Mean values and classes of harmonic functions

1984 ◽  
Vol 96 (3) ◽  
pp. 501-505 ◽  
Author(s):  
Thomas Ramsey ◽  
Yitzhak Weit
Keyword(s):  
Unit Circle ◽  
Harmonic Functions ◽  
Borel Measure ◽  
Mean Value ◽  
Finite Complex ◽  
Mean Values ◽  
Complex Borel Measure ◽  
Generalized Mean ◽  
Image Position

Let μ be a finite complex Borel measure supported on the unit circle.In this paper, we are concerned with the characterization of the sets of functions satisfying the generalized mean value equation of the form.and for all ξ ∈ , | ξ | = R for some fixed R > 0.

Multipliers of Fractional Cauchy Transforms and Smoothness Conditions

1998 ◽  
Vol 50 (3) ◽  
pp. 595-604 ◽  
Author(s):  
Donghan Luo ◽  
Thomas Macgregor
Keyword(s):  
Analytic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Borel Measure ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cauchy Transforms ◽  
The Family ◽  
Complex Borel Measure

AbstractThis paper studies conditions on an analytic function that imply it belongs to Mα, the set of multipliers of the family of functions given by where μ is a complex Borel measure on the unit circle and α > 0. There are two main theorems. The first asserts that if 0 < α < 1 and sup. The second asserts that if 0 < α < 1, ƒ ∈ H∞ and supt. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

scholarly journals Radial growth and exceptional sets for Cauchy–Stieltjes integrals

1994 ◽  
Vol 37 (1) ◽  
pp. 73-89 ◽  
Author(s):  
D. J. Hallenbeck ◽  
T. H. MacGregor
Keyword(s):  
Unit Circle ◽  
Radial Growth ◽  
Borel Measure ◽  
Local Conditions ◽  
Exceptional Sets ◽  
Image Position ◽  
Complex Valued

This paper considers the radial and nontangential growth of a function f given bywhere α>0 and μ is a complex-valued Borel measure on the unit circle. The main theorem shows how certain local conditions on μ near eiθ affect the growth of f(z) as z→eiθ in Stolz angles. This result leads to estimates on the nontangential growth of f where exceptional sets occur having zero β-capacity.

A theorem on positive harmonic functions

1949 ◽  
Vol 45 (2) ◽  
pp. 207-212 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Unit Circle ◽  
Harmonic Functions ◽  
Closed Interval ◽  
Positive Harmonic Function ◽  
Image Position ◽  
Function Conjugate ◽  
Positive Harmonic Functions

1. Let z = reiθ, and let h(z) denote a (regular) positive harmonic function in the unit circle r < 1. Then h(r) (1−r) and h(r)/(1 − r) tend to limits as r → 1. The first limit is finite; the second may be infinite. Such properties of h can be obtained in a straightforward way by using the fact that we can writewhere α(phgr) is non-decreasing in the closed interval (− π, π). Another method is to writewhere h* is a harmonic function conjugate to h. Then the functionhas the property | f | < 1 in the unit circle. Such functions have been studied by Julia, Wolff, Carathéodory and others.

scholarly journals MEAN VALUE CHARACTERIZATION OF 'USEFUL' INFORMATION MEASURES

1993 ◽  
Vol 24 (4) ◽  
pp. 383-394
Author(s):  
U. S. BRAKER ◽  
D. S. HOODA
Keyword(s):  
Mean Value ◽  
Comparison Theorems ◽  
Present Communication ◽  
Information Measures ◽  
Generalized Mean

In the present communication the generalized mean value characterization of 'useful' information and relativeinformation measures has been studied. Some comparison theorems related to these measures have also been proved.

scholarly journals On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties

2014 ◽  
Vol 34 (7) ◽  
pp. 2779-2793 ◽  
Author(s):  
Fausto Ferrari ◽  
◽  
Qing Liu ◽  
Juan Manfredi ◽  
Keyword(s):  
Heisenberg Group ◽  
Harmonic Functions ◽  
Mean Value

scholarly journals Asymptotic results for multiplexing subexponential on-off processes

1999 ◽  
Vol 31 (02) ◽  
pp. 394-421 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Aurel A. Lazar
Keyword(s):  
Queueing System ◽  
Random Variable ◽  
Mean Value ◽  
Activity Period ◽  
Arrival Process ◽  
Asymptotic Results ◽  
Fluid Queue ◽  
Arrival Processes ◽  
Image Position

Consider an aggregate arrival process A N obtained by multiplexing N on-off processes with exponential off periods of rate λ and subexponential on periods τon. As N goes to infinity, with λN → Λ, A N approaches an M/G/∞ type process. Both for finite and infinite N, we obtain the asymptotic characterization of the arrival process activity period. Using these results we investigate a fluid queue with the limiting M/G/∞ arrival process A t ∞ and capacity c. When on periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable Q t P observed at the beginning of the arrival process activity periods where ρ = 𝔼A t ∞ &lt; c; r (c ≤ r) is the rate at which the fluid is arriving during an on period. The asymptotic (time average) queue distribution lower bound is obtained under more general assumptions regarding on periods than regular variation. In addition, we analyse a queueing system in which one on-off process, whose on period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate 𝔼e t . This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value 𝔼e t .

scholarly journals A New Characterization of Rain and Clouds: Results from a Statistical Inversion of Count Data

2007 ◽  
Vol 64 (6) ◽  
pp. 2012-2028 ◽  
Author(s):  
A. R. Jameson
Keyword(s):  
Count Data ◽  
Drop Size ◽  
Mean Value ◽  
Probability Density Functions ◽  
Density Functions ◽  
Mean Values ◽  
Precise Meaning ◽  
Statistical Heterogeneity ◽  
Single Set

Most variables in meteorology are statistically heterogeneous. The statistics of data from several different locations, then, can be thought of as an amalgamation of information contained in several contributing probability density functions (PDFs) having different sets of parameters, different parametric forms, and different mean values. The frequency distribution of such data, then, will often be multimodal. Usually, however, in order to achieve better sampling, measurements of these variables over an entire set of data gathered at widely disparate locations are processed as though the data were statistically homogeneous, that is, as though they were fully characterized by just one PDF and one single set of parameters having one mean value. Is there, instead, a better way of treating the data in a manner that is consistent with this statistical heterogeneity? This question is addressed here using a statistical inversion technique developed by Tarantola based upon Bayesian methodology. Two examples of disdrometer measurements in real rain, one 16 h and the other 3 min long, reveal the presence of multiple mean values of the counts at all the different drop sizes. In both cases the heterogeneous rain can be decomposed into five–seven statistically homogeneous components, each characterized by its own steady drop size distribution. Concepts such as stratiform versus convective rain can be given more precise meaning in terms of the contributions each component makes to the rain. Furthermore, this discovery permits the explicit inclusion of statistical heterogeneity into some analytic theories.

scholarly journals Spline Parameterization of Complex Planar Domains for Isogeometric Analysis

2017 ◽  
Vol 47 (1) ◽  
pp. 18-35 ◽  
Author(s):  
Sangamesh Gondegaon ◽  
Hari K. Voruganti
Keyword(s):  
Unit Circle ◽  
Isogeometric Analysis ◽  
Harmonic Functions ◽  
Geometric Model ◽  
Mean Value ◽  
Mapping Method ◽  
Control Points ◽  
Computationally Efficient ◽  
B Spline ◽  
Domain Mapping

Abstract Isogeometric Analysis (IGA) involves unification of modelling and analysis by adopting the same basis functions (splines), for both. Hence, spline based parametric model is the starting step for IGA. Representing a complex domain, using parametric geometric model is a challenging task. Parameterization problem can be defined as, finding an optimal set of control points of a B-spline model for exact domain modelling. Also, the quality of parameterization, too has significant effect on IGA. Finding the B-spline control points for any given domain, which gives accurate results is still an open issue. In this paper, a new planar B-spline parameterization technique, based on domain mapping method is proposed. First step of the methodology is to map an input (non-convex) domain onto a unit circle (convex) with the use of harmonic functions. The unique properties of harmonic functions: global minima and mean value property, ensures the mapping is bi-jective and with no self-intersections. Next step is to map the unit circle to unit square to make it apt for B-spline modelling. Square domain is re-parameterized by using conventional centripetal method. Once the domain is properly parameterized, the required control points are computed by solving the B-spline tensor product equation. The proposed methodology is validated by applying the developed B-spline model for a static structural analysis of a plate, using isogeometric analysis. Different domains are modelled to show effectiveness of the given technique. It is observed that the proposed method is versatile and computationally efficient.

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