$ with finite Gaussian moment of an order larger than one, its expectation is given by a Loeb integral integral over a hyperfinite dimensional sphere. Some useful inequalities between high-dimensional spherical means of $f$ and its Gaussian mean are obtained in order to complete the above proof.