Convolution Operators with Fractional Measures Associated to Holomorphic Functions

Let be an open set in the complex plane and let be a holomorphic function on . Let K be a compact subset of with nonempty interior such that 0 K. Let be the Borel measure of R ⁴ C ² given by(E = _K E(z, (z))|z|^–2 d(z)where 0 < 2 and d(x ₁ + ix ₂) = dx ₁ dx ₂ denotes the Lebesgue measure on C. Let T be the convolution operator T f = * f. In this paper we characterize the type set E associated to T .     

On the choice of γ for singly-implicit RK or Rosenbrock methods

Following the slogan a picture is better than thousand data, for restricted Padé approximations with one realm-fold pole atz=1/ of orderm, we plot the error constants, the maximal error for negative realz as functions of, as well as the regions ofA- resp.A(0)-stability.     

A sandwich theorem and Hyers—Ulam stability of affine functions

It is shown that two real functionsf andg, defined on a real intervalI, satisfy the inequalitiesf(x + (1 – )y) g(x) + (1 – )g(y) andg(x + (1 – )y) f(x) + (1 – )f(y) for allx, y I and [0, 1], iff there exists an affine functionh: I such thatf h g. As a consequence we obtain a stability result of Hyers—Ulam type for affine functions.     

Extended Classical Asymptotic Expansions in the Case of Gaussian Limit Distribution

Let X, X ₁, X ₂,... be a sequence of independent and identically distributed random variables with common distribution function F. Denote by F _n the distribution function of centered and normed sum S _n . Let F belong to the domain of attraction of the standard normal law , that is, lim F _n (x)= (x), as n , uniformly in x . We obtain extended asymptotic expansions for the particular case where the distribution function F has the density p(x) = cx ^––1 ln(x), x > r, where 2, , c > 0, and r > 1. We write the classical asymptotic expansion (in powers of n ^–1/2) and then add new terms of orders n ^–/2 ln n, n ^–/2 ln^-1 n, etc., where 0.     

A result concerning the existence of certain finite Minkowski - 2 - Structures

Under a special assumption, a theorem concerning the order of Minkowski - 2 - Structures is proved. In particular it is shown that if is a sharply 4 - transitive set of permutations onn elements (n7, integer), such that contains the identical permutation and implies ^–1, then isn=11.     

Asymptotics of a Boundary Crossing Probability of a Brownian Bridge with General Trend

Let us consider a signal-plus-noise model h(z)+B ₀(z), z [0,1], where > 0, h: [0,1] , and B ₀ is a Brownian bridge. We establish the asymptotics for the boundary crossing probability of the weighted signal-plus-noise model for , that is P (sup _{z [0,1]} w(z)( h(z)+B ₀(z))>c), for , (1) where w: [0,1] [0, is a weight function and c>0 is arbitrary. By the large deviation principle one gets a result with a constant which is the solution of a minimizing problem. In this paper we show an asymptotic expansion that is stronger than large deviation. As byproduct of our result we obtain the solution of the minimizing problem occurring in the large deviation expression. It is worth mentioning that the probability considered in (1) appears as power of the weighted Kolmogorov test applied to the test problem H ₀: h 0 against the alternative K: h>0 in the signal-plus-noise model.     

A priori estimates of the solution of the dirichlet problem for the Monge-ampére equation in weight spaces

One proves that a priori boundedness of the norm of the solution of the problem det(U_xx)=f(x,u,u_x)>>0,u¦=0. The magnitudes of the exponents,() depends on whether the arguments u p occur or not in f (x,u,p).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 74–90, 1983.     

On the Space of Sequences of p-Bounded Variation and Related Matrix Mappings

The difference sequence spaces (), c(), and c ₀() were studied by Kzmaz. The main purpose of the present paper is to introduce the space bv _p consisting of all sequences whose differences are in the space _p , and to fill up the gap in the existing literature. Moreover, it is proved that the space bv _p is the BK-space including the space _p . We also show that the spaces bv _p and _p are linearly isomorphic for 1 p . Furthermore, the basis and the -, -, and -duals of the space bv _p are determined and some inclusion relations are given. The last section of the paper is devoted to theorems on the characterization of the matrix classes (bv _p : ), (bv : _p ), and (bv _p : ₁), and the characterizations of some other matrix classes are obtained by means of a suitable relation.     

Canonical Semigroups of States and Cocycles for the Group of Automorphisms of a Homogeneous Tree

Let G=A ut(T) be the group of automorphisms of a homogeneous tree and let d(v,gv) denote the natural tree distance. Fix a base vertex e in T. The function (g)=exp(–d(e,ge)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator d/d|₌₀=log() is conditionally positive definite but not positive definite. Hence, log() corresponds to a nontrivial cocycle (g): GH in some representation space H . In contrast with the case of PGL(2,), the representation is not irreducible.Let _o (g) be the derivative of the spherical function corresponding to the complementary series of A ut(T). We show that –d(e,ge) and _o (g) come from cohomologous cocycles. Moreover, _o is associated to one of the two (irreducible) special representations of A ut(T).     

Аппроксимация субга рмонических функций

The following statement is proved. Letu be a subharmonic function in the region and _u the associated measure. Then there exists a functionf holomorphic in and such that if _f is the associated measure of the function in ¦f¦, then ¦u(z)–ln¦f(z)¦ A¦ln s¦+B¦ln diam¦+ s(¦lns¦+1)+C. hold at every point z for which the setsD(z, t)={w: ¦w–z¦},t(0,s) lie in and satisfy(D(z, t))t both for= _u and for= _f . In the case where is an unbounded region, In diam should be replaced by ln ¦z¦. The constants, , do not depend on andu.

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A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

Positivity Preserving Forms have the Fatou Property

We prove that if (,D) is a positivity preserving form on L ² (E;m), and if (u _n)_n is a sequence in D() converging m-almost everywhere to u L ² (E;m), then (u,u) lim inf_n (u _n ,u _n ).     

Explicitly solvable model of Mössbauer scattering

An explicitly solvable model of Mössbauer scattering of rays by a nucleus bound in a harmonic-oscillator potential is constructed. The probability of elastic scattering, which is proportional to the Debye—Waller factor, is calculated in the framework of the explicitly solvable scattering problem. It is assumed that the rms deviation x of the nucleus and the photon wave numberk satisfykxE /E , whereE andE are typical energy levels of the photon and the oscillator states.St Petersburg State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 439–430, June, 1993.     

Fast Summation of Power Series with Coefficients Analytic at Infinity

We propose a fast summation algorithm for slowly convergent power series of the form _{j=j 0} z ^j _j j _i=1 ^s (j+ _i )^– _i , where R, _i 0 and _i C, 1is, are known parameters, and _j =(j), being a given real or complex function, analytic at infinity. Such series embody many cases treated by specific methods in the recent literature on acceleration. Our approach rests on explicit asymptotic summation, started from the efficient numerical computation of the Laurent coefficients of . The effectiveness of the resulting method, termed ASM (Asymptotic Summation Method), is shown by several numerical tests.     

Über geschlossene affine Zwangläufe in der Ebene

We consider integral coverings y:{1,2,..,} of an affine plane which occur when is moved under a continuous periodic affine motion(t):. One can distinguish normal points × , i.e. is constant in a certain neighborhood of x, and singular points. If (x) is the number of times x passes through its orbit (t)x all normal points x have (x)=1, and the set of all singular points consists of a number of isolated points and lines. If (x) is the tangent rotation number of the orbit of x all singular points lie on the moving pole curve.     

On the Self-Decomposability of Euler's Gamma Function

Let be Euler's Gamma function. We prove that, for all 0, > 0, > 0, > 0, the function (( + iz)/() ^{i z)} , z R ¹, is a self-decomposable characteristic function from the Thorin class and derive its explicit canonical form. Similarly to [1], we also describe several classes of Lévy-type stochastic processes related to .     

Non-Uniformly Elliptic Equations with Natural Growth in the Gradient

We prove the existence of bounded solutions for a class of nonlinear elliptic problems of type–div(a(x,u,Du))=H(x,u,Du)+f, uW ^1,p ₀()L (),where a(x,,)b(||)|| ^p , b is a continuous monotone decreasing function and |H(x,,)| k()|| ^p , k is a continuous monotone increasing function.     

Theorem on a convergence condition in the spaces

For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if f_n(x) f(x) and f _n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f _n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.