For sequences of rational functions, analytic in some domain, a theorem of Montel’s type is proved. As an application, sequences of rational functions of the best -approximation with an unbounded number of finite poles are considered.
The Dirac equation for spinor-valued fields on the Minkowski space of even dimension form a hyperbolic system of partial differential equations. In the paper, we are showing how to reconstruct the solution from initial data given on the upper sheet of the hyperboloid. In particular, we derive an integral formula expressing the value of in a chosen point as an integral over a compact cycle given by the intersection of the null cone with in the Minkowski space .
The main observation of this note is that the Lebesgue measure μ in the Turán-Nazarov inequality for exponential polynomials can be replaced with a certain geometric invariant ω ≥ μ, which can be effectively estimated in terms of the metric entropy of a set, and may be nonzero for discrete and even finite sets. While the frequencies (the imaginary parts of the exponents) do not enter the original Turán-Nazarov inequality, they necessarily enter the definition of ω.
For functions that are separately solutions of an elliptic homogeneous PDE with constant coefficients, we prove an analogue of Siciak's theorem for separately holomorphic functions.
Analytic interpolation problems arise quite naturally in a variety of engineering applications. This is due to the fact that analyticity of a (transfer) function relates to the stability of a corresponding dynamical system, while positive realness and contractiveness relate to passivity. On the other hand, the degree of an interpolant relates to the dimension of the pertinent system, and this motivates our interest in constraining the degree of interpolants. The purpose of the present paper is to...
Integrals of the Cauchy type extended over the boundary of a general compact set in the complex plane are investigated. Necessary and sufficient conditions on are established guaranteeing the existence of angular limits of these integrals at a fixed for all densities satisfying a Hölder-type condition at .