The uniqueness of the solution to inverse interpolation problems

关于逆q-Sturm-Liouville问题唯一性定理的研究

On Uniqueness Theorems for the Inverse q-Sturm-Liouville problems

本文建立了两个基于谱数据的逆q-Sturm-Liouville问题唯一性定理，分别模拟Levinson-Marchenko和Ashrafyan定理，并引入Gelfand-Levitan方法的q-模拟来证明。

Representer Theorems in Banach Spaces: Minimum Norm Interpolation, Regularized Learning and Semi-Discrete Inverse Problems

Constructing or learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation problem, regularized learning problem or, in general, a semi-discrete inverse problem, in certain functional spaces. The choice of an appropriate space is crucial for solutions of these problems. Motivated by sparse representations of the reconstructed functions such as compressed sensing and sparse learning, much of the recent research interest has been directed to considering these problems…

Fourier uniqueness in $\mathbb{R}^4$

We show an interrelation between the uniqueness aspect of the recent Fourier interpolation formula of Radchenko and Viazovska and the Heisenberg uniqueness study for the Klein-Gordon equation and the lattice-cross of critical density, studied by Hedenmalm and Montes-Rodriguez. This has been known since 2017.

谱Barron空间中的插值与反问题

Interpolation and inverse problems in spectral Barron spaces

研究谱Barron空间与正线性算子的关系，探讨插值与缩放关系，并通过反问题案例验证Tikhonov正则化误差界。

Self-mappings of the quaternionic unit ball: multiplier properties, Schwarz-Pick inequality, and Nevanlinna--Pick interpolation problem

We study several aspects concerning slice regular functions mapping the quaternionic open unit ball into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space. In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.

A Global Uniqueness on Spherically Stratified Dielectric Medium in Time-Harmonic Maxwell Equation with Interior Transmission Eigenvalues

A set of regularly distributed transmission eigenvalues generates a density function. We use such a density function inversely determines the form of the indicator function. Using the entire function theory, we reduce an uniqueness problem with interior transmission eigenvalues induced by time-harmonic Maxwell equation to an uniqueness problem in entire function theory. In such an inverse problem, the definite integral of the square root of refraction index is the main parameter.

Linearizing non-linear inverse problems and an application to inverse backscattering

We propose an abstract approach to prove local uniqueness and conditional Hölder stability to non-linear inverse problems by linearization. The main condition is that, in addition to the injectivity of the linearization $A$, we need a stability estimate for $A$ as well. That condition is satisfied in particular, if $A^*A$ is an elliptic pseudo-differential operator. We apply this scheme to show uniqueness and Hölder stability for the inverse backscattering problem for the acoustic equation near a constant sound speed.

Uniqueness, Lipschitz stability and reconstruction for the inverse optical tomography problem

In this paper, we consider the inverse problem of recovering a diffusion and absorption coefficients in steady-state optical tomography problem from the Neumann-to-Dirichlet map. We first prove a Global uniqueness and Lipschitz stability estimate for the absorption parameter provided that the diffusion is known. Then, we prove a Lipschitz stability result for simultaneous recovery of diffusion and absorption. In both cases the parameters belong to a known finite subspace with a priori known bounds. The proofs relies on a monotonicity result combined with the techniques of localized potentials.…