On Hankel conjugate functions

2004 ◽  
Vol 41 (1) ◽  
pp. 59-91 ◽  
Author(s):  
J. J. Betancor ◽  
K. Stempak ◽  
P. Vértesi
Keyword(s):  
Spectral Decomposition ◽  
Hankel Transform ◽  
Conjugate Functions ◽  
Boundary Values ◽  
Poisson Integrals ◽  
Poisson Semigroup ◽  
Adjoint Extension ◽  
Generalized Hilbert Transform ◽  
Conjugate Poisson Integrals ◽  
Classic Paper

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 > 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.

Cauchy and Poisson integral representations for ultradistributions of compact support and distributional boundary values

1981 ◽  
Vol 91 (1-2) ◽  
pp. 49-62 ◽  
Author(s):  
R. S. Pathak
Keyword(s):  
Analytic Function ◽  
Growth Condition ◽  
Compact Support ◽  
Analytic Functions ◽  
Convex Cone ◽  
Integral Representations ◽  
Boundary Values ◽  
Cauchy Integral ◽  
Boundedness Condition ◽  
Poisson Integrals

SynopsisUltradistributions of compact support are represented as the boundary values of Cauchy and Poisson integrals corresponding to tubular radial domains Tc' =ℝn + iC', C'⊂⊂C, where C is an open, connected, convex cone. The Cauchy integral of is shown to be an analytic function in TC' which satisfies a certain boundedness condition. Analytic functions which satisfy a specified growth condition in TC' have a distributional boundary value which can be used to determine an distribution.

scholarly journals A symmetrized conjugacy scheme for orthogonal expansions

2013 ◽  
Vol 143 (2) ◽  
pp. 427-443 ◽  
Author(s):  
Adam Nowak ◽  
Krzysztof Stempak
Keyword(s):  
Differential Operator ◽  
Higher Order ◽  
Riesz Transforms ◽  
Orthogonal System ◽  
Order Differential Operator ◽  
Orthogonal Expansions ◽  
Partial Derivatives ◽  
Poisson Integrals ◽  
Conjugate Poisson Integrals ◽  
Classical Shape

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.

scholarly journals Asymptotics of approximation of conjugate functions by Poisson integrals

2019 ◽  
Vol 22 (2) ◽  
pp. 235-243
Author(s):  
Yu. I. Kharkevych ◽  
K. V. Pozharska
Keyword(s):  
Upper Bound ◽  
Arbitrary Order ◽  
Conjugate Functions ◽  
Periodic Functions ◽  
Poisson Integrals

We obtain a decomposition of the upper bound for the deviation of Poisson integrals of conjugate periodic functions. The decomposition enables one to provide the Kolmogorov–Nikol'skii constants of an arbitrary order.

scholarly journals Asymptotics of approximation of functions by conjugate Poisson integrals

2020 ◽  
Vol 12 (1) ◽  
pp. 138-147
Author(s):  
I.V. Kal'chuk ◽  
Yu.I. Kharkevych ◽  
K.V. Pozharska
Keyword(s):  
Arbitrary Order ◽  
Extremal Problems ◽  
Asymptotic Distributions ◽  
Lipschitz Class ◽  
Periodic Functions ◽  
Approximation Of Functions ◽  
Poisson Integrals ◽  
Wide Range ◽  
Functional Classes ◽  
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.

X-Ray Analysis of Frozen Hydrated Sections:The Unconventional Approach

1982 ◽  
Vol 40 ◽  
pp. 754-757
Author(s):  
R. Beeuwkes ◽  
A. Saubermann ◽  
P. Echlin ◽  
S. Churchill
Keyword(s):  
Ratio Method ◽  
Frozen Sections ◽  
Technical Problems ◽  
Sample Handling ◽  
X Ray ◽  
Common Group ◽  
Ideal Method ◽  
Classic Paper ◽  
Physical Principles ◽  
The Ideal

Fifteen years ago, Hall described clearly the advantages of the thin section approach to biological x-ray microanalysis, and described clearly the ratio method for quantitive analysis in such preparations. In this now classic paper, he also made it clear that the ideal method of sample preparation would involve only freezing and sectioning at low temperature. Subsequently, Hall and his coworkers, as well as others, have applied themselves to the task of direct x-ray microanalysis of frozen sections. To achieve this goal, different methodological approachs have been developed as different groups sought solutions to a common group of technical problems. This report describes some of these problems and indicates the specific approaches and procedures developed by our group in order to overcome them. We acknowledge that the techniques evolved by our group are quite different from earlier approaches to cryomicrotomy and sample handling, hence the title of our paper. However, such departures from tradition have been based upon our attempt to apply basic physical principles to the processes involved. We feel we have demonstrated that such a break with tradition has valuable consequences.

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