Poisson Formula Research Articles

On the Summation of Series in Terms of Bessel Functions

In this article we deal with summation formulas for the series \sum_{n=1}^\infty\frac{J_\mu(nx)}{n^\nu} , referring partly to some results from our paper in J. Math. Anal. Appl. 247 (2000) 15–26. We show how these formulas arise from different representations of Bessel functions. In other words, we first apply Poisson's or Bessel's integral, then in the sequel we define a function by means of the power series representation of Bessel functions and make use of Poisson's formula. Also, closed form cases as well as those when it is necessary to take the limit have been thoroughly analyzed.

A unified framework for the Sussman, Moyal, and Janssen formulas

This paper derives three fundamental identities in the radar and sonar literature, namely, Sussman' identity for ambiguity functions, Moyal's formula which establishes the value of the inner product between two scattering functions, and Janseen's formula which establishes identities for mixed inner products between waveforms and Gabor wavelets. Starting from the fundamental convolution identity, we derive Sussman's identity. Following from an initial value theorem of Fourier analysis, we obtained Moyal's formula. Following from Poisson's sum formula and an initial value theorem, we also obtained Janssen's equality. The relationship between these three identities is as follows: Janssen's formula is a sampled-data version of Moyal's formula, and both follow from Sussman's identity. In turn, Sussman's identity is a consequence of the fundamental convolution identity.

A gaussian space of test functions

AbstractElsewhere we develop an aspect of analysis on the simplest space of type G/K when G = SL2(C) and K is the unitary subgroup, having to do with the analogue of Poisson's inversion formula and resulting additive zeta functions obtained by applying the Gauss transform. The present paper carries out systematically a number of foundational properties of gaussians on that space, which serve as test functions, give rise to explicit formulas, and lead immediately into the heat kernel, which is a gaussian. We prove approximation properties and the fact that the space of gaussians is dense in essentially anything space one wants. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Sur les fonctions harmoniques bornées sur les groupes de Lie résolubles connexes et l'existence de mesure invariante

LetG be a connected solvable Lie group with abelian derived group and μ a Borel measure onG. We define the μ-harmonic functions as the bounded Borel measurable solutions of the equation:h(g)=∫Gh(gy)μ(dy). We prove that if μ is a spread-out measure, bounded μ-harmonic functions are given by a Poisson formula, where the Poisson boundary is characterized in terms of the asymptotic behaviour of the right random walk of law μ onG. Moreover, we give a complete description of the Poisson boundary for certain groups. The Poisson formula remains essentially valid for adapted μ and left uniformly continuous bounded harmonic functions.

Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities

The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures, so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model and establish an analogue of the Poisson formula for (random) bounded harmonic functions. Under natural conditions on transition probabilities we identify these boundaries for several classes of groups with hyperbolic properties and prove the boundary triviality (i.e., the absence of non-constant random bounded harmonic functions) for groups of subexponential growth, in particular, for nilpotent groups.

Two complete and minimal systems associated with the zeros of the Riemann zeta function

Two complete and minimal systems associated with the zeros of the Riemann zeta function

A solution to the downward continuation effect on the geoid determined by Stokes' formula

The analytical continuation of the surface gravity anomaly to sea level is a necessary correction in the application of Stokes' formula for geoid estimation. This process is frequently performed by the inversion of Poisson's integral formula for a sphere. Unfortunately, this integral equation corresponds to an improperly posed problem, and the solution is both numerically unstable, unless it is well smoothed, and tedious to compute. A solution that avoids the intermediate step of downward continuation of the gravity anomaly is presented. Instead the effect on the geoid as provided by Stokes' formula is studied directly. The practical solution is partly presented in terms of a truncated Taylor series and partly as a truncated series of spherical harmonics. Some simple numerical estimates show that the solution mostly meets the requests of a 1-cm geoid model, but the truncation error of the far zone must be studied more precisely for high altitudes of the computation point. In addition, it should be emphasized that the derived solution is more computer efficient than the detour by Poisson's integral.

Existence of resonances in magnetic scattering

The Schrödinger operator with a compactly supported magnetic field is shown to produce infinitely many resonances, in any odd dimension⩾3. The proof is based on the Poisson formula for resonances and properties of the magnetic heat invariants.

Four Ways to Evaluate a Poisson Integral

Our integral is one of several known as the Poisson Integral; all are related in some way to Poisson's integral formula, which recovers an analytic function on the disk from its boundary values, a relationship we mention below. However, none of our methods involves complex analysis at all. The first one uses Riemann sums and relies on a trigonometric identity. The second method is based on a functional equation and involves a sequence of integral substitutions. The third method uses parametric differentiation and the half-angle substitution. We finish with an approach based on infinite series. It is interesting to see how wide a range of mathematical topics are exploited. These evaluations are suitable for an advanced calculus class and provide a very nice application of Riemann sums, functional equations, parametric differentiation, and infinite series. We begin with three elementary observations:

The Cauchy problem for the wave equation using the decomposition method

This article presents the Adomian decomposition series solution to the Cauchy problem for the wave equation, which is a much simpler solution than that given by the Poisson formula. Further, the method is made applicable to various types of boundary conditions.

Generalized series of Bessel functions

Known series of Bessel functions, currently available in handbooks, and many of Neumann type, are generalized to arbitrary order. The underlying result is a Poisson formula due to Titchmarsh. This formula gives rise to a Neumann series involving modified Bessel functions of integral order. The latter is the basis of many of the generalized series that follow. Included are examples of generalized trigonometric identities. The paper concludes by indicating the wide range of results that can be obtained.

The Poisson Formula for Groups with Hyperbolic Properties

The Poisson boundary of a group G with a probability measure „ is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded „-harmonic functions on G. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks. It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries for several classes of groups: word hyperbolic groups and discontinuous groups of isometries of Gromov hyperbolic spaces, groups with inflnitely many ends, cocompact lattices in Cartan-Hadamard manifolds, discrete subgroups of semisimple Lie groups.

The Beta Poisson dose-response model is not a single-hit model.

The choice of a dose-response model is decisive for the outcome of quantitative risk assessment. Single-hit models have played a prominent role in dose-response assessment for pathogenic microorganisms, since their introduction. Hit theory models are based on a few simple concepts that are attractive for their clarity and plausibility. These models, in particular the Beta Poisson model, are used for extrapolation of experimental dose-response data to low doses, as are often present in drinking water or food products. Unfortunately, the Beta Poisson model, as it is used throughout the microbial risk literature, is an approximation whose validity is not widely known. The exact functional relation is numerically complex, especially for use in optimization or uncertainty analysis. Here it is shown that although the discrepancy between the Beta Poisson formula and the exact function is not very large for many data sets, the differences are greatest at low doses--the region of interest for many risk applications. Errors may become very large, however, in the results of uncertainty analysis, or when the data contain little low-dose information. One striking property of the exact single-hit model is that it has a maximum risk curve, limiting the upper confidence level of the dose-response relation. This is due to the fact that the risk cannot exceed the probability of exposure, a property that is not retained in the Beta Poisson approximation. This maximum possible response curve is important for uncertainty analysis, and for risk assessment of pathogens with unknown properties.

A Poisson formula for harmonic functions on the Sierpi\nski gasket

We show a Poisson formula for bounded harmonic functions F on the Sierpinski gasket S. We construct a boundaryO~ S; ~ nU, a measurable action of a semigroup W on ~ and a map G : W! S, such that for every bounded harmonic function F on S

On some applications and properties of the random transects

A Monte Carlo model employing random transects is proposed to determine the distribution of properties of duplex materials. The model allows determination of the probability that the value of a specified property will be smaller than the average of the smallest values in several sets of measurements. The probability is determined without prior knowledge of the actual variation of the property as a function of the intercept from the second microstructural constituent. Some limitations of Poisson's formula to describe the fraction covered by overlapping objects are also discussed and a more general equation is proposed. An important property of the random transects is derived, concerning all systematic point-counting methods for determining the volume fraction of a selected constituent. It is established that for a duplex-phase microstructure without anisotropy, the correlation between the sampling results of the grid nodes does not depend on the amount of the second phase or on the size of the aggregates it forms. The only controlling parameter is the size of the largest grain from the second phase which would have been formed if no impingement existed during its growth.

A rapidly convergent modified Green' function for Laplace' equation in a rectangular region

A computationally useful expression for the modified Green' function in a rectangular region is derived. This can be used to obtain solutions of Laplace' equation or Poisson' equation for the electrical potential associated with any configuration of current sources that satisfies the condition of charge balance, and represents a generalization of a well–known result due to Newman applicable to an infinite channel. The starting point for the analysis is the very slowly converging Fourier–series expansion of the modified Green' function. This Fourier series can be separated into a Fourier series in one variable that can be summed in closed form, and a double Fourier series that can be partly summed with respect to one index and transformed by application of the Poisson summation formula and integration by residues. The result of applying the Poisson formula can be expressed either as a rapidly converging series of logarithmic functions, or as a rapidly converging Fourier series which can be integrated analytically with respect to the coordinates of the source point. The use of the modified Green' function is demonstrated first by determining potential distributions that arise from specification of uniform current density on two or more of the boundary surfaces. The potential in a cell with two electrodes embedded in opposite sides of the rectangle is then calculated, and circumstances in which this potential approaches the potential in an infinite channel are determined.

Mathematical basis, correlativity and selection of seismicity parameters

Using the analytic expression of seismicity parameters, the dependency and correlativity between the statistical parameters and seismic frequency or seismic intensity are discussed. The statistical parameters are divided into two kinds. One kind is the regional seismicity parameter, 17 parameters are analyzed in this paper. The other kind is the seismicity distribution parameter. They are the distribution parameters of time, space and magnitude. The existent base and rationality of distribution parameters depend on the application of distribution model. We analyze and draw an analogy between the natural probability, Poisson, Weibull distributions and multi-fractal analytic formula in time, space and magnitude. And some examples are given in this paper. The P value and H value of aftershock sequence attenuation, the U value and F value of earthquake swarm sequence and the entropy of information are discussed preliminarily. Another method about analyzing relationships among time-series curves are given. The resemblance relativity degree R Y 1 Y 2and the relativity degree G Y 1 Y 2 of relative change slope can be used as the determining values. At last, some preliminary ideas about sifting and using for seismicity parameters are advanced in this paper.

研究展望: 高速多重極展開法-粒子法への応用を中心として

We overview the Fast Multipole Method (FMM) and the Barnes-Hut tree method. These algorithms evaluate mutual gravitational interaction between N particles in O (N) or O (N log N) times, respectively. We present basic algorithms as well as recent developments, such as Anderson's method of using Poisson's formula, the use of FFT, and other optimization techniques. We also summarize the current states of two algorithms. Though FMM with O (N) scaling is theoretically preferred over O (N log N) tree method, comparisons of existing implementations proved otherwize.

Properties of the mappings that are close to the harmonic mappings

We continue studying the mappings that are close to the harmonic mappings (e-quasiharmonic mappings with e small). This study originates with the previous articles of the author. The results of the article include a theorem on connection between the notion of e-quasiharmonic mapping and the solutions to Beltrami systems, an analog to the arithmetic mean property of harmonic functions for e-quasiharmonic mappings, a theorem on stability in the Poisson formula for harmonic mappings in the ball, and a theorem on the local smoothing of e-quasiharmonic mappings with e small which preserves proximity to the harmonic mappings.

Convergence acceleration of Fourier series by analytical and numerical application of Poisson's formula

A form of the Poisson summation formula appropriate for the transformation of infinite Fourier series is derived and shown to be applicable to slowly converging series arising in the solution of Laplace's equation in a rectangular electrochemical cell. The result is a rapidly convergent Fourier series which is particularly useful in giving the potential close to the electrode surface. Results obtained from the analytical expression are found to be in close agreement with those resulting from numerical evaluation of the Fourier integrals (using Simpson's rule and the Euler transformation) and direct summation by using Goertzel's (1958) algorithm.