Meromorphic Approximation Research Articles

Meromorphic tangential approximation on the boundary of closed sets in Riemann surfaces

If a closed subset of a Riemann surface is a set of uniform meromorphic approximation, then its boundary is shown to be a set of tangential meromorphic approximation.

Comparison of EEG source localization using meromorphic approximation to fuzzy C-mean

Electroencephalography (EEG) is a neuroimaging technique for localizing active sources within the brain, from knowledge of electromagneticmeasurements outside the head. Recognition of point sources from boundary measurements is an ill-posed inverse problem. InEEG, measurements areonly accessible at electrode positions, the number of sources is not known a prior. This paper proposes a comparison between two approaches for EEGsource localization. First method based on Meromorphic approximation techniques in the complex plane and second one belongs to EEG’s methodwhich is processed using Fuzzy C-Means (FCM). Comparison results on simulated data are used to verify the superior of the Meromorphicapproximation with regarding to Fuzzy c-means, due to it provides the way for solving inverse problem of EEG source localization in 3D from boundarymeasurement based on Harmon function in the innermost layer .

On a characterization of Arakelian sets

Let K be a compact set in the complex plane, such that its complement in the Riemann sphere, (ℂ ∪ {∞}) / K, is connected. Also, let U ⊂ ℂ be an open set which contains K. Then there exists a simply connected open set V ⊂ ℂ such that K ⊂ V ⊂ U. We show that if K is replaced by a closed set F ⊂ ℂ, then the preceding result is equivalent to the fact that F is an Arakelian set in ℂ. This holds in more general case when ℂ is replaced by any simply connected open set Ω ⊂ ℂ. In the case of an arbitrary open set Ω ⊂ ℂ, the above extends to the one point compactification of Ω. If we do not require (ℂ ∪ {∞}) /K to be connected, we can demand that each component of (ℂ ∪ {∞}) / V intersects a prescribed set A containing one point in each component of (ℂ ∪ {∞}) / K. Using the previous result, we prove that again if we replace K by a closed set F, the latter is equivalent to the fact that F is a set of uniform meromorphic approximation with poles lying entirely in A.

An Index Formula in Connection with Meromorphic Approximation

Let \(\Phi \) be a continuous \(n\times n\) matrix-valued function on the unit circle \(\mathbb T \) such that the \((k-1)\)st singular value of the Hankel operator with symbol \(\Phi \) is greater than the \(k\)th singular value. In this case, it is well-known that \(\Phi \) has a unique superoptimal meromorphic approximant \(Q\) in \(H^{\infty }_{(k)}\); that is, \(Q\) has at most \(k\) poles in the unit disc \(\mathbb D \) (in the sense that the McMillan degree of \(Q\) in \(\mathbb D \) is at most \(k\)) and \(Q\) minimizes the essential suprema of singular values \(s_{j}\left((\Phi -Q)(\zeta )\right)\!, j\ge 0\), with respect to the lexicographic ordering. For each \(j\ge 0\), the essential supremum of \(s_{j}\left((\Phi -Q)(\zeta )\right)\) is called the \(j\)th superoptimal singular value of degree \(k\) of \(\Phi \). We prove that if \(\Phi \) has \(n\) non-zero superoptimal singular values of degree \(k\), then the Toeplitz operator \(T_{\Phi -Q}\) with symbol \(\Phi -Q\) is Fredholm and has index $$ \mathrm{ind}T_{\Phi -Q}=\dim \ker T_{\Phi -Q}=2k+\dim \mathcal E , $$ where \(\mathcal E =\{ \xi \in \ker H_{Q}: \Vert H_{\Phi }\xi \Vert _{2}=\Vert (\Phi -Q)\xi \Vert _{2}\}\) and \(H_{\Phi }\) denotes the Hankel operator with symbol \(\Phi \). This result can in fact be extended from continuous matrix-valued functions to the wider class of \(k\)-admissible matrix-valued functions, i.e. essentially bounded \(n\times n\) matrix-valued functions \(\Phi \) on \(\mathbb T \) for which the essential norm of the Hankel operator \(H_{\Phi }\) is strictly less than the smallest non-zero superoptimal singular value of degree \(k\) of \(\Phi \).

Meromorphic approximants to complex Cauchy transforms with polar singularities

We study AAK-type meromorphic approximants to functions of the form where is a rational function and is a complex measure with compact regular support included in , whose argument has bounded variation on the support. The approximation is understood in the -norm of the unit circle, . We dwell on the fact that the denominators of such approximants satisfy certain non-Hermitian orthogonal relations with varying weights. They resemble the orthogonality relations that arise in the study of multipoint Padé approximants. However, the varying part of the weight implicitly depends on the orthogonal polynomials themselves, which constitutes the main novelty and the main difficulty of the undertaken analysis. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of relative to the unit disc, that the approximants themselves converge in capacity to , and that the poles of attract at least as many poles of the approximants as their multiplicity and not much more.Bibliography: 35 titles.

On estimates for the ratio of errors in best rational approximation of analytic functions

Let E E be an arbitrary compact subset of the extended complex plane C ¯ \overline {\mathbb C} with nonempty interior. For a function f f continuous on E E and analytic in the interior of E E denote by ρ n ( f ; E ) \rho _n(f; E) the least uniform deviation of f f on E E from the class of all rational functions of order at most n n . In this paper we show that if f f is not a rational function and if K K is an arbitrary compact subset of the interior of E , E, then ∏ k = 0 n ( ρ k ( f ; K ) / ρ k ( f ; E ) ) , \prod _{k=0}^n (\rho _k(f; K) /\rho _k(f; E) ), the ratio of the errors in best rational approximation, converges to zero geometrically as n → ∞ n \to \infty and the rate of convergence is determined by the capacity of the condenser ( ∂ E , K ) (\partial E, K) . In addition, we obtain results regarding meromorphic approximation and sharp estimates of the Hadamard type determinants.

Universal meromorphic approximation on Vitushkin sets

The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λn} ⊂ ℂ, there exists a function ϕ, meromorphic on ℂ, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {nk} of ℕ such that \( \left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\} \) converges to f(z) uniformly on K.

Source localization in ellipsoids by the best meromorphic approximation in planar sections

We consider the inverse problem of localizing dipolar sources in an ellipsoid from boundary data, which we approach and constructively solve with techniques from harmonic and complex analysis. We use ellipsoidal harmonics to isolate the singular part of the solution, which we consider on a family of two-dimensional sections of the domain. We then use approximation theory to locate its singularities, and provide an algorithm which allows us to recover the sources from these singularities. We provide numerical illustrations related to the localization of pointwise dipolar sources in the human brain from electroencephalography (EEG).

2-D inverse problems for the Laplacian: A meromorphic approximation approach

We consider a classical inverse problem: detecting an insulating crack inside a homogeneous 2-D conductor, using overdetermined boundary data. Our method involves meromorphically approximating the complexified solution to the underlying Dirichlet–Neumann problem on the outer boundary of the conductor, and relating the singularities of the approximant (i.e., its poles) to the singular set of the approximated function (i.e., the crack). This approach was introduced in [L. Baratchart, J. Leblond, F. Mandréa, E.B. Saff, How can the meromorphic approximation help to solve some 2-D inverse problems for the Laplacian?, Inverse Problems 15 (1999) 79–90] when the crack is a real segment embedded in the unit disk. Here we show, more generally, that the best L 2 and L ∞ meromorphic approximants to the complexified solution on the outer boundary of the conductor have poles that accumulate on the hyperbolic geodesic arc linking the endpoints of the crack if the latter is analytic and “not too far” from a geodesic. The extension of the method to the case where the crack is piecewise analytic is briefly discussed. We provide numerical examples to illustrate the technique; as the computational cost is low, the results may be used to initialize a heavier local search. The bottom line of the approach is to regard the problem of “optimally” discretizing a potential using finitely many point masses as a regularization scheme for the underlying inverse potential problem. This point of view may be valuable in higher dimension as well.

Meromorphic approximation theorem in a Stein space

We prove the meromorphic version of the Weil–Oka approximation theorem in a reduced Stein space X and give some characterizations of meromorphically $\mathcal{O}(X)$ -convex open sets of X. As an application we prove that for every meromorphically $\mathcal{O}(X)$ -convex open set D of a reduced Stein space X with no isolated points there exists a family $\mathcal{F}$ of holomorphic functions on X such that the normality domain $D(\mathcal{F})$ of $\mathcal{F}$ coincides with D.

The langleSPPrangle Green function and SU(3) breaking inKell3decays

Using the 1/N_C expansion scheme and truncating the hadronic spectrum to the lowest-lying resonances, we match a meromorphic approximation to the < S P P > Green function onto QCD by imposing the correct large-momentum falloff, both off-shell and on the relevant hadron mass shells. In this way we determine a number of chiral low-energy constants of O(p^6), in particular the ones governing SU(3) breaking in the K_{l3} vector form factor at zero momentum transfer. The main result of our matching procedure is that the known loop contributions largely dominate the corrections of O(p^6) to f_{+}(0). We discuss the implications of our final value f_{+}^{K^0 \pi^-}(0)=0.984 \pm 0.012 for the extraction of V_{us} from K_{l3} decays.

Uniform approximation by meromorphic functions on Riemann surfaces

A closed subsetE of a Riemann surfaceS is called a set of uniform meromorphic approximation if every functionf continuous onE and holomorphic onE0 can be approximated uniformly onE by meromorphic functions onS. We show that ifE is a set of uniform meromorphic approximation, then so is\(E \cap \bar D\) for every compact parametric diskD. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A. G. Vitushkin.

Recovery of pointwise sources or small inclusions in 2D domains and rational approximation

We consider the inverse problems of locating pointwise or small size conductivity defaults in a plane domain, from overdetermined boundary measurements of solutions to the Laplace equation. We express these issues in terms of best rational or meromorphic approximation problems on the boundary, with poles constrained to belong to the domain. This approach furnishes efficient and original resolution schemes.

An Lp Analog to AAK Theory for p⩾2

We develop an Lp analog to AAK theory on the unit circle that interpolates continuously between the case p=∞, which classically solves for best uniform meromorphic approximation, and the case p=2, which is equivalent to H2-best rational approximation. We apply the results to the uniqueness problem in rational approximation and to the asymptotic behaviour of poles of best meromorphic approximants to functions with two branch points. As pointed out by a referee, part of the theory extends to every p∈[1, ∞] when the definition of the Hankel operator is suitably generalized; this we discuss in connection with the recent manuscript by V. A. Prokhorov, submitted for publication.

Asymptotics for Minimal Blaschke Products and Best L1 Meromorphic Approximants of Markov Functions

Let μ be a positive Borel measure with support suppμ = E ⊂ (−1, 1) and let $$\Delta_n = {{\rm inf}\atop {{B \in {\cal B}_n}}} \int_E \mid(x)\mid^2 d\mu(x)$$ where \({\cal B}_n\) is the collection of all Blaschke products of degree n. Denote by \(B_n \in \mathcal{B}_n\) a Blaschke product that attains the value Δn. We investigate the asymptotic behavior, as n → ∞, of the minimal Blaschke products Bn in the case when the measure μ with support E = [a, b] satisfies the Szeg’o condition: $$\int_a^b {\frac{{\log \left( {{{d\mu } \mathord{\left/ {\vphantom {{d\mu } {dx}}} \right. \kern-\nulldelimiterspace} {dx}}} \right)}} {{\sqrt {(x - a)(b - x)} }}dx > - \infty .} $$ At the same time, we shall obtain results related to the convergence of best L1 approximants on the unit circle to the Markov function $$f(z) = \frac{1} {{2\pi i}}\int_E {\frac{{d\mu (x)}} {{z - x}}}$$ by meromorphic functions of the form P/Q, where P belongs to the Hardy space H1 of the unit disk and Q is a polynomial of degree at most n. We also include in an appendix a detailed treatment of a factorization theorem for Hardy spaces of the slit disk, which may be of independent interest.

Best Meromorphic Approximation of Markov Functions on the Unit Circle

Let E \subset(-1,1) be a compact set, let μ be a positive Borel measure with support \supp μ =E , and let H p (G),1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z \colon |z|=1} . Let Δ n,p be the error in best approximation of the Markov function \frac{1}{2π i} ∈t_E \frac{d μ(x)}{z-x} in the space L p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H p (G),Q is a polynomial of degree at most n , Q\not \equiv 0 . We investigate the rate of decrease of Δ n,p ,1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1 - ∈fty.

On meromorphic approximation

Let G be a bounded N-connected domain, the boundary Γ of which consists of closed analytic Jordan curves. We assume that 0eG. For any nonnegative integers n and m, denote by ℳ n,m the class of all meromorphic functions on G that can be represented in the form h=p/qz m , where p belongs to the Smirnov class E ∞(G), q is a polynomial of degree at most n, q≢0. Theorems giving necessary and sufficient conditions for a function belonging to the class ℳ n,m to be an element of best approximation to a continuous function f on Γ in the space L ∞(Γ) by functions in the class ℳ n,m are proved. Some questions concerning orthogonal polynomials and the theory of Hankel operators are also considered.

How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian?

We exhibit new links between approximation theory in the complex domain and a family of inverse problems for the 2D Laplacian related to non-destructive testing.

On holomorphic factorization and meromorphic selectors

We show that, generically, a smooth Lorentzian 2 × 2 matrix valued function A, defined on the unit circle and possessing a so called exterior meromorphic section, can be extented into the unit disk in a way that admits a non-degenerate holomorphic factorization P ∗P , where the adjoint is taken with respect to the constant Lorentz matrix. This has implications for meromorphic approximation and Levi-flat surfaces with generalized disk fibers.

An Example in Tangential Meromorphic Approximation

It is known that all sets of meromorphic uniform approximation inCwhich satisfy an additional condition involving the Gleason parts of the algebraR(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, although it is a set of tangential approximation, does not satisfy this extra condition on parts and, thus, showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.