The scalar Nevanlinna-Pick interpolation problem with boundary conditions

We show that if the Nevanlinna-Pick interpolation problem is solvable by a function mapping into a compact subset of the unit disc, then the problem remains solvable with the addition of any number of boundary interpolation conditions, provided the boundary interpolation values have modulus less than unity. We give new, inductive proofs of the Nevanlinna-Pick interpolation problem with any finite number of interpolation points in the interior and on the boundary of the domain of interpolation (the right half plane or unit disc), with function values and any finite number of derivatives specified. Our solutions are analytic on the closure of the domain of interpolation. Our proofs only require a minimum of matrix theory and operator theory. We also give new, straightforward algorithms for obtaining minimal H_∞ norm solutions. Finally, some numerical examples are given.     

On the normability of Marcinkiewicz classes

We obtain simple necessary and sufficient conditions for the normability of Marcinkiewicz classes, which play an important role in various problems of analysis. These conditions are stated in terms of dilation exponents of the function generating the given class as well as in terms of interpolation for operators bounded in the couple (L ₁, L _∞).     

On exceptional values of entire and meromorphic functions

Letf(z) be meromorphic function of finite nonzero orderρ. Assuming certain growth estimates onf by comparing it withr ^ρ L(r) whereL(r) is a slowly changing function we have obtained the bounds for the zeros off(z) ?g (z) whereg (z) is a meromorphic function satisfyingT (r, g)=o {T(r, f)} asr → ∞. These bounds are satisfied but for some exceptional functions. Examples are given to show that such exceptional functions exist.     

Rational functions of best L p- approximation and analytic continuability

Sufficient conditions for the analytic (resp. meromorphic) continuability of a function f ∈ L _p([?1, l]), p > 0, in terms of rational functions of best weighted L _p- approximation with an unbounded number of finite poles are established.     

Mixed Löwner and Nevanlinna-Pick Interpolation

A mixed type, L?wner and Nevanlinna-Pick directional two-sided interpolation problem is considered. A necessary and sufficient condition for the problem to have a solution is established, in terms of properties of the Pick kernel to the problem. As well, a parametrization of the set of all real rational solutions of minimal degree is given. The corresponding Nevanlinna-Pick boundary-interior interpolation problem is also considered and a solvability condition for it is obtained. The approach to the problem is via functional Hilbert spaces.     

On compactness of the difference of composition operators

Let φ and ψ be analytic self-maps of the unit disc, and denote by C_φ and C_ψ the induced composition operators. The compactness and weak compactness of the difference T=C_φ−C_ψ are studied on H^p spaces of the unit disc and L^p spaces of the unit circle. It is shown that the compactness of T on H^p is independent of p∈[1,∞). The compactness of T on L¹ and M (the space of complex measures) is characterized, and examples of φ and ψ are constructed such that T is compact on H¹ but non-compact on L¹. Other given results deal with L^∞, weakly compact counterparts of the previous results, and a conjecture of J.E. Shapiro.     

Weighted Weierstrass' theorem with first derivatives

We characterize the set of functions which can be approximated by continuous functions with the norm ‖f_‖L^∞(w) for every weight w. This fact allows to determine the closure of the space of polynomials in L^∞(w) for every weight w with compact support. We characterize as well the set of functions which can be approximated by smooth functions with the norm

‖f_‖W1,∞(w₀,w₁):=‖f_‖L^∞(w₀)+‖f^′_‖L^∞(w₁),     

L2 − L∞ filtering for Markovian jump systems with time-varying delays and partly unknown transition probabilities

This paper considers the L₂ − L_∞ filtering problem for Markovian jump systems. The systems under consideration involve time-varying delays, disturbance signal and partly unknown transition probabilities. The aim of this paper is to design a filter, which is suitable for exactly known and partly unknown transition probabilities, such that the filtering error system is stochastically stable and a prescribed L₂ − L_∞ disturbance attenuation level is guaranteed. By using the Lyapunov-Krasovskii functional, sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). A numerical example is given to illustrate the effectiveness of the proposed main results. All these results are expected to be of use in the study of filter design for Markovian jump systems with partly unknown transition probabilities.     

Scalar conservation laws with general boundary condition and continuous flux function

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u₀ on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u₀∈L^∞(Ω), f∈L^∞(Q), a∈L^∞((0,T)×∂Ω). In the L¹-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.     

On the zeros of analytic functions belonging to Gevrey classes

For functionsf(z) ? 0, holomorphic in the unit disk u, infinitely differentiable in u, and belonging to a given Gevrey class on ?u, we establish sufficient conditions characterizing the sets K _f ^∞ = (z: ¦z¦ = 1,f ^(k) (z) = 0,k = 0, 1, 2, ... }. These conditions are close to the necessary condition due to L. Carleson and substantially more precise than the conditions given byA.-M. Chollet (see [1, 2]).     

Limit laws of entrance times for homeomorphisms of the circle

Given a homeomorphismf of the circle with irrational rotation number and a descending chain of renormalization intervalsj _n off, we consider for each interval the point process obtained by marking the times for the orbit of a point in the circle to enterJ _n. Assuming the point is randomly chosen by the unique invariant probability measure off, we obtain necessary and sufficient conditions which guarantee convergence in law of the corresponding point process and we describe all the limiting processes. These conditions are given in terms of the convergent subsequences of the orbit of the rotation number off under the Gauss transformation and under a certain realization of its natural extension. We also consider the case when the point is randomly chosen according to Lebesgue measure,f being a diffeomorphism which isC ¹-conjugate to a rotation, and we show that the same necessary and sufficient conditions guarantee convergence in this case.     

On directional blow-up for quasilinear parabolic equations with fast diffusion

We discuss blow-up at space infinity of solutions to quasilinear parabolic equations of the form ut=Δ?(u)+f(u) with initial data u₀∈L^∞(RN), where ? and f are nonnegative functions satisfying ?^″?0 and . We study nonnegative blow-up solutions whose blow-up times coincide with those of solutions to the O.D.E. v^′=f(v) with initial data ‖u_0‖L^∞(RN). We prove that such a solution blows up only at space infinity and possesses blow-up directions and that they are completely characterized by behavior of initial data. Moreover, necessary and sufficient conditions on initial data for blow-up at minimal blow-up time are also investigated.     

Orlicz spaces without extreme points

Orlicz function and sequence spaces unit balls of which have no extreme points are completely characterized for both (the Orlicz and the Luxemburg) norms. Their subspaces of order continuous elements, with the norms induced from the whole Orlicz spaces without extreme points in their unit balls are also characterized. The well-known spaces L₁ and c₀ with unit balls without extreme points are covered by our results. Moreover, a new example of a Banach space without extreme points in its unit ball is given (see Example 1). This is the subspace a(L₁+L_∞) of order continuous elements of the space L₁+L_∞ equipped with the norm whenever 0<a<∞ and μ(T)>1/a.     

On non-effective weights in Orlicz spaces

Given a weight w in Ω ⊂ ∝^N, |Ω| < ∞ and a Young function φ, we consider the weighted modular ∫_Ω ω(f(x))w(x)dx and the resulting weighted Orlicz space L_ω(w). For a Young function Ω ∉ Δ₂(∞) we present a necessary and sufficient conditions in order that L_ω(w) = L_ω(X_Ω) up to the equivalence of norms. We find a necessary and sufficient condition for ω in order that there exists an unbounded weight w such that the above equality of spaces holds. By way of applications we simplify criteria from [5] for continuity of the composition operator from L_ω into itself when ω Δ₂(∞) and obtain necessary and sufficient condition in order that the composition operator maps L_ω. continuously onto L_ω.     

Optimal control of a system governed by hyperbolic operator with an infinite number of variables

A distributed control problem for the hyperbolic operator with an infinite number of variables is considered. The controls are allowed to be in the space L₂(0, T; L₂(R^∞)) = L₂(Q). The necessary and sufficient condition for the control to be optimal is obtained and the set of inequalities that characterize this condition is also obtained.     

Extended Cesàro operators on Bergman spaces

We define an extended Cesàro operator T_g with holomorphic symbol g in the unit ball B of Cⁿ. For a large class of weights w we characterize those g for which T_g is bounded (or compact) from Bergman space L^p_a,w(B) to L^q_a,w(B), 0<p,q<∞. In addition, we obtain some results about equivalent norms, the norm of point evaluation functionals, and the interpolation sequences on L^p_a,w(B).     

On the sequences of zeros of holomorphic solutions of linear second-order differential equations

We find necessary and sufficient conditions under which a finite or infinite sequence of complex numbers is the sequence of zeros of a holomorphic solution of the linear differential equation f″ + a ₀ f = 0 with a meromorphic coefficient a ₀ that has second-order poles. In addition, we present a criterion for all solutions of second-order linear equations to be meromorphic.     

Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces

Let T be an injective bilateral weighted shift onl ² thought as "multiplication by λ" on a space of formal Laurent series L²(β). (a) If L²(β) is contained in a space of quasi-analytic class of functions, then the point spectrum σ_p(T^?) of T^? contains a circle and the cyclic invariant subspaceM _f of T generated by f is simply invariant (i.e., ∩{(T^k M _f)^?: k ≥ 0}= {0}) for each f in L²(β); (b) If L²(β) contains a non-quasi-analytic class of functions (defined on a circle г) of a certain type related with the weight sequence of T, then there exists f in L²(ß) such thatM _f is a non-trivial doubly invariant subspace (i.e., (TM _f)^? =M _f); furthermore, if г ? σ_p(T^*), then σ_p (T^*) = г and f can be chosen so that σ_p([T∣M _f]^*) = г?{α}, for some α ε г. Several examples show that the gap between operators satisfying (a) and operators satisfying (b) is rather small.