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

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

We introduce a method for the fast numerical approximation of linear, second-order parabolic partial differential equations (PDEs for short) with time-independent coefficients based on model order reduction techniques and the Laplace transform. We start by applying this transform to the evolution problem, thus yielding a time-independent boundary value problem solely depending on the complex Laplace variable. In an offline stage, we judiciously sample the Laplace variable and numerically solve the corresponding collection of high-fidelity or full-order problems. Next, we apply a proper orthogonal decomposition (POD) to this collection of solutions in order to obtain a reduced basis in the Laplace domain. We project the linear parabolic problem onto this basis and then, using any suitable time-stepping method, we solve the evolution problem. A key insight to justify the implementation and analysis of the proposed method consists of using Hardy spaces of analytic functions and establishing, through the Paley-Wiener theorem, an isometry between the solution of the time-dependent problem and its Laplace transform. As a result, one may conclude that computing a POD with samples taken in the Laplace domain produces an exponentially accurate reduced basis for the time-dependent problem. Numerical experiments illustrate the performance of the method in terms of accuracy and, in particular, speed-up when compared to the solution obtained by solving the full-order model.

The object of this paper is to present generalizations of some recent results ot S.S. Miller and P.T. Mocanu on the univalence and starlikeness of Kummer' confluent hypergeometric function. Some applications involving generalized hypergeometric functions associated with the Hardy space of analytic functions are also considered.

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.

We study the Cauchy singular integral operator SwI on (−1, 1), where |w| is a generalized Jacobi weight. This operator is considered in pairs of weighted spaces of continuous functions, where the weights u and v are generalized Jacobi weights with nonnegative exponents such that |w| = u/v. We introduce a certain polynomial approximation space which is well appropriate to serve as domain of definition of SwI. A description of this space in terms of smoothness properties shows that it can be viewed as a limit case of weighted Besov spaces of continuous functions. We use our results to prove necessary and sufficient conditions for the continuity of operators awI + SbwI and \(\varrho^{-1}(aw\varrho I\,+\,bSw\varrho I),\,\varrho^{-1}\,\in\,b^{-1}\Pi\) (Π: set of all algebraic polynomials), in certain pairs of Ditzian-Totik type Besov spaces.

The theory of Hardy spaces has close connections to many branches of mathematics including Fourier analysis, harmonic analysis, singular integrals, potential theory and operator theory, and has found essential applications in robust control engineering. For each application, the ability to represent elements of these classes by series or integral formulas is of utmost importance. This self-contained text provides an introduction to a wide range of representation theorems and provides a complete description of the representation theorems with direct proofs for both classes of Hardy spaces: Hardy spaces of the open unit disc and Hardy spaces of the upper half plane. With over 300 exercises, many with accompanying hints, this book is ideal for those studying Advanced Complex Analysis, Function Theory or Theory of Hardy Spaces. Advanced undergraduate and graduate students will find the book easy to follow, with a logical progression from basic theory to advanced research.

Some new function spaces and their applications to Harmonic analysis

Traces of functions in Hardy and Besov spaces on Lipschitz domains with applications to compensated compactness and the theory of Hardy and Bergman type spaces

A series of results similar to Marcinkiewicz's theorem on the interpolation of operators is put forward. The difference from the classical forms of this theorem is that spaces of integrable functions are replaced by some function classes that are extensions of various Hardy spaces. Some applications of these results to the extension of Carleson's embedding theorem and the Hardy-Littlewood inequalities for analytic functions in Hardy classes are presented. Bibliography: 41 titles.

In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity.

Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases

In the Hardy space Hp(Dϱ), 1 ≤ p ⪯ ∞, of functions analytic in the disk Dϱ = {z ∈ ℂ}: z < ϱ, we denote by NHp(Dϱ), N > 0, the class of functions whose Lp-norm on the circle γϱ = {z ∈ ℂ: z = ϱ} does not exceed the number N and by ∂Hp(Dϱ) the class consisting of the derivatives of functions from 1Hp(Dϱ). We consider the problem of the best approximation of the class ∂Hp(Dϱ) by the class NHp(DR)N > 0, with respect to the Lp-norm on the circle γr, 0 < r < ρ < R. The order of the best approximation as N → +∞ is found: $$\varepsilon (\partial {H^p}({D_\rho }),N{H^p}){)_{{L^p}({\Gamma _r})}} \asymp {N^{ - \beta }}^{/\alpha }{\ln ^{1/\alpha }}N,\alpha = \frac{{\ln R - \ln \rho }}{{\ln R - \ln r}},\beta = 1 - \alpha.$$ In the case where the parameter N belongs to some sequence of intervals, the exact value of the best approximation and a linear method implementing it are obtained. A similar problem is considered for classes of functions analytic in annuli.

A result on the necessity and sufficiency of conditions for the Fredholmness of singular integral operators in weighted Lp -spaces on the interval (0, 1) of the real axis ℝ is proved together with a respective Fredholm index formula. These operators are composed by the Cauchy singular integral operator, Mellin operators and operators of multiplication by piecewise continuous functions.

A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the -algebra generated by all multiplication operators by piecewise quasicontinuous (PQC) functions, by the Cauchy singular integral operator and by the unitary weighted shift operators , acting on the space over the unit circle . Here G denotes a discrete amenable group of orientation-preserving piecewise smooth homeomorphisms with finite sets of discontinuities for their derivatives , which acts topologically freely on , where is the interior of the nonempty closed set composed by all common fixed points for all shifts , with boundary of zero Lebesgue measure. A Fredholm symbol calculus for the -algebra is constructed and a Fredholm criterion for the operators is established.

in this paper we give a mathematical formulation of the Riemann Hilbert problem. We then proceed to show scaler and the solution of scaler of Riemann-Hilbert problem arises in the theory to this. To do this we formula and solve some problems and solution. the Solution of Scaler Riemann Hilbert problem depend to Holder Condition, pemelj formula, and Hardy spaces