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We study infinite products of reproducing kernels with view to their use in dynamics (of iterated function systems), in harmonic analysis, and in stochastic processes. On the way, we construct a new family of representations of the Cuntz relations. Then, using these representations we associate a fixed filled Julia set with a Hilbert space. This is based on analysis and conformal geometry of a fixed rational mapping $R$ in one complex variable, and its iterations.
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A bounded linear operator $T$ on a Hilbert space is said to be homogeneous if $varphi(T)$ is unitarily equivalent to $T$ for all $varphi$ in the group M{o}b of bi-holomorphic automorphisms of the unit disc. A projective unitary representation $sigma$ of M{o}b is said to be associated with an operator T if $varphi(T)= sigma(varphi)^star T sigma(varphi)$ for all $varphi$ in M{o}b. In this paper, we develop a M{o}bius equivariant version of the Sz.-Nagy--Foias model theory for completely non-unitary (cnu) contractions. As an application, we prove that if T is a cnu contraction with associated (projective unitary) representation $sigma$, then there is a unique projective unitary representation $hat{sigma}$, extending $sigma$, associated with the minimal unitary dilation of $T$. The representation $hat{sigma}$ is given in terms of $sigma$ by the formula $$ hat{sigma} = (pi otimes D_1^+) oplus sigma oplus (pi_star otimes D_1^-), $$ where $D_1^pm$ are the two Discrete series representations (one holomorphic and the other anti-holomorphic) living on the Hardy space $H^2(mathbb D)$, and $pi, pi_star$ are representations of M{o}b living on the two defect spaces of $T$ defined explicitly in terms of $sigma$. Moreover, a cnu contraction $T$ has an associated representation if and only if its Sz.-Nagy--Foias characteristic function $theta_T$ has the product form $theta_T(z) = pi_star(varphi_z)^* theta_T(0) pi(varphi_z),$ $zin mathbb D$, where $varphi_z$ is the involution in M{o}b mapping $z$ to $0.$ We obtain a concrete realization of this product formula %the two representations $pi_star$ and $pi$ for a large subclass of homogeneous cnu contractions from the Cowen-Douglas class.
The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A.M. Whitneys density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Polya frequency functions, and Polya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.
We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space.
We investigate connections between the geometry of linear subspaces and the convergence of the alternating projection method for linear projections. The aim of this article is twofold: in the first part, we show that even in Euclidean spaces the convergence of the alternating method is not determined by the principal angles between the subspaces involved. In the second part, we investigate the properties of the Oppenheim angle between two linear projections. We discuss, in particular, the question of existence and uniqueness of consistency projections in this context.
This work is devoted to the study of Bessel and Riesz systems of the type $big{L_{gamma}mathsf{f}big}_{gammain Gamma}$ obtained from the action of the left regular representation $L_{gamma}$ of a discrete non abelian group $Gamma$ which is a semidirect product, on a function $mathsf{f}in ell^2(Gamma)$. The main features about these systems can be conveniently studied by means of a simple matrix-valued function $mathbf{F}(xi)$. These systems allow to derive sampling results in principal $Gamma$-invariant spaces, i.e., spaces obtained from the action of the group $Gamma$ on a element of a Hilbert space. Since the systems $big{L_{gamma}mathsf{f}big}_{gammain Gamma}$ are closely related to convolution operators, a connection with $C^*$-algebras is also established.
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