Classical Moment Problem Research Articles

On the physical measures in weakly coupled EuclideanP(ϕ)2-theories

I t has been recent ly shown by N~wMA~ (1) tha t the Schwinger functions of the weakly coupled IP(~)2-theories are the moments 02 a measure t~ .a on the a-algebra generated by the cyl inder sets on ~'(R~). La te r FR(~HLICH (~) proved a somewhat s t ronger result of this type , i .e . t ha t the above-mentioned Schwinger functions are the moments of a unique and Euclidean invar iant measure on the ~-algebra generated by the cyl inder sets on 5~'(R2). The purpose of this paper is to give a much simpler proof of FrShlich result. This will be accomplished by making use of results due to BWREZANSKII (a) on the infinitedimensional generalizat ion of the classical power moment problem, and of a proposi t ion proved in a recent paper by GLIM~ and JAFF]~ (4). This last proposi t ion shows tha t the n-point Sehwinger functions of weakly coupled P(~0)2-theories are generalized funct ions on the nth tensor product of the Sobolev space H_~(R~). In addi t ion we remark tha t the present proof, as FrShlich's one, does not require a par t icu lar way of approaching the infinite-volume l imit , as i t will clearly appear in the following. As a l ready mentioned, we will app ly to the case under discussion a result due to

Estimation of the parameters of the logistic distribution from grouped samples

Abstract The classical Hamburger moment problem asks for the existence of a bounded, increasing function having certain prescribed moments. Thus, given a real sequence one seeks a function 1p such that

The Classical Moment Problem and the Calculation of Thermal Averages

The physical information contained in the first 2n moments of the single-particle spectral weight function of a fermionic many-body system is investigated. The approach is based on the mathematical theory of the classical moment problem. Under consideration are the thermal as well as dynamical properties of the system. Using this information, approximate n-pole single-particle thermal Green's functions and the corresponding spectral weight functions are constructed. It is shown that these approximations are not unique and depend on a real parameter. This dependence is used for the calculation of the rigorous error bounds of the approximate thermal averages.

General Approach to the Line-Shape Problem in Nuclear-Magnetic-Resonance Spectra

The description of the macroscopic magnetization of an arbitrary sample is first reduced to the calculation of two similar magnetic-moment autocorrelation functions $G(t)$. The nature of the dynamical evolution of the magnetic-moment operator $m(t)$ needed in these correlation functions is then expressed in terms of an infinite set of time-independent orthogonal vectors in a generalized Hilbert space, such that the desired correlations are just the projection of the evolving magnetic-moment operator onto the first of the orthogonal vectors. Equations of motion of the coefficients of the expansion are obtained and formally solved in Laplace-transform form, yielding the transform of $G(t)$ as a ratio of infinite-order determinants. The formalism is related to the procedures of Zwanzig and Mori and, more generally, to the "classical moment problem" of mathematical analysis. Various practical approximations to the rigorous results are examined.

On moment sequences and renewal sequences

then it is a renewal sequence if and only if it is bounded. Since a Stieltjes moment sequence satisfies condition (1) and since it is a bounded sequence if and only if it is a HausdorR moment sequence, one is naturally led to consider the relationship between moment sequences and renewal sequences. In this note we use Loewner’s theory of monotone matrix functions to investigate the solutions of the renewal equations associated with the three classical moment problems and we obtain results which include those of T. Kaluza.

One-sided approximation of functions

where :Yl denotes the orthogonal complement of Y (see [I]). Another example of this type of duality appears in one of the classical moment problems. Letfbe Bore1 measurable and bounded on [a,b], h,, . . ., h, a CebySev system on [a,b] and for a given positive Bore1 measure yO defined on [a,b] denote by C (y,,) the class of all positive Bore1 measures y which satisfy f.bhkdy=[hkdyo k= 1,2 ,..., n. We then have

The Classical Moment Problem, by N. I. Akhiezer. Translated from the 1961 Russian edition by N. Kemmer. Hafner Publishing Company, New York, 1965. 253 pages. $10.00 (U.S.)

The Classical Moment Problem, by N. I. Akhiezer. Translated from the 1961 Russian edition by N. Kemmer. Hafner Publishing Company, New York, 1965. 253 pages. $10.00 (U.S.)

Moment Synthesis of Array Factors With Nonuniform Spacing and Amplitude Parameters

A theory of nonuniform arrays based upon the classical moment problem is developed in this paper. We suppose that a given pattern can be represented as a Fourier‐Stieltjes transform of a cumulative current distribution over a finite (normalized) aperture interval, which we then approximate by a step‐function distribution whose “treads and risers” are determined as follows. The moments of the step‐distribution are equated to the moments of the continuous distribution belonging to the given pattern, producing a set of nonlinear equations for the positions (treads) and amplitudes (risers) of the steps. The former tum out to be roots of an orthogonal polynomial, of degree equal to number of array elements, belonging to the (continuous) cumulative distribution; the latter can then be found from a resulting set of linear equations and are the so‐called Christoffel numbers. An equivalent development, which helps to illuminate the basic ideas, is to approximate the kernel of the Fourier‐Stieltjes transform by a Lagrange interpolation polynomial. Both of these approaches will produce, in general, an array factor with both nonuniform spacings and amplitudes, and will be said to be optimum in a moment sense: i.e., the original distribution and its approximation agree in their first 2M moments if M is the number of elements used for the approximation.The equivalence of the moment and interpolation methods implies a mean‐square minimality for the polynomial factor of an error representation for the latter method. But the remaining factor of the error representation involves a parameter which cannot in general be determined explicitly; this error form is therefore of little use for large apertures (compared to the number of elements involved). Practical application of moment synthesis would therefore have to rely for an error assessment largely upon either statistical estimation or pattern computation. These facts notwithstanding, the moment method can be used to produce remarkable pattern approximations in the near main‐beam region, and, if the number of elements is sufficiently large, acceptable pattern behavior everywhere. Its chief drawback is that a large number of elements requires the computation of zeros of a orrespondingly large degree polynomial.

On the Accuracy of Gaussian Approximation to the Distribution Functions of Sums of Independent Variables

On the Accuracy of Gaussian Approximation to the Distribution Functions of Sums of Independent Variables

Stieltjes's Moment Problem

The Classical Moment Problem And some Related Questions in Analysis. By N. I. Akhiezer. Translated by N. Kemmer. (University Mathematical Monographs.) Pp. x + 253. (Edinburgh and London: Oliver and Boyd, 1965.) 70s.

On the Question of Finding a General Distribution from the Distribution of a Statistic

Let X be a real random variable with the distribution function $F(x) = {\bf P}(X < x)$ and $\xi = (x_1 , \cdots ,x_n )$ the corresponding sample of size n ($x_i $ being independent replicas of X). A statistic $Q(\xi )$ is called definite if it is homogeneous of positive dimension and the level surfaces $Q(\xi ) = {\text{const.}}$ are continuous, piecewise-smooth and star-finite regions. A statistic $Q(\xi )$ is called defining in a class K of distribution functions $F(x)$, if the distribution $F_Q (x) = {\bf P}(Q < x)$, induced by $F(x)$, determines $F(x)$ in the class K. A definite statistic cannot be defining in general for the class K of all distribution functions, but it is defining in certain rather wide classes of symmetric distribution densities. Three theorems are proved to this effect. The problem can be given as a generalization of the classical moment problem, putting $Q(\xi ) = x_1^2 + \cdots + x_n^2 $.