J-inner matrix functions, interpolation and inverse problems for canonical systems, IV: Direct and inverse bitangential input scattering problems

Bitangential input scattering problems are formulated and analyzed for canonical integral systems. Special attention is paid to the case when the input scattering matrix is ap×q matrix valued function of Wiener class. Formulas for the solution of the inverse input scattering problem are obtained by reproducing kernel Hilbert space methods. A number of illustrative examples are presented. Additional examples for the case when the input scattering matrix is of Wiener class/rational will be presented in a future publication.     

Trace formulas for a class of Toeplitz-like operators II

Let P_T denote the orthogonal projection of L²(R¹, dΔ) onto the space of entire functions of exponential type ? T which are square summable on the line with respect to the measure $\text{dΔ(γ) = ¦ h(γ)¦}{}^2\text{dγ}$, and let G denote the operator of multiplication by a suitably restricted complex valued function g. It is shown that if $\text{(γ}{}^2\text{+ 1)}{}^{\mathrm{?1}}\text{log}\text{¦ h(γ)¦}$ is summable, if $\text{¦ h ¦}{}^{\mathrm{?2}}$ is locally summable, and if $\text{h}\text{h}{}^{\mathrm{\#}}$ belongs to the span in L^∞ of e^?iyTH^∞:T ? 0, in which h is chosen to be an outer function and h^#(γ) agrees with the complex conjugate of h(γ) on the line, then

$\text{lim trace}\text{T↑∞}\text{{(P}{}_{\mathrm{T}}\text{GP}{}_{\mathrm{T}}\text{)}{}^{\mathrm{n}}\text{? P}{}_{\mathrm{T}}\text{G}{}^{\mathrm{n}}\text{P}{}_{\mathrm{T}}\text{}}$

exists and is independent of h for every positive integer n. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special case h = 1 using a different formalism. It also extends earlier results of the author which were established under more stringent conditions on h. The conclusions are based in part upon a preliminary study of a more general class of projections.     

An extremal problem in the theory of hardy functions

It is proved that ifj is an inner function and ρ(T)=sup|∫(e^iγT/j(γ))f(γ)dγ| overf in the unit ball ofH ¹, then eitherρ ≡ 1 for allT≧0, or elseρ(T) ↓ 0 exponentially fast asT ↑ ∞. The inner functionsj corresponding to each alternative are classified.     

J-inner matrix functions, interpolation and inverse problems for canonical systems, I: Foundations

This is the first of a planned sequence of papers on inverse problems for canonical systems of differential equations. It is devoted largely to foundational material (much of which is of independent interest) on the theory of assorted classes of meromorphic matrix valued functions. Particular attention is paid to the structure of J-inner functions and connections with bitangential interpolation problems and reproducing kernel Hilbert spaces. Some new characterizations of regular, singular and strongly regular J-inner functions in terms of the associated reproducing kernel Hilbert spaces are presented.D. Z. Arov wishes to thank the Weizmann Institute of Science for hospitality and support; H. Dym wishes to thank Renee and Jay Weiss for endowing the chair which supports his research.     

On the zeros of some continuous analogues of matrix orthogonal polynomials and a related extension problem with negative squares

A new proof of a recent theorem of Ellis, Gohberg, and Lay, which identifies the number of roots of a “continuous” matrix orthogonal polynomial in the open upper halfplane with the number of negative eigenvalues of a related integral operator is presented. A related extension problem is then formulated and solved in assorted classes of functions which are analytic in the open upper half plane, apart from a finite number of poles. A discrete analogue of this extension problem is also formulated and solved. © 1994 John Wiley & Sons, Inc.     

On the inverse spectral problem for the string equation

The theory of Hilbert spaces of entire functions is used to deduce the changes in the mass distribution of a string effected by a small change in its spectral function. The formula of Gelfand-Levitan emerges as a special case.     

Electrokinetic characterization of superparamagnetic nanoparticle–aptamer conjugates: design of new highly specific probes for miniaturized molecular diagnostics

With the view of designing new nanoparticle (NP)–aptamer conjugates and proving their suitability as biorecognition tools for miniaturized molecular diagnostics, new maghemite–silica core–shell NP–aptamer conjugates were characterized for the first time in terms of grafting rate and colloidal stability under electrophoretic conditions using capillary electrophoresis. After the grafting rate (on the order of six to 50) of the lysozyme-binding aptamer had been estimated, the electrophoretic stability and peak dispersion of the resulting oligonucleotide–NP conjugates were estimated so as to determine the optimal separation conditions in terms of buffer pH, ionic strength and nature, as well as temperature and electric field strength. The effective surface charge density of the NPs was close to zero for pH lower than 5, which led to some aggregation. The NPs were stable in the pH range from 5 to 9, and an increase in electrophoretic mobility was evidenced with increasing pH. Colloidal stability was preserved at physiological pH for both non-grafted NPs and grafted NPs in the 10–100 mM ionic strength range and in the 15–60 °C temperature range. A strong influence of the nature of the buffer counterion on NP electrophoretic mobility and peak dispersion was evidenced, thus indicating some interactions between buffer components and NP–aptamer conjugates. Whereas an electric field effect (50–900 V cm^?1) on NP electrophoretic mobility was evidenced, probably linked to counterion dissociation, temperature seems to have an appreciable effect on the zeta potential and aptamer configuration as well. This information is crucial for estimating the potentialities of such biorecognition tools in electrophoretic systems.     

Bernoulli thinning of a Markov renewal process

A Bernoulli thinning of a Markov renewal process is investigated. The properties of the thinned process are considered and are related to the properties of the original process. The parameters, moments and equilibrium of the thinned process are determined in terms of the parameters defining the underlying Markov renewal process. Results are illustrated by examples. © 1998 John Wiley & Sons, Ltd.