Nevanlinna–Pick interpolation on distinguished varieties in the bidisk

This article treats Nevanlinna–Pick interpolation in the setting of a special class of algebraic curves called distinguished varieties. An interpolation theorem, along with additional operator theoretic results, is given using a family of reproducing kernels naturally associated to the variety. The examples of the Neil parabola and doubly connected domains are discussed.     

The Nevanlinna–Pick problem on the closed unit disk: Minimal norm rational solutions of low degree

For the Nevanlinna–Pick interpolation problem with n$n$ interpolation conditions (interior and boundary), we construct a family of rational solutions of degree at most n−1$n-1$. We also establish necessary and sufficient conditions for the existence and the uniqueness of a solution with the minimally possible H^∞$H^{\infty }$-norm and construct a family of minimal-norm rational solutions of degree at most n−1$n-1$ in the indeterminate case. Finally, we supplement a result of Ruscheweyh and Jones showing that in case the interpolation nodes and the target values are all unimodular, any rational solution of degree at most n−1$n-1$ is necessarily a finite Blaschke product.     

Nevanlinna–Pick Interpolation and Factorization of Linear Functionals

If \mathfrakA{\mathfrak{A}} is a unital weak-* closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property \mathbbA₁(1){\mathbb{A}_1(1)}, then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has \mathbbA₁(1){\mathbb{A}_1(1)}, and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-* closed subalgebras of H ^∞ acting on Hardy space or on Bergman space.     

Two-component generalized Ragnisco–Tu equation and the Riemann–Hilbert problem

Theoretical and Mathematical Physics - Using the Riemann–Hilbert approach, we investigate the two-component generalized Ragnisco–Tu equation. The modified equation is integrable in the...     

Discrete-Time Dichotomous Well-Posed Linear Systems and Generalized Schur–Nevanlinna–Pick Interpolation

We introduce a class of matrix-valued functions W called “L²- regular”. In case W is J-inner, this class coincides with the class of “strongly regular J-inner” matrix functions in the sense of Arov–Dym. We show that the class of L²-regular matrix functions is exactly the class of transfer functions for a discrete-time dichotomous (possibly infinite-dimensional) input-state-output linear system having some additional stability properties. When applied to J-inner matrix functions, we obtain a state-space realization formula for the resolvent matrix associated with a generalized Schur–Nevanlinna–Pick interpolation problem. Communicated by Daniel Alpay Submitted: August 20, 2006; Accepted: September 13, 2006     

Composition operators between Bloch-type spaces in the polydisc

We characterize those holomorphic mappings (?) from the polydisc Dn in Cn to itself for which the induced composition operators C(?) are bounded (or compact) from the Bloch-type space Bωto Bμ(respectively, from the little Bloch-type space Bω,0 to Bμ,0), where ωis a normal function on [0,1) and μis a nonnegative function on [0,1) with μ(tn) > 0 for some sequence {tn}n=1∞(?)[0,1) satisfying limn→∞tn = 1.     

Kotani–Last problem and Hardy spaces on surfaces of Widom type

This is a small theory of non almost periodic ergodic families of Jacobi matrices with purely (however) absolutely continuous spectrum. The reason why this effect may happen is that under our “axioms” we found an analytic condition on the resolvent set that is responsible for (exactly equivalent to) this effect.     

Weighted Fourier–Laplace transforms in reproducing kernel Hilbert spaces on the sphere

We study the action of a weighted Fourier–Laplace transform on the functions in the reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier–Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier–Laplace transforms: fractional derivatives and Laplace–Beltrami derivatives.     

Existence and uniqueness of solutions for Fokker–Planck equations on Hilbert spaces

We consider a stochastic differential equation in a Hilbert space with time-dependent coefficients for which no general existence and uniqueness results are known. We prove, under suitable assumptions, the existence and uniqueness of a measure valued solution, for the corresponding Fokker–Planck equation. In particular, we verify the Chapman–Kolmogorov equations and get an evolution system of transition probabilities for the stochastic dynamics informally given by the stochastic differential equation.     

The Gleason''''s problem for some polyharmonic and hyperbolic harmonic function spaces

LetΩ Rn be a bounded convex domain with C2 boundary. For 0相似文献   

The Cauchy problem for the generalized Zakharov–Kuznetsov equation on modulation spaces

We consider the Cauchy problem for the generalized Zakharov–Kuznetsov equation ${?}_{t}u+{?}_{x_1}\mathrm{\Delta }u={?}_{x_1}(u^{m+1})$ on three and higher dimensions. We mainly study the local well-posedness and the small data global well-posedness in the modulation space $M_{2,1}^0({\mathbb{R}}^n)$ for $m\ge 4$ and $n\ge 3$. We also investigate the quartic case, i.e., $m=3$.     

Finding the coefficients in the new representation of the solution of the Riemann–Hilbert problem using the Lauricella function

The solution of the Riemann–Hilbert problem for an analytic function in a canonical domain for the case in which the data of the problem is piecewise constant can be expressed as a Christoffel–Schwartz integral. In this paper, we present an explicit expression for the parameters of this integral obtained by using a Jacobi-type formula for the Lauricella generalized hypergeometric function F _D ^(N). The results can be applied to a number of problems, including those in plasma physics and the mechanics of deformed solids.     

Maurer–Cartan moduli and models for function spaces

We set up a formalism of Maurer–Cartan moduli sets for _L∞$L_{\infty }$ algebras and associated twistings based on the closed model category structure on formal differential graded algebras (a.k.a. differential graded coalgebras). Among other things this formalism allows us to give a compact and manifestly homotopy invariant treatment of Chevalley–Eilenberg and Harrison cohomology. We apply the developed technology to construct rational homotopy models for function spaces.     

The Gleason's problem on normal weight general function spaces in the unit ball of ■n

In this paper,we first discuss the boundedness of certain integral operator T_ton the normal weight general function space F (p,μ,s) in the unit ball B_n of ■ⁿ.As an application of this operator,we prove that the Gleason’s problem is solvable on F (p,μ,s).