Subordinations for Functions with Positive Real Part

Some sufficient conditions are determined for certain first order differential subordinations to imply the corresponding analytic solution is subordinate to a rational, exponential, or sine function. By applying these results, we also obtain sufficient conditions for normalized analytic functions to be in certain well known subclasses of starlike functions.     

R-Boundedness of Smooth Operator-Valued Functions

In this paper we study R-boundedness of operator families , where X and Y are Banach spaces. Under cotype and type assumptions on X and Y we give sufficient conditions for R-boundedness. In the first part we show that certain integral operator are R-bounded. This will be used to obtain R-boundedness in the case that is the range of an operator-valued function which is in a certain Besov space . The results will be applied to obtain R-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems. Mark Veraar is supported by the Alexander von Humboldt foundation. His visit to Helsinki, which started this project, was funded by the Finnish Centre of Excellence in Analysis and Dynamics Research.     

Convolution Kernels of 2D Fourier Multipliers Based on Real Analytic Functions

In this paper, estimates are proven for convolution kernels associated to multipliers from a reasonably general class of compactly supported two-dimensional functions constructed out of real analytic functions. These estimates are both for overall decay rate and decay rate in specific directions. The estimates are sharp for a certain range of exponents appearing in the theorems.     

Strictly Positive Definite Functions on a Real Inner Product Space

If converges for all with all coefficients , then the function is positive definite on H×H for any inner product space H. Set K={k: a _k >0}. We show that is strictly positive definite if and only if K contains the index 0 plus an infinite number of even integers and an infinite number of odd integers.     

Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures

Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure \(F: \varOmega \to B(H)\) has an integral representation of the form

$$ F(E) =\sum_{k=1}^{m} \int _{E} G_{k}(\omega )\otimes G_{k}(\omega )\, d \mu (\omega ) $$

for some weakly measurable maps \(G_{k}\ (1\leq k\leq m) \) from a measurable space \(\varOmega \) to a Hilbert space ℋ and some positive measure \(\mu \) on \(\varOmega \). Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.

     

Representation of Reproducing Kernels and the Lebesgue Constants on the Ball

For the weight function (1−x²)^μ−1/2 on the unit ball, a closed formula of the reproducing kernel is modified to include the case −1/2<μ<0. The new formula is used to study the orthogonal projection of the weighted L² space onto the space of polynomials of degree at most n, and it is proved that the uniform norm of the projection operator has the growth rate of n^(d−1)/2 for μ<0, which is the smallest possible growth rate among all projections, while the rate for μ0 is n^μ+(d−1)/2.     

Mixed Fractional Differential Equations and Generalized Operator-Valued Mittag-Leffler Functions

Mathematical Notes - We introduce the most general mixed fractional derivatives and integrals from three points of views: probability, the theory of operator semigroups, and the theory of...     

Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Let E be a separable infinite-dimensional Hilbert space, and let denote the algebra of all functions that are holomorphic. If is a subalgebra of , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra , the disk algebra and the Wiener algebra are all infinite. Submitted: October 10, 2007., Revised: January 11, 2008., Accepted: January 12, 2007.     

On the Strongly Supermedian Functions and Kernels

We answer to a question of P.J. Fitzsimmons and R.K. Getoor concerning the characterization of a strongly supermedian function by the property of being universally supermedian. Based on the Revuz correspondence we also prove (alternatively to the probabilistic approach of Fitzsimmons and Getoor) that the Radon–Nikodym derivative between the Revuz measures of a semiregular excessive kernel and the regular strongly supermedian kernel generating it, does not depend on the kernel. Mathematics Subject Classifications (2000) Primary 60J45, 31D05; Secondary 60J35, 60J40.     

The Refined Schwarz-Pick Estimates for Positive Real Part Holomorphic Functions in Several Complex Variables

In this article,the refined Schwarz-Pick estimates for positive real part holomorphic functions■are given,where k is a positive integer,and G is a balanced domain in complex Banach spaces.In particular,the results of first order Fréchet derivative for the above functions and higher order Frechet derivatives for positive real part holomorphic functions■are sharp for G=B,where B is the unit ball of complex Banach spaces or the unit ball of complex Hilbert spaces.Their results reduce to the classic...     

Vectorial Slepian Functions on the Ball

Abstract

Due to the uncertainty principle, a function cannot be simultaneously limited in space as well as in frequency. The idea of Slepian functions, in general, is to find functions that are at least optimally spatio-spectrally localized. Here, we are looking for Slepian functions which are suitable for the representation of real-valued vector fields on a three-dimensional ball. We work with diverse vectorial bases on the ball which all consist of Jacobi polynomials and vector spherical harmonics. Such basis functions occur in the singular value decomposition of some tomographic inverse problems in geophysics and medical imaging. Our aim is to find band-limited vector fields that are well-localized in a part of a cone whose apex is situated in the origin. Following the original approach towards Slepian functions, the optimization problem can be transformed into a finite-dimensional algebraic eigenvalue problem. The entries of the corresponding matrix are treated analytically as far as possible. For the remaining integrals, numerical quadrature formulae have to be applied. The eigenvalue problem decouples into a normal and a tangential problem. The number of well-localized vector fields can be estimated by a Shannon number which mainly depends on the maximal radial and angular degree of the basis functions as well as the size of the localization region. We show numerical examples of vectorial Slepian functions on the ball, which demonstrate the good localization of these functions and the accurate estimate of the Shannon number.     

Bi-Inner Dilations and Bi-Stable Passive Scattering Realizations of Schur Class Operator-Valued Functions

Let S(U; Y) be the class of all Schur functions (analytic contractive functions) whose values are bounded linear operators mapping one separable Hilbert space U into another separable Hilbert space Y , and which are defined on a domain , which is either the open unit disk or the open right half-plane . In the development of the Darlington method for passive linear time-invariant input/state/output systems (by Arov, Dewilde, Douglas and Helton) the following question arose: do there exist simple necessary and sufficient conditions under which a function has a bi-inner dilation mapping into ; here U ₁ and Y ₁ are two more separable Hilbert spaces, and the requirement that Θ is bi-inner means that Θ is analytic and contractive on Ω and has unitary nontangential limits a.e. on ∂Ω. There is an obvious well-known necessary condition: there must exist two functions and (namely and ) satisfying and for almost all . We prove that this necessary condition is also sufficient. Our proof is based on the following facts. 1) A solution ψ _r of the first factorization problem mentioned above exists if and only if the minimal optimal passive realization of θ is strongly stable. 2) A solution ψ _l of the second factorization problem exists if and only if the minimal *-optimal passive realization of θ is strongly co-stable (the adjoint is strongly stable). 3) The full problem has a solution if and only if the balanced minimal passive realization of θ is strongly bi-stable (both strongly stable and strongly co-stable). This result seems to be new even in the case where θ is scalar-valued.      

On Holomorphic Extendability of Functions from Part of the Lie Sphere to the Lie Ball

We describe domains to which the Hua Loo-Keng-type integral for the Lie ball extends holomorphically. We obtain a criterion for the possibility of holomorphic extension of functions from part of the Lie sphere to the Lie ball.     

Eigenvalue Distribution of Positive Definite Kernels on Unbounded Domains

We study eigenvalues of positive definite kernels of L² integral operators on unbounded real intervals. Under the assumptions of integrability and uniform continuity of the kernel on the diagonal the operator is compact and trace class. We establish sharp results which determine the eigenvalue distribution as a function of the smoothness of the kernel and its decay rate at infinity along the diagonal. The main result deals at once with all possible orders of differentiability and all possible rates of decay of the kernel. The known optimal results for eigenvalue distribution of positive definite kernels in compact intervals are particular cases. These results depend critically on a 2-parameter differential family of inequalities for the kernel which is a consequence of positivity and is a differential generalization of diagonal dominance.     

The Heat Kernels of the Real Second Classical Domains

L3t[1]istherealsecondclassicaldomains.ThemapdefinedbyW~(I T)--'(I--T)carriesRillontotheunboundeddomainWhichisinvariantfinderthelineartransformationW=ATA',whereAisannxnupertrianglarmatrixwithpositivediagonalelements.SuchtransformationsformaLiegroupRill(n),whichisequivalenttoRillunderthemapdefinedbyW=AA'.TheAiscalledthehoro-hypercirclecoordinateofRRII.Itiswell-knownthatinRillthereisanillvariantmetricdsZ=tr[(I--KW)--'dW]'.InRHI,,whichisequaltothemetricdsZ=fir[(W--'dW)'].Inthehoro-h…     

Reproducing Kernels and Extremal Functions in Dirichlet-Type Spaces

Some identities are established for reproducing kernels and extremal functions in Dirichlet-type spaces. In certain cases, these identities are characteristic. A new proof of Aleman's theorem on extremal functions as contractive multipliers is obtained. In contrast to the original one, this proof can be extended to vector-valued spaces. Bibliography: 11 titles.     

Subharmonic Functions in the Unit Ball

For functions u subharmonic in the unit ball B_N of , this paper compares the growth of the repartition function of their Riesz measure μ with the growth of u near the boundary of B_N. Cases under study are: and , with A, B, γ positive constants and if N=2 or if N≥ 3. This paper contains several integral results, as for instance: when ∫_BN u⁺(x)[-ω^′(|x|²)]dx < +∞ for some positive decreasing C¹ function ω, it is proved that .