Context. Passive microwave systems are usually modeled by scattering matrices reflecting the phys... more Context. Passive microwave systems are usually modeled by scattering matrices reflecting the physical properties of the device. In a finite dimensional context, these matrices are rational and their McMillan degree corresponds to the number of state variable. Stability of the system implies analyticity in the right halfplane, so that the poles of the matrix lies in the left half-plane. For a conservative device, the evaluation of the matrix at any imaginary point is unitary. Such a stable rational matrix is called LQMD lossless LQLJ referring to the fact that energy is neither created nor dissipated in the underlying system. In addition, electrical networks possess a reciprocity property which translates into symmetry of the scattering matrix, a far reaching property for rational matrices. In practice, devices are measured thanks to high precision measurement instruments (Frequency Response Analysers). Measurements can be viewed as evaluations of the rational transfers at some frequ...
We derive a criterion for uniqueness of a critical point in rational approximation of degree 1. A... more We derive a criterion for uniqueness of a critical point in rational approximation of degree 1. Although narrowly restricted in scope because it deals with degree 1 only, this criterion is interesting because it addresses a large class of functions. The method elaborates on the topological approach in [15] and [12].
For ∂Ω the boundary of a bounded and connected strongly Lipschitz domain in R d with d ≥ 3, we pr... more For ∂Ω the boundary of a bounded and connected strongly Lipschitz domain in R d with d ≥ 3, we prove that any field f ∈ L 2 (∂Ω; R d) decomposes, in an unique way, as the sum of three silent vector fields-fields whose magnetic potential vanishes in one or both components of R d \ ∂Ω. Moreover, this decomposition is orthogonal if and only if ∂Ω is a sphere. We also show that any f in L 2 (∂Ω; R d) is uniquely the sum of two silent fields and a Hardy function, in which case the sum is orthogonal regardless of ∂Ω; we express the corresponding orthogonal projections in terms of layer potentials. When ∂Ω is a sphere, both decompositions coincide and match what has been called the Hardy-Hodge decomposition in the literature.
Let E ⊂ (−1, 1) be a compact set, let µ be a positive Borel measure with support supp µ = E, and ... more Let E ⊂ (−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 ≤ ∞, be the Hardy space of analytic functions on the open unit disk G with circumference = {z: |z| = 1}. Let n, p be the error in best approximation of the Markov function 1 2πi E dµ(x) z − x in the space L p () 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 ≡ 0. We investigate the rate of decrease of n, p , 1 ≤ p ≤ ∞, and its connection with n-widths. The convergence of the best meromorphic approximants and the limiting Date
Heldermann Verlag where Bn is the collection of all Blaschke products of degree n. Denote by Bn E... more Heldermann Verlag where Bn is the collection of all Blaschke products of degree n. Denote by Bn E Bn a Blaschke product that attains the value ~n. We investigate the asymptotic behavior, as n-t 00, of the minimal Blaschke products Bn in the case when the measure p. with support E = [a, b) satisfies the Szegc:> condition: (b log(dp.{dx) dx >-00. la ";(x-a)(b-x) At the same time, we shall obtain results related to the convergence of best £1 approximants on the unit circle to the Markov function f(z)=~ {2 1r1 1 E Z-x by meromorphic functions of the form P {Q, where P belongs to the Hardy space HI 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.
Surface Acoustic Wave Filters, Unitary Extensions and Schur Analysis
We study the concept of a mixed matrix in connection with lin- ear fractional transformations of ... more We study the concept of a mixed matrix in connection with lin- ear fractional transformations of lossless and passive matrix-valued rational functions, and show that they can be parametrized by sequences of elemen- tary chain matrices. These notions are exemplified on a model of a Surface Acoustic Wave filter for which a state-space realization is carried out in de- tail. The issue of optimal design of such filters -as yet unsolved- naturally raises a Darlington synthesis problem with both symmetry and interpolation constraints whith control on the McMillan degree. As a partial answer, we give necessary and sucient conditions for symmetric Darlington synthesis to be possible without increasing the McMillan degree for a symmetric ra- tional contractive matrix which is strictly contractive in at least one point of the unit circle .
Journal of Computational and Applied Mathematics, 1999
Let f be a Markov function with deÿning measure supported on (−1; 1), i.e., f(z) = (t − z) −1 d (... more Let f be a Markov function with deÿning measure supported on (−1; 1), i.e., f(z) = (t − z) −1 d (t); ¿0, and supp() ⊆ (−1; 1). The uniqueness of rational best approximants to the function f in the norm of the real Hardy space H 2 (V); V := C\ D = {z ∈ C | |z|¿1}, is investigated. It is shown that there exist Markov functions f with rational best approximants that are not unique for inÿnitely many numerator and denominator degrees n − 1 and n, respectively. In the counterexamples, which have been constructed, the deÿning measures are rather rough. But there also exist Markov functions f with smooth deÿning measures such that the rational best approximants to f are not unique for odd denominator degrees up to a given one.
Let E be a closed subset of the open unit disk G ¼ fz : jzjo1g; and let m be a positive Borel mea... more Let E be a closed subset of the open unit disk G ¼ fz : jzjo1g; and let m be a positive Borel measure with support supp m ¼ E: Denote by A p the restriction on E of the closed unit ball of the Hardy space H p ðGÞ; 1pppN: In this paper we investigate orthogonality properties of the extremal functions associated with the Kolmogorov, Gelfand, and linear n-widths of A p in L q ðm; EÞ; 1pqoN; qpp:
We study diagonal multipoint Padé approximants to functions of the form F (z) = Z dλ(t) z − t + R... more We study diagonal multipoint Padé approximants to functions of the form F (z) = Z dλ(t) z − t + R(z), where R is a rational function and λ is a complex measure with compact regular support included in R, whose argument has bounded variation on the support. Assuming that interpolation sets are such that their normalized counting measures converge sufficiently fast in the weak-star sense to some conjugate-symmetric distribution σ, we show that the counting measures of poles of the approximants converge to b σ, the balayage of σ onto the support of λ, in the weak * sense, that the approximants themselves converge in capacity to F outside the support of λ, and that the poles of R attract at least as many poles of the approximants as their multiplicity and not much more.
This paper deals with the identification of linear constant dynamical systems when formalized as ... more This paper deals with the identification of linear constant dynamical systems when formalized as a rational approximation problem. The criterion is the l 2 norm of the transfer function, which is of interest in a stochastic context. The problem can be expressed as nonlinear optimization in a Hilbert space, but standard algorithms are usually not well adapted. Here, we present a generic recursive procedure to find a local optimum of the criterion in the case of scalar systems. Our methods are borrowed from differential theory mixed with a bit of classical complex analysis. To our knowledge, the algorithm described in this paper is the first that ensures convergence to a local minimum. Finally, we discuss a number of unsettled issues.
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Papers by L. Baratchart