arXiv subjects
Publications and source records attributed to Laurent Baratchart.
We derive lower bounds in best rational approximation of given degree to finite Blaschke products, in the Hardy space $H^2$ of the unit disk. We first consider approximation to $z^N$, and then move on to more general Blaschke products whose zeros are bounded away from the circle. The latter case depends on Fourier coefficients estimates for Blaschke products which are of independent interest.
This paper deals with the stability of linear periodic difference delay systems, where the value at time $t$ of a solution is a linear combination with periodic coefficients of its values at finitely many delayed instants $t-τ_1,\ldots,t-τ_N$. We establish a necessary and sufficient condition for exponential stability of such systems when the coefficients have Hölder-continuous derivative, that generalizes the one obtained for difference delay systems with constant coefficients by Henry and Hale in the 1970s. This condition may be construed as analyticity, in a half plane, of the (operator valued) harmonic transfer function of an associated linear control system.
This note states and proves an integral representation formula of the ``variation-of-constant'' type for continuous solutions of linear non-autonomous difference delay systems, in terms of a Lebesgue-Stieltjes integral involving a fundamental solution and the initial data of the system. This gives a precise and correct version of several formulations appearing in the literature, and extends them to the time-varying case. This is of importance for further stability studies of various kinds of delay systems.
We give a sufficient condition for exponential stability of a network of lossless telegrapher's equations, coupled by linear time-varying boundary conditions. The sufficient conditions is in terms of dissipativity of the couplings, which is natural for instance in the context of microwave circuits. Exponential stability is with respect to any $L^p$-norm, $1\leq p\leq\infty$. This also yields a sufficient condition for exponential stability to a special class of linear time-varying difference delay systems which is quite explicit and tractable. One ingredient of the proof is that $L^p$ exponential stability for such difference delay systems is independent of $p$, thereby reproving in a simpler way some results from [Y. Chitour, G. Mazanti, and M. Sigalotti, $\it {Netw. Heterog. Media}$, 11 (2016), pp. 563--601].
We study inverse problems for the Poisson equation with source term the divergence of an $\mathbf{R}^3$-valued measure, that is, the potential $Φ$ satisfies $$ ΔΦ= \text{div} \boldsymbolμ, $$ and $\boldsymbolμ$ is to be reconstructed knowing (a component of) the field grad $Φ$ on a set disjoint from the support of $\boldsymbolμ$. Such problems arise in several electro-magnetic contexts in the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We develop methods for recovering $\boldsymbolμ$ based on total variation regularization. We provide sufficient conditions for the unique recovery of $\boldsymbolμ$, asymptotically when the regularization parameter and the noise tend to zero in a combined fashion, when it is uni-directional or when the magnetization has a support which is sparse in the sense that it is purely 1-unrectifiable. Numerical examples are provided to illustrate the main theoretical results.
Performing a stability analysis during the design of any electronic circuit is critical to guarantee its correct operation. A closed-loop stability analysis can be performed by analysing the impedance presented by the circuit at a well-chosen node without internal access to the simulator. If any of the poles of this impedance lie in the complex right half-plane, the circuit is unstable. The classic way to detect unstable poles is to fit a rational model on the impedance. In this paper, a projection-based method is proposed which splits the impedance into a stable and an unstable part by projecting on an orthogonal basis of stable and unstable functions. When the unstable part lies significantly above the interpolation error of the method, the circuit is considered unstable. Working with a projection provides one, at small cost, with a first appraisal of the unstable part of the system. Both small-signal and large-signal stability analysis can be performed with this projection-based method. In the small-signal case, a low-order rational approximation can be fitted on the unstable part to find the location of the unstable poles.
In Geomagnetism it is of interest to separate the Earth's core magnetic field from the crustal magnetic field. However, measurements by satellites can only sense the sum of the two contributions. In practice, the measured magnetic field is expanded in spherical harmonics and separation into crust and core contribution is achieved empirically, by a sharp cutoff in the spectral domain. In this paper, we derive a mathematical setup in which the two contributions are modeled by harmonic potentials $Φ_0$ and $Φ_1$ generated on two different spheres $\mathbb{S}_{R_0}$ (crust) and $\mathbb{S}_{R_1}$ (core) with radii $R_1 R_0$, we show that it becomes possible if the magnetization $\mathbf{m}$ generating $Φ_0$ is localized in a strict subregion of $\mathbb{S}_{R_0}$. Beyond unique recoverability, we show in this case how to numerically reconstruct characteristic features of $Φ_0$ (e.g., spherical harmonic Fourier coefficients). An alternative way of phrasing the results is that knowledge of $\mathbf{m}$ on a nonempty open subset of $\mathbb{S}_{R_0}$ allows one to perform separation.
We prove that a $IR n+1$-valued vector field on IR n is the sum of the traces of two harmonic gradients, one in each component of $IR n+1 \ IR n$ , and of a $IR n$-valued divergence free vector field. We apply this to the description of vanishing potentials in divergence form. The results are stated in terms of Clifford Hardy spaces, the structure of which is important for our study.
In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain $Ω\subset\RR^n$, with $L^\infty$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}(Ω)$, $r\textgreater{}n$. We show that uniqueness of the Robin coefficient on a subpart of the boundary given Cauchy data on the complementary part, does hold in dimension $n=2$ but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context.
We consider an overdetermined problem for Laplace equation on a disk with partial boundary data where additional pointwise data inside the disk have to be taken into account. After reformulation, this ill-posed problem reduces to a bounded extremal problem of best norm-constrained approximation of partial L2 boundary data by traces of holomorphic functions which satisfy given pointwise interpolation conditions. The problem of best norm-constrained approximation of a given L2 function on a subset of the circle by the trace of a H2 function has been considered in [Baratchart \& Leblond, 1998]. In the present work, we extend such a formulation to the case where the additional interpolation conditions are imposed. We also obtain some new results that can be applied to the original problem: we carry out stability analysis and propose a novel method of evaluation of the approximation and blow-up rates of the solution in terms of a Lagrange parameter leading to a highly-efficient computational algorithm for solving the problem.
We derive some lower bounds in rational approximation of given degree to functions in the Hardy space $H^2$ of the disk. We apply these to asymptotic errors rates in approximation to Blaschke products and to Cauchy integrals on geodesic arcs. We also explain how to compute such bounds, either using Adamjan-Arov-Krein theory or linearized errors, and we present a couple of numerical experiments on several types of functions. We dwell on the Adamjan-Arov-Krein theory and a maximin principle developed in the article "An L^p analog of AAK theory for p \textgreater{}= 2", by L. Baratchart and F. Seyfert, in the Journal of Functional Analysis, 191 (1), pp. 52-122, 2012.
We study Hardy classes on the disk associated to the equation $\bar\d w=α\bar w$ for $α\in L^r$ with $2\leq r<\infty$. The paper seems to be the first to deal with the case $r=2$. We prove an analog of the M.~Riesz theorem and a topological converse to the Bers similarity principle. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth domains of the Dirichlet problem with weighted $L^p$ boundary data for 2-D isotropic conductivity equations whose coefficients have logarithm in $W^{1,2}$. In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for $W^{1,2}_0$-functions.
In this paper we investigate the question of uniform convergence of Padé approximants to elliptic functions that can be represented as Cauchy integrals of Dini-continuous non-vanishing densities given on 3-point Chebotarëv continua.
Given non-collinear points a_1, a_2, a_3, there is a unique compact, say Δ, that has minimal logarithmic capacity among all continua joining a_1, a_2, and a_3. For h be a complex-valued non-vanishing Dini-continuous function on Δ, we consider f_h(z) := (1/πi)\int_Δh(t)/(t-z) dt/w^+(t), where w(z) := \sqrt{\prod_{k=0}^3(z-a_k)} and w^+ the one-sided value according to some orientation of Δ. In this work we present strong asymptotics of diagonal Padé approximants to f_h and describe the behavior of the spurious pole and the regions of locally uniform convergence from a generic perspective.
We study Hardy spaces $H^p_ν$ of the conjugate Beltrami equation $\bar{\partial} f=ν\bar{\partial f}$ over Dini-smooth finitely connected domains, for real contractive $ν\in W^{1,r}$ with $r>2$, in the range $r/(r-1)<p<\infty$. We develop a theory of conjugate functions and apply it to solve Dirichlet and Neumann problems for the conductivity equation $\nabla.(σ\nabla u)=0$ where $σ=(1-ν)/(1+ν)$. In particular situations, we also consider some density properties of traces of solutions together with boundary approximation issues.
Let f be holomorphically continuable over the complex plane except for finitely many branch points contained in the unit disk. We prove that best rational approximants to f of degree n, in the L^2-sense on the unit circle, have poles that asymptotically distribute according to the equilibrium measure on the compact set outside of which f is single-valued and which has minimal Green capacity in the disk among all such sets. This provides us with n-th root asymptotics of the approximation error. By conformal mapping, we deduce further estimates in approximation by rational or meromorphic functions to f in the L^2-sense on more general Jordan curves encompassing the branch points. The key to these approximation-theoretic results is a characterization of extremal domains of holomorphy for f in the sense of a weighted logarithmic potential, which is the technical core of the paper.
We consider the symmetric Darlington synthesis of a p x p rational symmetric Schur function S with the constraint that the extension is of size 2p x 2p. Under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the minimal McMillan degree of the extension. In particular, we show that it is generically given by the number of zeros of odd multiplicity of I-SS*. A constructive characterization of all such extensions is provided in terms of a symmetric realization of S and of the outer spectral factor of I-SS*. The authors's motivation for the problem stems from Surface Acoustic Wave filters where physical constraints on the electro-acoustic scattering matrix naturally raise this mathematical issue.
We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents $1<\infty$. We analyse their boundary behaviour and certain density properties of their traces. We derive on the way an analog of the Fatou theorem for the Dirichlet and Neumann problems associated with the equation ${div}(σ\nabla u)=0$ with $L^p$-boundary data.