On unconditionally convergent basis expansions in the space of continuous functions

[0,1], - H .

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This paper was written during the author's scholarship at the State University of Odessa in the USSR.     

A generalized Laplace transform of generalized functions

- . . . , .     

Convergence of sequential and asynchronous nonlinear paracontractions

Summary We establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces. Applications to the solution of linear systems of equations with convex constraints are outlined. A first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented.Research supported in part by Sonderforschungsbereich 343 Diskrete Strukturen in der MathematikResearch supported in part by NSF Grant DMS-9007030 and by Sonderforschungsbereich 343 Diskrete Strukturen in der Mathematik, Fakultät für Mathematik at the Universität BielefeldResearch supported in part by U.S. Air Force Grant AFOSR-88-0047, by NSF Grants DMS-8901860 and DMS-9007030, and by Sonderforschungsbereich 343 Diskrete Strukturen in der Mathematik, Fakultät für Mathematik at the Universität Bielefeld     

On cubature formulas on the basis of lattices

Q (.. , L). Q . P(S_r(2)) — 2 (S _r(2) (r — ). , M(P(S _r(m=sup{t(·)t(·)₁:t P(S _r(2)),t 0}. , /4+(1)M(P(S _r(2)))/r ²15/17+(1)(r+). (Q), Q L.     

Wave operators and similarity of certain non-normal one-parameter groups

This paper starts from a self-adjoint Schrödinger operator H(0) for three particles. If the interaction is dilation-analytic, H(0) has an analytic continuation H() (>0). G(t,) (–(±,a,) defined as strong limits, when t±, of t-dependent operators. The wave operators establish transformations under which the subgroups are similar to unitary groups. The scattering matrix determined by G(t,) is diagonal with respect to a.This work was supported in part by the National Science Foundation under grant DMS-8301096.     

Role of information in the stochastic zero-sum differential game

In this paper, a linear-quadratic Gaussian zero-sum differential game is studied. Maneuverability is defined to measure players' strength. It is shown that a more maneuverable player would prefer a more observable information system. An example is given to show that a more controllable player might not prefer more observable measurements in the stochastic environment.The research reported in this paper was made possible through support extended to the Division of Engineering and Applied Physics, Harvard University, by the US Office of Naval Research under the Joint Services Electronics Program by Contract No. N00014-75-c-0648 and by the National Science Foundation under Grant No. GK31511.     

Containment problems in high-dimensional spaces

For any integersn, d 2, let(n, d) be the largest number such that every setP ofn points inR ^d contains two pointsx, y P satisfying |box_d(x, y) P| (n, d), where box_d(x, y) means the smallest closed box with sides parallel to the axes, containingx andy. We show that, for any integersn, , which improves the lower bound due to Grolmusz [9] by a short self-contained proof.Partially supported by the Grant in Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan and by the Grant for Basic Science Research Projects of the Sumitomo Foundation.     

How non-uniform can a uniform sample be: a histogram approach

Summary Let _n be the empirical probability measure associated with n i.i.d. random vectors each having a uniform distribution in the unit square S of the plane. After _n is known, take the worst partition of the square into kn rectangles R _i, each with its short side at least times as long as the long side, and let Z= n|_n(R _j)–(R _j)|. We prove distribution inequalities for Z implying the right half of c _p,(n,k)^p/2 EZ ^p C _p,(n,k ^p/2, p > 0. (The left half follows easily by considering non-random partitions.) Similar results are obtained in other dimensions, and for population distributions other than uniform, and our results are related to data based histogram density estimation.Supported by NSF Grant MCS 8201128Supported by NSF Grant DMS-8401996     

A joint central limit theorem for the sample mean and regenerative variance estimator

Let {V(k) :K1} be a sequence of independent, identically distributed random vectors in ^d with mean vector . The mappingg is a twice differentiable mapping from ^d to ¹. Setr=g(). A bivariate central limit theorem is proved involving a point estimator forr and the asymptotic variance of this point estimate. This result can be applied immediately to the ratio estimation problem that arises in regenerative simulation. Numerical examples show that the variance of the regenerative variance estimator is not necessarily minimized by using the return state with the smallest expected cycle length.This research was supported by Army Research Office Contract DAAG29-84-K-0030. The first author was also supported by National Science Foundation Grant ECS-8404809 and the second author by National Science Foundation Grant MCS-8203483.     

Bijectivity of -endomorphisms of ℬ(H) and the unilateral shift

LetH be a separable infinite-dimensional complex Hilbert space. We prove that if : (H)(H) is a^*-preserving ring homomorphism whose range contains a rank-one operator and an operator with dense range, then is an isometric linear or conjugate-linear algebra automorphism of (H). In particular, if the unilateral shift is contained in the range of a^*-endomorphism of (H), then is bijective.Research partially supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA 1652 and K&H Bank Ltd., Universitas Foundation.     

Rational Approximation of Analytic Functions Having Generalized Orders of Rate of Growth

Let f be holomorphic on a domain G C¯ and _n be the error in best approximation of f in the supremum norm on a compact set E G by rational functions of order n. We obtain results characterizing the degree of decrease of the best approximation _n in terms connected with the condenser (E,F), F=C¯ \ G¯, and the rate of growth of the maximum modulus of f(z). In particular, if f has a generalized order (, , f) in the domain G, thenlim sup_n (n)/ (log (1/log⁺((_{0 1} ....... _n)^1/n(n+1) ))) (, , f),where = exp (1/C(E,F)), C(E,F) is the capacity of the condenser (E,F).     

Limit periodic Schur algorithms,the case 441-1441-1441-1

We show that if a function bounded in the unit disk gives rise to a limit periodic Schur algorithm with ||=1, d _n ,< then the function can be continued analytically to a meromorphic function in the entire plane except, possibly, for an essential singularity at z=–1. The lpS provides a very good approximation to the function.This research was supported in part by the US National Science Foundation under Grant DMS-9103141.     

A splitting theorem for compact complex homogeneous spaces with a symplectic structure

In this note we prove a splitting theorem for compact complex homogeneous spaces with a cohomology 2 class [] such that the top power [ ⁿ ]0.Dedicated to Professor W. C. Hsiang on the occasion of his 60th birthdayPartially supported by NSF Grant DMS-9401755.     

Connectivity properties of Mandelbrot's percolation process

Summary In 1974, Mandelbrot introduced a process in [0, 1]² which he called canonical curdling and later used in this book(s) on fractals to generate self-similar random sets with Hausdorff dimension D(0,2). In this paper we will study the connectivity or percolation properties of these sets, proving all of the claims he made in Sect. 23 of the Fractal Geometry of Nature and a new one that he did not anticipate: There is a probability p _c(0,1) so that if p<p _c then the set is duslike i.e., the largest connected component is a point, whereas if pp _c (notice the =) opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles. More succinctly put the system has a first order phase transition.Work supported by the NSF under Grant #DMR-83-14625Work supported by the DOE under Grant #DE-AC02-83-ER13044Work supported by the NSF under Grant #DMS-85-05020     

The role of Fourier modes in extension theorems of Hartogs-Chirka type

We generalize Chirkas theorem on the extension of functions holomorphic in a neighbourhood of (F)(D×D) – where D is the open unit disc and (F) is the graph of a continuous D-valued function F – to the bidisc. We extend holomorphic functions by applying the Kontinuitätssatz to certain continuous families of analytic annuli, which is a procedure suited to configurations not covered by Chirkas theorem.Mathematics Subject classification (2000): 32D15Work supported by NSF Grant DMS-0072237.     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

Existence of strong solutions for Itô's stochastic equations via approximations

Summary Given strong uniqueness for an Itô's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on any probability space by using, for example, Euler's polygonal approximations. Stochastic equations in ^d and in domains in ^d are considered.Research supported by the Hungarian National Foundation of Scientific Research No. 2990.Supported in part by NSF Grant DMS-9302516     

An algebraic test forA 0-stability

The stability of a large class of numerical methods to solve initial value problems of ordinary differential equations is governed by a two-variable polynomial (,) when the method is applied toy'=qy. Here=hq, whereh is the stepsize. This class of methods includes Runge-Kutta methods, linear multistep methods, predictor-corrector methods, composite multistep methods and linear multistep-multiderivative methods. An algebraic test is given to determineA ₀-stability of such methods in a finite number of operations (additions, subtractions, multiplications and divisions). It is shown that the number of multiplications and divisions is of order 1/8²(⁴ +O(³)), where is the degree of (,) in the variable and the degree in the variable. The test has been implemented for multistep-multiderivative methods in a symbol manipulation language. For Enright's second derivativek-step methods it is proved that the methods areA ₀-stable if and only ifk<8.Supported by the Swiss National Foundation Grant No. 82.524.077. On leave from Institute of Mathematics, Ruhr-University Bochum, D-463 Germany.     

Isomorphismen von rechtseiträumen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation and whose set of point pairs P² is equipped with a congruence relation , such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism of two rectangular spaces (P,G, , ) and (P,G, , ) we mean a bijection of the point setP onto P which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, , ) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P is in fact an isomorphism from (P,G, , ) onto (P,G, , ) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, , ) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, , ). By a motion of(P. G,, ) we mean a bijection ofP which maps lines onto lines, preserves parallelism and satisfies the condition((x), (y)) (x,y) for allx, y P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, , ) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

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Herrn Professor Burau zum 85. Geburtstag gewidmet     

Marcinkiewicz integral on hardy spaces

In this paper we prove that the Marcinkiewicz integral is an operator of type (H ¹,L ¹) and of type (H ^1,,L ^1,). As a corollary of the results above, we obtain again the the weak type (1,1) boundedness of , but the smoothness condition assumed on is weaker than Stein's condition.The research was supported partly by Doctoral Programme Foundation of Institution of Higher Education (Grant No. 98002703) of China.The author was supported partly by NSF of China (Grant No. 19971010).The author was supported partly by NSF of China (Grant No. 19131080).