Schur–Agler and Herglotz–Agler classes of functions: Positive-kernel decompositions and transfer-function realizations

We discuss transfer-function realization for multivariable holomorphic functions mapping the unit polydisk or the right polyhalfplane into the operator analogue of either the unit disk or the right halfplane (Schur/Herglotz functions over either the unit polydisk or the right polyhalfplane) which satisfy the appropriate stronger contractive/positive real part condition for the values of these functions on commutative tuples of strict contractions/strictly accretive operators (Schur–Agler/Herglotz–Agler functions over either the unit polydisk or the right polyhalfplane). As originally shown by Agler, the first case (polydisk to disk) can be solved via unitary extensions of a partially defined isometry constructed in a canonical way from a kernel decomposition for the function (the lurking-isometry method). We show how a geometric reformulation of the lurking-isometry method (embedding of a given isotropic subspace of a Kre?n space into a Lagrangian subspace—the lurking-isotropic-subspace method) can be used to handle the second two cases (polydisk to halfplane and polyhalfplane to disk), as well as the last case (polyhalfplane to halfplane) if an additional growth condition at ∞ is imposed. For the general fourth case, we show how a linear-fractional-transformation change of variable can be used to arrive at the appropriate symmetrized nonhomogeneous Bessmertny? long-resolvent realization. We also indicate how this last result recovers the classical integral representation formula for scalar-valued holomorphic functions mapping the right halfplane into itself.     

Inequalities of Littlewood--Paley Type for n-Harmonic Functions on the Polydisk

For n-harmonic functions on the unit polydisk in the space $\mathbb{C}^n $ we define g-functions of Littlewood--Paley type and establish L ^p-inequalities related to them. In the present paper, the main theorems deal with the extension of results of Littlewood, Paley, and Flett to the polydisk and their generalizion to fractional derivatives of arbitrary order. This gives an answer to a question posed by Littlewood.     

On the Fekete and Szegö problem for starlike mappings of order <Emphasis Type="Italic">α</Emphasis>

Let S _α ^* be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szegö inequality for the class S _α ^* , and then we generalize this result to the unit ball in a complex Banach space or on the unit polydisk in C ⁿ .     

Fekete and Szeg problem in one and higher dimensions

In this paper, we establish the Fekete and Szeg inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in C~n.     

Bergman Spaces and Carleson Measures on Homogeneous Isotropic Trees

Hastings studied Carleson measures for non-negative subharmonic functions on the polydisk and characterized them by a certain geometric condition relative to Lebesgue measure σ. Cima & Wogen and Luecking proved analogous results for weighted Bergman spaces on the unit ball and other open subsets of \(\mathbb {C}^{n}\). We consider a similar problem on a homogeneous tree, and study how the characterization and properties of Carleson measures for various function spaces depend on the choice of reference measure σ.     

Contractive Hilbert modules and their dilations

In this note, we show that a quasi-free Hilbert module R defined over the polydisk algebra with kernel function k(z,w) admits a unique minimal dilation (actually an isometric co-extension) to the Hardy module over the polydisk if and only if S ^?1(z, w)k(z, w) is a positive kernel function, where S(z,w) is the Szegö kernel for the polydisk. Moreover, we establish the equivalence of such a factorization of the kernel function and a positivity condition, defined using the hereditary functional calculus, which was introduced earlier by Athavale [8] and Ambrozie, Englis and Müller [2]. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. The proof works for a wider class of Hilbert modules in which the Hardy module is replaced by more general quasi-free Hilbert modules such as the classical spaces on the polydisk or the unit ball in ? ^m . Some consequences of this more general result are then explored in the case of several natural function algebras.     

“Compactly” supported frames for spaces of distributions on the ball

Frames are constructed on the unit ball B ^d in ${\mathbb{R}^d}$ consisting of smooth functions with small shrinking supports. The new frames are designed so that they can be used for decomposition of weighted Triebel–Lizorkin and Besov spaces on B ^d with weight ${w_\mu(x):=(1-|x|^2)^{\mu-1/2}, \mu}$ half integer,?μ?≥ 0.     

Schur coefficients in several variables

In the paper we study weakly continuous Schur-class-valued maps and their associated Schur coefficient families, that we call functional Schur coefficients. A case of special interest is the family of the “slices” through the polytorus of an n-variable function in the unit ball of H^∞(Dn), which is shown to be a weakly continuous map from the polytorus into the Schur class. The continuity properties of its functional Schur coefficients are used to characterize the rational inner functions in the polydisk algebra. As a consequence we obtain extensions in several variables of the Schur-Cohn test on zeroes of polynomials. This provides in particular a necessary and sufficient condition of stability for multi-dimensional AR filters.     

Composition operators induced by symbols defined on a polydisk

Suppose φ is a holomorphic mapping from the polydisk Dm into the polydisk Dn, or from the polydisk Dm into the unit ball Bn, we consider the action of the associated composition operator Cφ on Hardy and weighted Bergman spaces of Dn or Bn. We first find the optimal range spaces and then characterize compactness. As a special case, we show that if

     

The sharp estimates of homogeneous expansions for the generalized class of close-to-quasi-convex mappings

In this paper, the class of strongly close-to-quasi-convex mappings of type α and order β is introduced in the unit ball of complex Banach space or unit polydisk in ${\mathbb{C}}^n$, and we obtain the sharp estimates of homogeneous expansions for this class. These results generalize many known results.     

Reverse estimates in growth spaces

Let H(B _d ) denote the space of holomorphic functions on the unit ball B _d of ${{\mathbb{C}}^d}$ . Given a radial doubling weight w, we construct functions ${f, g\in H(B_1)}$ such that |f| + |g| is comparable to w. Also, we obtain similar results for B _d , d ≥ 2, and for circular, strictly convex domains with smooth boundary. As an application, we study weighted composition operators and related integral operators on growth spaces of holomorphic functions.     

The Estimates of All Homogeneous Expansions for a Subclass of ε Quasi-convex Mappings in Several Complex Variables

The authors obtain the estimates of all homogeneous expansions for a subclass of ε quasi-convex mappings on the unit ball in complex Banach spaces. Moreover, the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in ?ⁿ are also obtained. Especially, the above estimates are only sharp for a subclass of starlike mappings, quasi-convex mappings and quasi-convex mappings of type \(\mathbb{A}\). The results are the generalization of many known results.     

Shannon-type sampling for multivariate non-bandlimited signals

In this paper, starting from a function analytic in a neighborhood of the unit disk and based on Bessel functions, we construct a family of generalized multivariate sinc functions, which are radial and named radial Bessel-sinc (RBS) functions being time-frequency atoms with nonlinear phase. We obtain a recursive formula for the RBS functions in R d with d being odd. Based on the RBS function, a corresponding sampling theorem for a class of non-bandlimited signals is established. We investigate a class of radial functions and prove that each of these functions can be extended to become a monogenic function between two parallel planes, where the monogencity is taken to be of the Clifford analysis sense.     

The Julia-Wolff-Carathéodory theorem in polydisks

The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the non-tangential limit of both a bounded holomorphic function and its derivative at a given boundary point of the unit disk in the complex plane. This theorem has been generalized by Rudin to holomorphic maps between unit balls inC ⁿ and by the author to holomorphic maps between strongly (pseudo)convex domains. Here we describe Julia-Wolff-Carathéodory theorems for holomorphic maps defined in a polydisk and with image either in the unit disk, or in another polydisk, or in a strongly convex domain. One of the main tools for the proof is a general version of the Lindelöf principle valid for not necessarily bounded holomorphic functions.     

Bounded Toeplitz products on Bergman spaces of the unit ball

We consider the question for which square integrable analytic functions f and g on the unit ball the densely defined products are bounded on the weighted Bergman spaces. We prove results analogous to those we obtained in the setting of the unit disk and the polydisk.     

Hölder norm estimate for the Hilbert transform in Clifford analysis

Let Ω ? ? ⁿ be a Jordan domain with d-summable boundary Γ. The main gol of this paper is to estimate the Hölder norm of a fractal version of the Hilbert transform in the Clifford analysis context acting from Hölder spaces of Clifford algebra valued functions defined on Γ. The explicit expression for the upper bound of the norm provided here is given in terms of the Hölder exponents, the diameter of Γ and certain d-sum (d > d) of the Whitney decomposition of Ω. The result obtained is applied to standard Hilbert transform for domains with left Ahlfors-David regular surface.     

Essential norms of composition operators between Bloch type spaces in the polydisk

In 2010, Ruhan Zhao obtained the essential norms of composition operators between Bloch type spaces in the disk by the nth power of the induced analytic function. This paper will generalize Zhao’s results to the polydisk. Unlike the case of the composition operators on the unit disk, the essential norms are different for the cases ${p \in (0,1)}$ and p ≥ 1.     

Explicit bounds for multidimensional linear recurrences with restricted coefficients

This note employs path counting techniques to extend recent results on bounds for odd order linear recurrences to higher dimensions. The results imply optimal zero-free polydisks for multivariable power series with 0, 1 coefficients. Among the applications is a result that states that the optimal zero-free polydisk has radius approximately 1/(v+1), for large dimensions v.