Hilbertraum-Approximation und Jacobi-Matrizen

Zusammenfassung SeiH ein Hilbertraum mit Norm und Skalarprodukt (,). Seien _n ,n=0, 1, 2, ... undf Elemente H, s, i natürliche Zahlen und >0 derart dass gilt: 1) _n =1,2)( _f ), _k )=0 für j–k >s, 3) ( _f , _n ) fürj k, 4) (f, _f =0 fürj>i. Seienf _n undf ^* die Projektionen vonf auf die durch ₀,..., _n bezw. ₀,₁,₂,... aufgespannten abgeschlossenen Unterräume. Das Hauptresultat der Arbeit besagt dass für hinreichend kleines KonstantenC>0 und 0<q<1 existieren mit: f _n -f ^*Cq ⁿ . Dieses Resultat steht in enger Beziehung zu gewissen unendlichen MatrizenA=(a _jk ), die charakterisiert sind durch: (*) es existiertm>0 so dassa _jk =0 für |j-k|>m. Das Hauptresultat wird auf unendliche lineare GleichungssystemeAf=0,Af=g angewandt, woA eine Matrix mit der Eigenschaft (*) ist, deren Diagonalen gewissen Wachstumsbedingungen genügen.

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LetH be a Hilbert space with norm and scalar product (,). Let _n ,n=0, 1, 2, ... andf be elements H, i, s integers and >0 such that: 1) _n =1,2)( _f ), _k )=0 for j–k >s, 3) ( _f , _n ) forj k, 4) (f, _f =0 forj>i. Letf _n andf ^* be the projections off onto the closed subspaces spanned by ₀,..., _n and ₀,₁,₂ .... respectively. The main result says that for sufficiently small there are constantsC>0 and 0<q<1 with f _n -f ^*Cq ⁿ . This result is closely related to certain infinite matricesA=(a _jk ) with the property: (*) there exists anm>0 such thata _jk =0 for |j-k|>m. The result is applied to infinite systems of linear equationsAf=0 andAf=g, whereA is a matrix with property (*), whose diagonals satisfy certain growth conditions.

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Dem Andenken von Professor Eduard Stiefel gewidmet     

Local isometric immersions of two-dimensional metrics inE 4 with prescribed Gaussian torsion

Suppose that in a domain R(, B) of variables (r, ): (0 r , ₁ +B(r–r ₀ ) ₂–B(r–r₀), where > 0, B > 0, ₁ < 0 < ₂ are numbers) a metric ds² = dr² +G(r, )d ² and a function k(r, ) are given. The problem of isometrically immersing ds² in E ⁴ with prescribed Gaussian torsion is considered. The following is proved: The class C ⁵ metric ds ² is locally realized in the form of a class C ³ surface F ² whose Gaussian torsion is the prescribed class C ³ function (r, ).Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 38–47, 1992.     

The integrability of conjugate functions

For any functionf of L(0, 2), we prove that there is a function L(0, 2) such that ¦(x)¦ = ¦f(x)¦ almost everywhere and L(0, 2), where is the conjugate of.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 461–465, October, 1968.     

Dilation-analytic wave operators for three particles

Let H(0) be a dilation-analytic three-particle Schrödinger operator with analytic continuation H() (>0). Let a be zero or the energy of a two-particle bound state. Let- (a) be the Laplace operator representing the kinetic energy of the relative motion of fragments scattered in channel a. By recent results, wave operators W (±, a, ) with conjugates W (±, a, ) exist such that W (±, a, ) W (±, a, ) is a projection P (a, ) commuting with H () while [H ()-a]W (±, a, ) equals-W(±, a, ) (a) e²ⁱ. This paper shows that the wave operators transform dilation-analytic functions of particle coordinates into dilation-analytic functions. Specifically, if the left shoulder of the spectrum of P (a,) H () does not sweep across eigenvalues of H() when , then W(-, a, ) and W (+, a, ) are dilation analytic in [, ]. If the right shoulder does not sweep across eigenvalues, W(+, a, ) and W(-, a, ) are dilation analytic in [,]. A semisimple eigenvalue of H () embedded in the spectrum of P (a, ) H () does not prevent the wave operators from being dilation analytic in an interval [, ] with as an interior point.This work was supported in part by the National Science Foundation under grant DMS-8301096.     

A characterization of multivariate quasi-interpolation formulas and its applications

Summary Let be a compactly supported function on ^s andS () the linear space withgenerator ; that is,S () is the linear span of the multiinteger translates of . It is well known that corresponding to a generator there are infinitely many quasi-interpolation formulas. A characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasi-interpolants. We introduce a generalization of interpolation called -interpolation and a notion of higher order quasi-interpolation called -approximation. A characterization of -approximants similar to that of quasi-interpolants is obtained with similar applications. Among these applications are estimating least-squares approximants without matrix inversion, surface fitting to incomplete or semi-scattered discrete data, and constructing generators with one-point quasi-interpolation formulas. It will be seen that the exact values of the generator at the multi-integers ^s facilitates the above study. An algorithm to yield this information for box splines is discussed.Supported by the National Science Foundation and the U.S. Army Research Office     

High order approximation of implicitly defined maps

Summary Approximations for the function implicitly defined by (u)=(u, (u)) are obtained via the iterative scheme _n(u)=(u, _n–1(u)). In this paper the uniform convergence of high order derivatives of _n to the corresponding derivatives of is proved. This result yields a high order approximation theorem for the input-output map generated by a nonlinear control system, using linear combinations of iterated integrals of the control.Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Asymptote of the eigenvalues of a completely continuous operator

It is proved that if(x) is the majorant of the s-numbers of a completely continuous operator A (i.e.,'(x)- 0, S_n(A) (n)) and if there are found numbers [0, 1] and r₀ > 0 such that r⁰ (r)/(r) will be monotonic in (r₀,), then for some > 0,((x) will be a majorant of the eigenvalues of A.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 487–492, October, 1973.     

On Interpolation of the Fourier Maximal Operator in Orlicz Spaces

Let and be positive increasing convex functions defined on [0, ). Suppose satisfies the 2-condition, that is, (t)2 (C1t) for sufficiently large t, and has some nice properties. If -1(u)log(u+1) C2-1(u) for sufficiently large uthen we have S*(f) L CfL for all f L ([-, ])where S*(f) is the majorant function of partial sums of trigonometric Fourier series and fL is the Orlicz norm of f. This result is sharp.     

Eine Charakterisierung unendlich oft teilbarer Verteilungen auf [0, ∞)

A function : [0, )(-,0] is the logarithm of the Laplace transform of an infinitely divisible probability measure on [0,) if and only if (0)=0, is continuous and conditionally positiv semi-definite.     

On the Existence and Regularity of Mass-Minimizing Currents with an Elastic Boundary

We study the following variational problem. For a compact manifold S₀ embedded in the Euclidean space we consider deformations of S₀. They are represented by Lipschitz continuous homeomorphisms of S₀ whose images are embedded manifolds. We introduce an energy of a deformation which depends on the first derivative of the curvature of (S₀) and the mass of a mass minimizing current which is bounded by (S₀). In this paper it is shown that an energy minimizing deformation of (S₀) exists. Moreover, in the case that S₀ has codimension 1, (S₀) is an embedded C^3a -submanifold, if is of the class C^2,1.     

Quasifractional Approximants in the Theory of Composite Materials

We study the effective heat conductivity of regular arrays of perfectly conducting spheres embedded in a matrix with unit conductivity. Quasifractional approximants allow us to derive an approximate analytical solution, valid for all values of the spheres volume fraction [0, _max] (_max is the maximum volume fraction of a spheres). As a starting point we use a perturbation approach for 0 and an asymptotic solution for _max. Three different spatial arrangements of the spheres, simple cubic, body centred and face centred cubic arrays, are considered. Results obtained give a good agreement with numerical data.     

Lévy homeomorphic parametrization and exponential families

Summary Let P={P : } be an exponential family of probability distributions with the canonical parameter and consider the one to one mapping : P . It is shown that, under mild regularity assumptions, and ^–1 are continuous with respect to the Lévy metric in P and Euclidean metric in .     

Regular homomorphisms of generalized projective planes

A generalized projective plane is an incidence structure together with a relation distant on the set of points and also on the set of lines, such that any two distant points A,B (lines a,b) have a unique common line (A,B) (common point (a,b)) and three further axioms hold. Every commutative ring with 1 supplies a model. A homomorphism of into an incidence structure is called regular if the following condition and its dual are valid: A distant B and c IA,B implies c=(A,B). We shall prove the following two theorems. Let be a generalized projective plane satisfying a richness condition called (U). Let M I m. If and are regular homomorphisms of such that X = M X = M for each point X of the line m then A = B A = B for any two points A,B. If is a projective plane over a commutative ring such that (U) holds then the surjective regular homomorphisms of are induced by the ideals of the ring; in particular, the image of under a regular homomorphism is again a projective plane over a ring, and preserves distant.     

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.     

Generalization of the theorem of Marcinkiewicz on entire characteristic functions of probability distributions

The relation is studied between the distribution of the zeros and the order of growth of entire analytic functions for which ¦p(z)¦ (i Imz) for Imz 0, in particular, of entire characteristic functions of probability distributions. The main result is the following: if ₁ is the exponent of convergence of the sequence of zeros of such a function of order which lie in a half plane Imz d > 0, then the inequality ₁ < implies the inequality p 3. This estimate is precise.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 94–103, 1979.The author is grateful to I. V. Ostrovskii for posing the problem and for his constant assistance with the work.     

Bounded projections and duality on spaces of holomorphic functions in the unit ball ofℂ n

In this paper three Banach spacesA ₀(),A andA ¹() of functions holomorphic in the unit ballB of ⁿ are defined. We exhibit bounded projections fromC ₀(B) ontoA ₀(), fromL ¹(B) ontoA ¹(), and fromL(B) ontoA(). Using these projections, we show thatA ₀()* A ¹() andA ¹()* A().Supported in part by the National Natural Science Foundation of China.     

Stresses due to a nucleus of thermo-elastic strain (i) in an infinite elastic solid with spherical cavity and (ii) in a solid elastic sphere

In this paper solutions in series form for the stresses due to a nucleus of thermo-elastic strain in an infinite elastic solid in the presence of a spherical cavity and also in an elastic solid sphere have been found.

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Zusammenfassung Die thermischen Spannungen in einem festen Körper unendlicher Ausdehnung, welcher einen sphärischen Hohlraum enthält, sind bei einer Temperatur von 0°C in Gegenwart eines erhitzten Elementes, das sich in endlichem Abstand vom Hohlraum befindet, hergeleitet worden, wobei zahlenmässige Angaben für die Spannungen und Verschiebungen an der Oberfläche des Hohlraums gemacht werden können. Die Ergebnisse sind mit den entsprechenden, für den zweidimensionalen Fall gültigen Zahlenwerten verglichen worden. Ferner was es möglich, auch für das Problem einer festen Kugel von der Temperatur 0°C und einem erhitzten Kern in ihrem Innern eine Lösung zu finden.

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Nomenclature x, y, z Cartesian coordinates; - r, , spherical polar coordinates; - u _x ,u _y ,u _z components of displacement in Cartesian coordinates; - u _r ,u ,u components of displacement in spherical coordinates; - _r , , , , _r , components of stress in spherical coordinates; - E coefficient of elasticity in stress; - G coefficient of elasticity in shear; - coefficient of linear expansion; - Poisson's ration The following nomenclature has been used in this paper:     

Commutative Jacobi fields in Fock space

A jacobi field is understood to be a family (Ã()) of commuting selfadjoint operatorsÃ() acting in a Fock space, having a Jacobi structure, and depending linearly on the test functions . In this article, we give a spectral representation of such a family and outline its applications to the theory of distributions on an infinite dimensional space.This article is dedicated to the memory of my dear teacher Mark G. KreinThe work is partially supported by Fundamental Research Foundation of Ukraine, grant 1.4/62.     

The arithmetic of the characteristic Pólya functions

In the framework of the theory of D. Kendall's delphic semigroups are considered problems of divisibility in the semigroup of convex characteristic functions on the semiaxis (0,). Letn ()={:₁¦₁1 or ₁=}, and I_o()={: ₁¦ ₁ N()}. The following results are proved: 1) The semigroup is almost delphic in the sense of R. Davidson. 2) N() is a set of the type G which is dense in (in the topology of uniform convergence on compacta). 3) The class I_o() contains only the function identically equal to one.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 717–725, May, 1977.The author thanks I. V. Ostrovskii for the formulation of the problem and valuable remarks.     

Mengenwertige Masse und Fortsetzungen

Let (, _i) be a probability space for i=1,2 with and : ^m a correspondence, i.e. () is a non-void subset of ^m for all . We give necessary and sufficient conditions under which it holds, that ₂ extends ₁. iff _A d₂ is equal to _A d₁ for all A, where _A d_i is the set of all integrals _A f d_i of functions f: ^m with f()() _i.-a.e.