arXiv subjects

Hiroyuki Chihara

Publications and source records attributed to Hiroyuki Chihara.

At least 19 recordsLinked to original sources

arXiv2025

The matrix weighted real-analytic double fibration transforms

We show that the real-analytic matrix-weighted double fibration transform determines the analytic wavefront set of a vector-valued function. We apply this result to show that the matrix weighted ray transform is injective on a two-dimensional, non-trapping, real-analytic Riemannian manifold with strictly convex boundary. Additionally, we show that a real-analytic Higgs field can be uniquely determined from the nonabelian ray transform on real-analytic Riemannian manifolds of any dimension with a strictly convex boundary point.

math.AP↗

arXiv2024

The symbol of the normal operator for the d-plane transform on the Euclidean space

We directly compute the symbol of the normal operator for the d-plane transform on the Euclidean space. We show that this symbol is the product of the symbol of the power of the Laplacian of order -d/2 and a constant given by an invariant integral over excess-dimensional spaces. This leads to an alternative derivation of the filtered backprojection formula for the d-plane transform.

math.AP↗

arXiv2024

Microlocal analysis of double fibration transforms with conjugate points

We study the structure of normal operators of double fibration transforms with conjugate points. Examples of double fibration transforms include Radon transforms, $d$-plane transforms on the Euclidean space, geodesic X-ray transforms, light-ray transforms, and ray transforms defined by null bicharacteristics associated with real principal type operators. We show that, under certain stable conditions on the distribution of conjugate points, the normal operator splits into an elliptic pseudodifferential operator and several Fourier integral operators, depending on the degree of the conjugate points. These problems were first studied for geodesic X-ray transforms by Stefanov and Uhlmann (Analysis \& PDE, {\bf 5} (2012), pp.219--260). After that Holman and Uhlmann (Journal of Differential Geometry, {\bf 108} (2018), pp.459--494) proved refined results according to the degree of regular conjugate points.

math.AP↗

arXiv2024

Geodesic X-ray transform and streaking artifacts on simple surfaces or on spaces of constant curvature

The X-ray transform on the plane or on the three-dimensional Euclidean space can be considered as the measurements of CT scanners for normal human tissue. If the human body contains metal regions such as dental implants, stents in blood vessels, metal bones, etc., the beam hardening effect for the energy level of the X-ray causes streaking artifacts in its CT image. More precisely, if there are two strictly convex metal regions contained in the cross-section of normal human tissue, then streaking artifacts occur along the common tangent lines of the two regions. In this paper we study the geodesic X-ray transform and streaking artifacts on nontrapping simple compact Riemannian manifolds with strictly convex boundaries. We show that the streaking artifacts result from the propagation of conormal singularities on the boundary of metal regions along the common tangent geodesics under the strong and seemingly strange assumption that the manifolds are two dimensional or spaces of constant curvature. This condition ensures that every Jacobi field takes the form of the product of a scalar function and parallel transport along the geodesic. Our results clarify the geometric meaning of the theory, which was imperceptible in the known results on the Euclidean space.

math.AP↗

arXiv2022

Bargmann transform on the space of hyperplanes

We introduce a Bargmann transform on the space of hyperplanes by applying the Plancherel formula of the Radon transform to the definition of the Bargmann transform on the Euclidean space. Some basic facts on microlocal analysis are also discussed.

math.FA↗

arXiv2021

Microlocal analysis of d-plane transform on the Euclidean space

We study the basic properties of d-plane transform on the Euclidean space as a Fourier integral operator, and its application to the microlocal analysis of streaking artifacts in its filtered back-projection. The d-plane transform is defined by integrals of functions on the n-dimensional Euclidean space over all the d-dimensional planes, where 0<d<n. This maps functions on the Euclidean space to those on the affine Grassmannian G(d,n). This is said to be X-ray transform if d=1 and Radon transform if d=n-1. When n=2 the X-ray transform is thought to be measurements of CT scanners. In this paper we obtain concrete expression of the canonical relation of the d-plane transform and quantitative properties of the filtered back-projection of the product of the images of the d-plane transform. The latter one is related to the metal streaking artifacts of CT images, and some generalization of recent results of Park-Choi-Seo (2017) and Palacios-Uhlmann-Wang (2018) for the X-ray transform on the plane.

math.FA↗

arXiv2020

Inversion of higher dimensional Radon transforms of seismic-type

We study integral transforms mapping a function on the Euclidean space to the family of its integration on some hypersurfaces, that is, a function of hypersurfaces. The hypersurfaces are given by the graphs of functions with fixed axes of the independent variables, and are imposed some symmetry with respect to the axes. These transforms are higher dimensional version of generalization of the parabolic Radon transform and the hyperbolic Radon transform arising from seismology. We prove the inversion formulas for these transforms under some vanishing and symmetry conditions of functions.

math.CA↗

arXiv2020

Inversion of seismic-type Radon transforms on the plane

We study integral transforms mapping a function on the Euclidean plane to the family of its integration on plane curves, that is, a function of plane curves. The plane curves we consider in the present paper are given by the graphs of functions with a fixed axis of the independent variable, and are imposed some symmetry with respect to the axes. These transforms contain the parabolic Radon transform and the hyperbolic Radon transform arising from seismology. We prove the inversion formulas for these transforms under some vanishing and symmetry conditions of functions.

math.CA↗

arXiv2019

Bargmann-type transforms and modified harmonic oscillators

We study some complete orthonormal systems on the real-line. These systems are determined by Bargmann-type transforms, which are Fourier integral operators with complex-valued quadratic phase functions. Each system consists of eigenfunctions for a second-order elliptic differential operator like the Hamiltonian of the harmonic oscillator. We also study the commutative case of a certain class of systems of second-order differential operators called the non-commutative harmonic oscillators. By using the diagonalization technique, we compute the eigenvalues and eigenfunctions for the commutative case of the non-commutative harmonic oscillators. Finally, we study a family of functions associated with an ellipse in the phase plane. We show that the family is a complete orthogonal system on the real-line.

math.FA↗

arXiv2018

Holomorphic Hermite functions in Segal-Bargmann spaces

We study systems of holomorphic Hermite functions in the Segal-Bargmann spaces, which are Hilbert spaces of entire functions on the complex Euclidean space, and are determined by the Bargmann-type integral transform on the real Euclidean space. We prove that for any positive parameter which is strictly smaller than the minimum eigenvalue of the positive Hermitian matrix associated with the transform, one can find a generator of holomorphic Hermite functions whose anihilation and creation operators satisfy canonical commutation relations. In other words, we find the necessary and sufficient conditions so that some kinds of entire functions can be such generators. Moreover, we also study the complete orthogonality, the eigenvalue problems and the Rodrigues formulas.

math.FA↗

arXiv2018

Holomorphic Hermite functions and ellipses

In 1990 van Eijnghoven and Meyers introduced systems of holomorphic Hermite functions and reproducing kernel Hilbert spaces associated with the systems on the complex plane. Moreover they studied the relationship between the family of all their Hilbert spaces and a class of Gelfand-Shilov functions. After that, their systems of holomorphic Hermite functions have been applied to studying quantization on the complex plane, combinatorics, and etc. On the other hand, the author recently introduced systems of holomorphic Hermite functions associated with ellipses on the complex plane. The present paper shows that their systems of holomorphic Hermite functions are determined by some cases of ellipses, and that their reproducing kernel Hilbert spaces are some cases of the Segal-Bargmann spaces determined by the Bargmann-type transforms introduced by Sjoestrand.

math.FA↗

arXiv2017

Hermite expansions of some tempered distributions

We compute Hermite expansions of some tempered distributions by using the Bargmann transform. In other words, we calculate the Taylor expansions of the corresponding entire functions. Our method of computations seems to be superior to the direct computations in the shifts of singularities and the higher dimensional cases.

math.CA↗

arXiv2014

Fourth-order dispersive systems on the one-dimensional torus

We present the necessary and sufficient conditions of the well-posedness of the initial value problem for certain fourth-order linear dispersive systems on the one-dimensional torus. This system is related with a dispersive flow for closed curves into compact Riemann surfaces. For this reason, we study not only the general case but also the corresponding special case in detail. We apply our results on the linear systems to the fourth-order dispersive flows. We see that if the sectional curvature of the target Riemann surface is constant, then the equation of the dispersive flow satisfies our conditions of the well-posedness.

math.AP↗

arXiv2013

A fourth-order dispersive flow into Kähler manifolds

We discuss a short-time existence theorem of solutions to the initial value problem for a fourth-order dispersive flow for curves parametrized by the real line into a compact Kähler manifold. Our equations geometrically generalize a physical model describing the motion of a vortex filament or the continuum limit of the Heisenberg spin chain system. Our results are proved by using so-called the energy method. We introduce a bounded gauge transform on the pullback bundle, and make use of local smoothing effect of the dispersive flow a little.

math.AP↗

arXiv2009

Bounded Berezin-Toeplitz operators on the Segal-Bargmann space

We discuss the boundedness of Berezin-Toeplitz operators on a generalized Segal-Bargmann space (Fock space) over the complex $n$-space. This space is characterized by the image of a global Bargmann-type transform introduced by Sjöstrand. We also obtain the deformation estimates of the composition of Berezin-Toeplitz operators whose symbols and their derivatives up to order three are in the Wiener algebra of Sjöstrand. Our method of proofs is based on the pseudodifferential calculus and the heat flow determined by the phase function of the Bargmann transform.

math.FA↗

arXiv2008

Schroedinger flow into almost Hermitian manifolds

We present a short-time existence theorem of solutions to the initial value problem for Schroedinger maps of a closed Riemannian manifold to a compact almost Hermitian manifold. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty, we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle, and essentially eliminate the loss of one derivative from the partial differential equation of the Schroedinger map.

math.AP↗

arXiv2008

A third order dispersive flow for closed curves into almost Hermitian manifolds

We discuss a short-time existence theorem of solutions to the initial value problem for a third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations geometrically generalize a physical model describing the motion of vortex filament. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty, we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle, and eliminate the loss of one derivative from the partial differential equation of the dispersive flow.

math.AP↗

arXiv2008

Gain of analyticity for semilinear Schroedinger equations

We discuss gain of analyticity phenomenon of solutions to the initial value problem for semilinear Schroedinger equations with gauge invariant nonlinearity. We prove that if the initial data decays exponentially, then the solution becomes real-analytic in the space variable and a Gevrey function of order 2 in the time variable except in the initial plane. Our proof is based on the energy estimates developed in our previous work and on fine summation formulae concerned with a matrix norm.

math.AP↗