Hardy Space Of Analytic Functions Research Articles

On the characterization properties of certain hypergeometric functions in the open unit disk

Our purpose in the present investigation is to study certain geometric properties such as the close-to-convexity, convexity, and starlikeness of the hypergeometric function z{}_{1}F_{2}(a;b,c;z) in the open unit disk. The usefulness of the main results for some familiar special functions like the modified Sturve function, the modified Lommel function, the modified Bessel function, and the {}_{0}F_{1}(-;c;z) function are also mentioned. We further consider a boundedness property of the function _{1}F_{2}(a;b,c;z) in the Hardy space of analytic functions. Several corollaries and special cases of the main results are also pointed out.

Linear combinations of composition operators induced by linear fractional maps on the infinite polydisk

The aim of this paper is to characterize the compactness of linear combinations of composition operators induced by linear fractional maps on the Hardy space of analytic functions over the infinite polydisk.

Blaschke–singular–outer factorization of free non-commutative functions

By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner–outer factorization. Here, a bounded analytic function is called inner or outer if multiplication by this function defines an isometry or has dense range, respectively, as a linear operator on the Hardy Space, H2, of analytic functions in the complex unit disk with square-summable Taylor series. This factorization can be further refined; any inner function θ decomposes uniquely as the product of a Blaschke inner function and a singular inner function, where the Blaschke inner contains all the vanishing information of θ, and the singular inner factor has no zeroes in the unit disk.We prove an exact analogue of this factorization in the context of the full Fock space, identified as the Non-commutative Hardy Space of analytic functions defined in a certain multi-variable non-commutative open unit ball.

Co-Isometric Weighted Composition Operators on Hilbert Spaces of Analytic Functions

We obtain a necessary and sufficient condition for a weighted composition operator to be co-isometric on a general weighted Hardy space of analytic functions in the unit disk whose reproducing kernel has the usual natural form. This turns out to be equivalent to the property of being unitary. The result reveals a dichotomy identifying a specific family of weighted Hardy spaces as the only ones that support non-trivial operators of this kind.

Control problems for energy harvester model and interpolation in Hardy space

AbstractThree control problems for the system of two coupled differential equations governing the dynamics of an energy harvesting model are studied. The system consists of the equation of an Euler–Bernoulli beam model and the equation representing the Kirchhoff's electric circuit law. Both equations contain coupling terms representing the inverse and direct piezoelectric effects. The system is reformulated as a single evolution equation in the state space of 3‐component functions. The control is introduced as a separable forcing term on the right‐hand side of the operator equation. The first control problem deals with an explicit construction of that steers an initial state to zero on a time interval [0, T]. The second control problem deals with the construction of such that the voltage output is equal to some given function (with being given as well). The third control problem deals with an explicit construction of both the force profile, , and the control, , which generate the desired voltage output . Interpolation theory in the Hardy space of analytic functions is used in the solution of the second and third problems.

$L_{p;r} $ spaces: Cauchy Singular Integral, Hardy Classes and Riemann-Hilbert Problem in this Framework

In the present work the space $L_{p;r} $ which is continuously embedded into $L_{p} $ is introduced. The corresponding Hardy spaces of analytic functions are defined as well. Some properties of the functions from these spaces are studied. The analogs of some results in the classical theory of Hardy spaces are proved for the new spaces. It is shown that the Cauchy singular integral operator is bounded in $L_{p;r} $. The problem of basisness of the system $left{Aleft(tright)e^{{mathop{rm int}} }; Bleft(tright)e^{-{mathop{rm int}} } right}_{nin Z_{+} }, $ is also considered. It is shown that under an additional condition this system forms a basis in $L_{p;r} $ if and only if the Riemann-Hilbert problem has a unique solution in corresponding Hardy class ${ H}_{p;r}^{+} times { H}_{p;r}^{+} $.

Analytic variable exponent Hardy spaces

‎We introduce a variable exponent version of the Hardy space of analytic functions on the unit disk‎. ‎We then show some properties of the space and give an example of a variable exponent $p(\cdot)$ that satisfies the $\log$-Hölder condition and $H^{p(\cdot)}\neq H^q$ for every constant exponent $q \in (1, \infty)$‎. ‎We also consider a variable exponent version of the Hardy space on the upper-half plane.‎

Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: A hundred decimal digits for the dimension of E2

We prove that the algorithm of [19] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set E2 (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 decimal places, using points of period up to 25.The method for establishing rigorous dimension bounds involves the holomorphic extension of mappings associated to the allowed continued fraction digits, an appropriate disc which is contracted by these mappings, and an associated transfer operator acting on the Hilbert Hardy space of analytic functions on this disc. We introduce methods for rigorously bounding the approximation numbers for the transfer operators, showing that this leads to effective estimates on the Taylor coefficients of the associated determinant, and hence to explicit bounds on the Hausdorff dimension.

HARDY SPACE OF LOMMEL FUNCTIONS

In this work we present some geometric properties (like star-likeness and convexity of order <TEX>${\alpha}$</TEX> and also close-to-convexity of order (<TEX>$1+{\alpha}$</TEX>)/2) for normalized of Lommel functions of the first kind. In order to prove our main results, we use the technique of differential subordinations and some inequalities. Furthermore, we present some applications of convexity involving Lommel functions associated with the Hardy space of analytic functions, i.e., we obtain conditions for the function <TEX>$h_{{\mu},{\upsilon}}(z)$</TEX> to belong to the Hardy space <TEX>$H^p$</TEX>.

Hardy space of generalized Struve functions

In this paper, we present some immediate applications of convexity involving Struve functions associated with the Hardy space of analytic functions, i.e. we obtain conditions for the functionto belong to the Hardy space

Restrictions and Extensions of Semibounded Operators

We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.

Essential spectra of quasi-parabolic composition operators on Hardy spaces of analytic functions

In this work we study the essential spectra of composition operators on Hardy spaces of analytic functions which might be termed as “quasi-parabolic.” This is the class of composition operators on H2 with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form φ(z)=z+ψ(z), where ψ∈H∞(H) and ℑ(ψ(z))>ϵ>0. We especially examine the case where ψ is discontinuous at infinity. A new method is devised to show that this type of composition operator fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.

Solid hulls of quasi-Banach spaces of analytic functions and interpolation

Abstract We study solid hulls of abstract Hardy spaces of analytic functions on the unit disk. Interpolation of sublinear operators is used to description of solid hulls of quasi-Banach spaces generated by interpolation methods associated to classical Hardy spaces H p for 0 p 1 . We specialize our results to solid hulls of Hardy–Orlicz and Hardy–Lorentz spaces and we are able to characterize multipliers from Hardy–Lorentz spaces to any solid sequence space. We give, for example, a complete description of multipliers with the domain in Hardy–Lorentz–Zygmund spaces.

A characterization of upper triangular trace class matrices

As a consequence of the vector-valued Hardy inequality it is given a characterization of upper triangular trace class matrices completely similar to that of classical Hardy space of analytic functions H 1 , as may be found for instance in Pavlović's book. To cite this article: N. Popa, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Hardy spaces for the strip

In this paper we shall study Hardy spaces of analytic functions in a strip S . Our main result is on one hand an intrinsic characterization of the spaces and on the second that polynomials are dense. We also present an orthogonal (in H 2 ( S ) ) basis of polynomials.

Properties of Removable Singularities for Hardy Spaces of Analytic Functions

Removable singularities for Hardy spaces Hp(Ω) = {f ∈ Hol(Ω): |f|p ⩽ u in Ω for some harmonic u}, 0 < p < ∞ are studied. A set E = Ω is a weakly removable singularity for Hp(Ω\E) if Hp(Ω\E) ⊂ Hol(Ω), and a strongly removable singularity for Hp(Ω\E) if Hp(Ω\E) = Hp(Ω). The two types of singularities coincide for compact E, and weak removability is independent of the domain Ω. The paper looks at differences between weak and strong removability, the domain dependence of strong removability, and when removability is preserved under unions. In particular, a domain Ω and a set E ⊂ Ω that is weakly removable for all Hp, but not strongly removable for any Hp(Ω\E), 0 < p < ∞, are found. It is easy to show that if E is weakly removable for Hp(Ω\E) and q > p, then E is also weakly removable for Hq(Ω\E). It is shown that the corresponding implication for strong removability holds if and only if q/p is an integer. Finally, the theory of Hardy space capacities is extended, and a comparison is made with the similar situation for weighted Bergman spaces.

Banach envelopes of vector valued H p spaces

In the paper we describe the Banach envelopes of Hardy spaces of analytic functions of several variables on polydiscs taking values in quasi-Banach spaces.

Hausdorff matrices and composition operators

We consider Hausdorff matrices as operators on Hardy spaces of analytic functions. When the generating sequence of the matrix is the moment sequence of a measure $\mu$, we find conditions on $\mu$ such that the matrix represents a bounded operator. The results unify and extend some known special cases of operators on Hardy spaces such as the Cesaro and generalized Cesaro operators.

Best Meromorphic Approximation of Markov Functions on the Unit Circle

Let E \subset(-1,1) be a compact set, let μ be a positive Borel measure with support \supp μ =E , and let H p (G),1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z \colon |z|=1} . Let Δ n,p be the error in best approximation of the Markov function \frac{1}{2π i} ∈t_E \frac{d μ(x)}{z-x} in the space L p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H p (G),Q is a polynomial of degree at most n , Q\not \equiv 0 . We investigate the rate of decrease of Δ n,p ,1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1 - ∈fty.

Invariant Subspaces and Nevanlinna–Pick Kernels

A theorem of Beurling–Lax–Halmos represents a subspace M of H2C(D)—the Hardy space of analytic functions with values in the Hilbert space E and square summable power series—invariant for multiplication by z as ΦH2F, where F is a subspace of E and Φ is an inner function with values in L(F, E). When the Hardy space is replaced by the Hilbert space H(k) determined by a Nevanlinna–Pick kernel k, such as the Dirichlet kernel or the row contraction kernel on the ball in Cd, the BLH Theorem survives with F an auxiliary Hilbert space and Φ a L(F, E) valued function which is inner in the sense that the operator MΦ of multiplication by Φ is a partial isometry. Under mild additional hypotheses, when E=C, Mz, the operator of multiplication by z, is cellularly indecomposable and has the codimension one property; however, if M is invariant for Mz, M⊖MzM need not be a cyclic subspace for Mz restricted to M.