James Pickering
We give a set of test functions for the interpolation problem on $H_1^\infty$ , the constrained interpolation problem studied by Davidson, Paulsen, Raghupathi and Singh. We show that this set of test functions is minimal.
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A time-variant analogue of an interpolation problem equivalent to the relaxed commutant lifting problem is introduced and studied. In a somewhat less general form the problem already appears in the analysis of the set of all solutions to the three chain completion problem. The interpolants are upper triangular operator matrices of which the columns induce contractive operators. The set of all solutions of the problem is described explicitly. The results presented are time-variant analogues of the main theorems in [A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to the relaxed commu…
Given a finite set $σ$ of the unit disc $\mathbb{D}$ and a holomorphic function $f$ in $\mathbb{D}$ which belongs to a class $X$ we are looking for a function $g$ in another class $Y$ which minimizes the norm $|g|_{Y}$ among all functions $g$ such that $g_{|σ}=f_{|σ}$. Generally speaking, the interpolation constant considered is $c(σ,\, X,\, Y)={sup}{}_{f\in X,\,\parallel f\parallel_{X}\leq1}{inf}\{|g|_{Y}:\, g_{|σ}=f_{|σ}\} \,.$ When $Y=H^{\infty}$, our interpolation problem includes those of Nevanlinna-Pick (1916), Caratheodory-Schur (1908). Moreover, Carleson's free interpolation (1958) has…
Recent results of Davidson-Paulsen-Raghupathi-Singh give necessary and sufficient conditions for the existence of a solution to the Nevanlinna-Pick interpolation problem on the unit disk with the additional restriction that the interpolant should have the value of its derivative at the origin equal to zero. This concrete mild generalization of the classical problem is prototypical of a number of other generalized Nevanlinna-Pick interpolation problems which have appeared in the literature (for example, on a finitely-connected planar domain or on the polydisk). We extend the results of Davidson…
We study a general metric constrained interpolation problem in a de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with a contractive multiplier $S$ between two Fock spaces along with its commutative counterpart, a de Branges-Rovnyak space associated with a Schur multiplier on the Drury-Arveson space of the unit ball of $\mathbb{C}^n$.
Towards a More General Type of Univariate Constrained Interpolation With Fractal Splines
本文提出更一般的单变量约束分形插值方法，通过选择参数使插值曲线严格位于给定线性或二次样条之上，并利用Peano核定理分析近似性质。
Along this work we study an indefinite abstract smoothing problem. After establishing necessary and sufficient conditions for the existence of solutions to this problem, the set of admissible parameters is discussed in detail. Then, its relationship with a linearly constrained interpolation problem is analyzed.
Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and problems in function theory, vol 146, pp 83-69, Naukova Dumka, Keiv, 1987) for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with an operator-valued Schur class function $S$. A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is…
Jim Agler revolutionized the area of Pick interpolation with his realization theorem for what is now called the Agler-Schur class for the unit ball in $\mathbb C^d$. We discuss an extension of these results to algebras of functions arising from test functions and the dual notion of a family of reproducing kernels, as well as the related interpolation theorem. When working with test functions, one ideally wants to use as small a collection as possible. Nevertheless, in some situations infinite sets of test functions are unavoidable. When this is the case, certain topological considerations come…
We consider an overdetermined problem for Laplace equation on a disk with partial boundary data where additional pointwise data inside the disk have to be taken into account. After reformulation, this ill-posed problem reduces to a bounded extremal problem of best norm-constrained approximation of partial L2 boundary data by traces of holomorphic functions which satisfy given pointwise interpolation conditions. The problem of best norm-constrained approximation of a given L2 function on a subset of the circle by the trace of a H2 function has been considered in [Baratchart \& Leblond, 1998]. I…
It is shown that rational dilation fails on broad collection of distinguished varieties associated to constrained subalgebras of the disk algebra of the form C + B A(D), where B is a finite Blaschke product with two or more zeros. This is accomplished in part by finding a minimal set of test functions. In addition, an Agler-Pick interpolation theorem is given and it is proved that there exist Kaijser-Varopoulos style examples of non-contractive unital representations where the generators are contractions.
A seminal result of Agler characterizes the so-called Schur-Agler class of functions on the polydisk in terms of a unitary colligation transfer function representation. We generalize this to the unit ball of the algebra of multipliers for a family of test functions over a broad class of semigroupoids. There is then an associated interpolation theorem. Besides leading to solutions of the familiar Nevanlinna-Pick and Caratheodory-Fejer interpolation problems and their multivariable commutative and noncommutative generalizations, this approach also covers more exotic examples.
We develop a dilation theory for row contractions subject to constraints determined by sets of noncommutative polynomials. Under natural conditions on the constraints, we have uniqueness for the minimal dilation. A characteristic function is associated with any (constrained) row contraction. For pure constrained row contraction, we show that the characteristic function is a complete unitary invariant and provide a model. We also show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp. commuting) row contraction can be express…
Complete discrete Schoenberg-Delsarte theory for homogeneous spaces
本文提出了一种关于同质空间的完整离散Schoenberg-Delsarte理论，扩展了Schoenberg的经典特征化，应用于受限排列问题和Delsarte理论，并证明了特定正定函数必须来自多项式。
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