Conjugate Poisson Integrals Research Articles

Representation of Weierstrass integral via Poisson integrals

In our research, we have presented a second-order linear partial differential equation in polar coordinates. Considering this differential equation on the unit disk, we have obtained a one-dimensional heat equation. It is well-known that the heat equation can be solved taking into account the boundary condition for the general solution on the unit circle. In our paper, the boundary-value problem is solved using the well-known method called the separation of variables. As a result, the general solution to the boundary-value problem is presented in terms of the Fourier series. Then the expressions for the Fourier coefficients are used to transform the Fourier series expansion for the general solution to the boundary-value problem into the so-called Weierstrass integral, which is represented via the so-called Weierstrass kernel. A representation of the Weierstrass kernel via the infinite geometric series is derived by a way allowing a complicated function to be parameterized via a simplified function. The derivation of the corresponding parametrization is based on two well-known integrals. As a result, a complicated function of the natural argument is represented in the form of a double integral that contains a simplified function of the same natural argument. So, the double-integral representation of the Weierstrass kernel has been derived. To obtain this result, the integral representation of the so-called Dirac delta function is taken into account. The expression found for the Weierstrass kernel is substituted into the expression for the Weierstrass integral. As a result, it was found that the Weierstrass integral can be considered a double-integral that contains the Poisson and conjugate Poisson integrals.

Representation of Weierstrass integral via Poisson integrals

In our research, we have presented a second-order linear partial differential equation in polar coordinates. Considering this differential equation on the unit disk, we have obtained a one-dimensional heat equation. It is well-known that the heat equation can be solved taking into account the boundary condition for the general solution on the unit circle. In our paper, the boundary-value problem is solved using the well-known method called the separation of variables. As a result, the general solution to the boundary-value problem is presented in terms of the Fourier series. Then the expressions for the Fourier coefficients are used to transform the Fourier series expansion for the general solution to the boundary-value problem into the so-called Weierstrass integral, which is represented via the so-called Weierstrass kernel. A representation of the Weierstrass kernel via the infinite geometric series is derived by a way allowing a complicated function to be parameterized via a simplified function. The derivation of the corresponding parametrization is based on two well-known integrals. As a result, a complicated function of the natural argument is represented in the form of a double integral that contains a simplified function of the same natural argument. So, the double-integral representation of the Weierstrass kernel has been derived. To obtain this result, the integral representation of the so-called Dirac delta function is taken into account. The expression found for the Weierstrass kernel is substituted into the expression for the Weierstrass integral. As a result, it was found that the Weierstrass integral can be considered a double-integral that contains the Poisson and conjugate Poisson integrals.

Asymptotics of approximation of functions by conjugate Poisson integrals

Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha $, i.e. the class of continuous $ 2\pi $-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0<\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha $. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.

Fourier Spectrum Characterizations of Hardy Spaces $$H^p$$ on Tubes for $$0<p<1$$

As continuation of the study of Fourier spectrum characterization of higher-dimensional Hardy spaces $$H^p(T_{\Gamma })$$ on tubes for $$1\le p\le \infty $$ , this paper aims to obtain analogous Fourier spectrum characterizations and integral representation formulas of higher-dimensional Hardy spaces $$H^p(T_{\Gamma })$$ on tubes for the index range $$0< p < 1$$ . For $$1\le p\le \infty $$ , the $$H^p(T_{\Gamma })$$ are well understood via the Poisson and conjugate Poisson integrals. However, for $$0< p < 1$$ , those integrals are no longer defined that requires more delicate analysis.

A symmetrized conjugacy scheme for orthogonal expansions

We establish a symmetrization procedure in the context of general orthogonal expansions associated with a second-order differential operator L, a Laplacian. Combining with a unified conjugacy scheme from an earlier paper by Nowak and Stempak permits, using a suitable embedding, a differential-difference Laplacian $\mathbb{L}$ to be associated with the initially given orthogonal system of eigenfunctions of L, so that the resulting extended conjugacy scheme has the natural classical shape. This means, in particular, that the related partial derivatives decomposing $\mathbb{L}$ are skew-symmetric in an appropriate L2 space and they commute with Riesz transforms and conjugate Poisson integrals. The results also shed new light on the issue of defining higher-order Riesz transforms for general orthogonal expansions.

Hilbert transform and related topics associated with the differential Jacobi operator on (0, +∞)

We define Hilbert transform and conjugate Poisson integrals associated with the Jacobi differential operator on (0, +∞). We prove that these operators are bounded in the appropriate Lebesgue spaces L p , 1 < p < +∞. In this study, the tools used are the Littlewood–Paley g-functions associated with the Poisson semigroup and the supplementary Poisson semigroup which we introduce in this paper.

Hilbert Transforms Associated with Dunkl-Hermite Polynomials

We consider expansions of functions in L p (R,|x| 2k dx), 1 p < +1 with respect to Dunkl-Hermite functions in the rank-one setting. We actually define the heat-diffusion and Poisson integrals in the one-dimensional Dunkl setting and study their properties. Next, we define and deal with Hilbert transforms and conjugate Poisson integrals in the same setting. The formers occur to be Calderon-Zygmund operators and hence their mapping properties follow from general results.

Riesz transforms for Jacobi expansions

We define Riesz transforms and conjugate Poisson integrals associated with multi-dimensional Jacobi expansions. Under a slight restriction on the type parameters, we prove that these operators are bounded in L p , 1 < p < ∞, with constants independent of the dimension. Our tools are suitably defined g-functions and Littlewood-Paley-Stein theory, involving the Jacobi-Poisson semigroup and modifications of it.

Riesz transforms for multi-dimensional Laguerre function expansions

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of type α are defined and investigated. It is proved that for any multi-index α = ( α 1 , … , α d ) such that α i ⩾ − 1 / 2 , the appropriately defined Riesz–Laguerre transforms R j α , j = 1 , 2 , … , d , are Calderón–Zygmund operators in the sense of the associated space of homogeneous type, hence their mapping properties follow from the general theory. Similar results are obtained for all higher order Riesz–Laguerre transforms. The conjugate Poisson integrals are shown to satisfy a system of equations of Cauchy–Riemann type and to recover the Riesz–Laguerre transforms on the boundary.

Riesz transforms and conjugacy for Laguerre function expansions of Hermite type

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of Hermite type with index α are defined and investigated. It is proved that for any multi-index α = ( α 1 , … , α d ) such that α i ⩾ − 1 / 2 , α i ∉ ( − 1 / 2 , 1 / 2 ) , the appropriately defined Riesz transforms R j α , j = 1 , 2 , … , d , are Calderón–Zygmund operators, hence their mapping properties follow from a general theory. Similar mapping results are obtained in one dimension, without excluding α ∈ ( − 1 / 2 , 1 / 2 ) , by means of a local Calderón–Zygmund theory and weighted Hardy's inequalities. The conjugate Poisson integrals are shown to satisfy a system of Cauchy–Riemann type equations and to recover the Riesz–Laguerre transforms on the boundary. The two specific values of α, ( − 1 / 2 , … , − 1 / 2 ) and ( 1 / 2 , … , 1 / 2 ) , are distinguished since then a connection with Riesz transforms for multi-dimensional Hermite function expansions is established.

Higher Riesz transforms and imaginary powers associated to the harmonic oscillator

We consider higher order Riesz transforms for the multi-dimensional Hermite function expansions. The Riesz transforms occur to be Calderon--Zygmund operators hence their mapping properties follow by using results from a general theory. Then we investigate higher order conjugate Poisson integrals showing that at the boundary they approach appropriate Riesz transforms of a given function. Finally, we consider imaginary powers of the harmonic oscillator by using tools developed for studying Riesz transforms.

On Hankel conjugate functions

We consider some aspects of harmonic analysis of the differential operator . Spectral decomposition of its self-adjoint extension is given in terms of the Hankel transform Hν. We present a fairly detailed analysis of the corresponding Poisson semigroup {Pt}t &gt; 0: this is given in a weighted setting with Ap-weights involved. Then, we consider conjugate Poisson integrals of functions from Lp(w), w ∈ Ap, 1 ≦ p &lt; ∞. Boundary values of the conjugate Poisson integrals exist both in Lp(w) and a.e., and the resulting mapping is called the generalized Hilbert transform. Mapping properties of that transform are then proved. All this complements, in some sense, the analysis of conjugacy for the modified Hankel transform Hν which was initiated in the classic paper of Muckenhoupt and Stein [15], then continued in a series of papers by Andersen, Kerman, Rooney and others.

On Riesz transforms for Laguerre expansions

We prove that Riesz transforms and conjugate Poisson integrals associated with the multi-dimensional Laguerre semigroup are bounded in L p, 1<p<∞ . Our main tools are appropriately defined square functions and the Littlewood–Paley–Stein theory.

On Hankel conjugate functions

We consider some aspects of harmonic analysis of the differential operator Cv=−d2/dx2+{v2−1/4)/a?2,v>−1 . Spectral decomposition of its self-adjoint extension is given in terms of the Hankel transform Hν. We present a fairly detailed analysis of the corresponding Poisson semigroup {Pt}t > 0: this is given in a weighted setting with Ap-weights involved. Then, we consider conjugate Poisson integrals of functions from Lp(w), w ∈ Ap, 1 ≦ p < ∞. Boundary values of the conjugate Poisson integrals exist both in Lp(w) and a.e., and the resulting mapping is called the generalized Hilbert transform. Mapping properties of that transform are then proved. All this complements, in some sense, the analysis of conjugacy for the modified Hankel transform Hν which was initiated in the classic paper of Muckenhoupt and Stein [15], then continued in a series of papers by Andersen, Kerman, Rooney and others.

Images of non-tangential sectors under Cauchy transforms

Let μ be a complex measure on the circle. We denote byPμ andQμ the Poisson and conjugate Poisson integrals of μ in the unit disk, respectively. In this note, we study the relative asymptotic growth ofPμ andQμ near the support of μ. In particular, we show that on μ-almost every radius,Qμ grows no faster thanPμ. We also discuss related questions on the boundary behavior of analytic functions.

Poisson integrals and Riesz transforms for Hermite function expansions with weights

We consider expansions with respect to the multi-dimensional Hermite functions which are eigenfunctions of the harmonic oscillator L=− Δ+| x| 2. For the heat-diffusion and Poisson semigroups corresponding to a self-adjoint extension L of L we investigate their boundary behaviour and mapping properties. All this is done for functions from L p ( w), 1⩽ p<∞, w∈ A p . Then Riesz transforms and conjugate Poisson integrals are considered. The Riesz transforms occur to be Calderón–Zygmund operators hence their mapping properties follow by using results from a general theory.