Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

On the convergence of powers of interval matrices (2)

Summary Let be a real irreduciblen×n interval matrix. Then a necessary and sufficient condition is given for the sequence of the powers of an interval matrix to converge to a matrix which is not the null matrix. In addition a criterion for is proved to decide whether the limit matrix satisfies the condition of symmetry .     

Hankel operators on the weighted Bergman spaces with exponential type weights

Let denote the closed subspace of consisting of analytic functions in the unit disk . For certain class of subharmonic , the Hankel operatorH _b on with symbol is studied. Criteria for boundedness and compactness of such kind of Hankel operators are presented.R. Rochberg's research was partially supported by a grant from the National Science Foundation.     

Un théorème de Vitali-Hahn-Saks pour les semimartingales

Summary Let (, , P) be a complete probability space; let _t0 be an increasing right-continuous family of -complete sub--fields of ; let be a sequence of semimartingales. Assume that for all positive t and for all bounded predictable processes H, the r.v.'s converge in probability to a limit J(t, H) when n tends to infinity. Then there exists a semimartingale X such that, for all t and H, J(t, H)= .     

Finite sublattices in the lattice of clones

The lattice of clones of functions over a k-element set is studied. It is shown that every lattice which is a countable direct product of finite lattices is embedded (up to isomorphism) in and, hence, in for k 4. This directly implies that every finite and any countable residually finite lattice is embedded in , k 4, and that no nontrivial quasi-identity holds in , k 4. A number of particular lattices (which are free in some lattice varieties) embeddable in , k 4,are presented. Translated fromAlgebra i Logika, Vol. 33, No. 5, pp. 514–549, September–October, 1994.     

Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Let be a reductive Lie algebra over C. We say that a -module M is a generalized Harish-Chandra module if, for some subalgebra , M is locally -finite and has finite -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when is a Cartan subalgebra. We also review the recent determination of which reductive in subalgebras are essential to a classification. Finally, we present in detail the emerging picture for the case when is a principal 3-dimensional subalgebra.     

Product-integration with the Clenshaw-Curtis and related points

Summary This paper is concerned with the theoretical properties of a productintegration method for the integral , wherek is absolutely integrable andf is continuous. The integral is approximated by , where the points are given byx _ni =cos(i/n, 0in, and where the weightsw _ni are chosen to make the rule exact iff is any polynomial of degree n. The principal result is that ifkL _p [–1, 1] for somep>1, then the rule converges to the exact result asn for all continuous (or indeed R-integrable) functionsf, and moreover that the sum of the absolute values of the weights converges to the least possible value, namely . A limiting expression for the individual weights is also obtained, under certain assumptions. The results are exteded to other point sets of a similar kind, including the classical Chebyshev points.     

On the tail behaviour of the distribution function of multiple stochastic integrals

Summary Let F _n (u) denote the empirical distribution function of a sample of i.i.d. random variables with uniform distribution on [0, 1]. Define , and consider the integrals where f is a bounded measurable function. We give a good upper bound on the probability . An analogous estimate is given for multiple integrals with respect to a Poisson process.     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

A stability test for real polynomials

Summary By the argument principle the number of zeros inside of the unit circle of a real polynomial , can be estimated by the variation of the argument ofp _n (exp(it)) ift varies from 0 to . This variation has its maximal value n iff then distinct zeros of are separated by then–1 distinct zeros of . Now from Sturm sequence techniques in connection with special properties of the Chebyshev polynomials there results a very simple stability test.Dedicated to Professor Dr. Wilhelm Niethammer on his sixtieth birthday (December 2, 1993)     

Quadrature formulas obtained by variable transformation

Quadrature formulas suitable for evaluation of improper integrals such as are obtained by means of variable transformations =tanhu and =erfu, and subsequent use of trapezoidal quadrature rule. Error analysis is carried out by the method of contour integral, and the results are confirmed on several concrete examples. Similar formulas are also obtained to accelerate the convergence of infinite integrals by means of variable transformations =sinhu and =tanu.     

Numerical stability of the cyclic Richardson iteration

Summary The success of the cyclic Richardson iteration depends on the proper ordering of the acceleration parameters. We give a rigorous error analysis to show that, with the proper ordering, the relative error in the iterative method, when properly terminated, is not larger than the error incurred in stable direct methods such as Cholesky factorization. For both the computed approximation tou=L ^–1f satisfies cond (L)u2^–t and this bound is attainable. We also show that the residual norm is bounded by L cond . This bound is attainable for a small cycle lengthN. Our analysis suggests that for a larger cycle lengthN the residuals are bounded by . We construct a theoretical example in which this bound is attainable. However we observed in all numerical tests that ultimately the residual norms were of order . We explain why in practice even the factor is never encountered. Therefore the residual stopping criterion for the Richardson iteration appears to be very reliable and the method itself appears to be stable.The author gratefully acknowledges partial support from ONR Contract N00014-85-K-0180On leave from the University of Krakow, Poland, during the spring semester 1989     

A modification of the uniqueness criterion for the solution of the Watson problem for a half-plane

It is proved that a known theorem yielding the solution of the Watson problem for a half-plane in terms of the Ostrovskii function remains valid if the Ostrovskii function is replaced by the function 0} r^x /m (x)$$ " align="middle" border="0"> , where for x [n, n+1) the function m(x)=m_n, or by the function .Translated from Matematicheskie Zametki, Vol. 14, No. 5, pp. 609–614, November, 1973.     

On dualities between function spaces

By [6], the dualities between and , whereX andW are two sets and (i.e., the mappings satisfying for all and all index setsI), can be represented with the aid of functions . Here we show that they can be also represented with the aid of functions , whereR = (–, +). As an application, we show that every duality is completely determined by a suitable duality between 2 ^{X ×R} and 2 ^{W ×R} (i.e., a mapping 2 ^{X ×R} 2 ^{W ×R} satisfying for all {M _i} _iI 2 ^{X ×R} and all index setsI), applied to the epigraphs of the functions .     

The space of minimal prime ideals in a 0-distributive semilattice

LetS be a 0-distributive semilattice and be its minimal spectrum. It is shown that is Hausdorff. The compactness of has been characterized in several ways. A representation theorem (like Stone's theorem for Boolean algebras) for disjunctive, 0-distributive semilattices is obtained.     

Bounded solutions for 119-01119-01119-01-problem in pseudo-Siegel domains

Summary We study the problem of the existence of bounded solutions for the equation on pseudo-Siegel domains when the data satisfies the condition .

---

Sunto Si studia il problema dell'esistenza di soluzioni limitate per l'equazione sui domini pseudo-Siegel quando il dato soddisfa alla condizione .

---

---

Work done within the Project 40% M.U.R.S.T. «Geometria Reale e Complessa».     

On a theorem of Seidel and Walsh

Given a sequence ( _n ) _n in with there are functions such that , is a dense subset of , and the set of functions with this property is residual in . We will show that in and some related Banach spaceX there are functionsf with is dense in , and we will give a sufficient condition when the set of such functions is residual inX.     

Positive elements in the algebra of the quantum moment problem

Summary Let denote the extended Weyl algebra, , the Weyl algebra. It is well known that every element of of the formA=B _k ^* B _k is positive. We prove that the converse implication also holds: Every positive elementA in has a quadratic sum factorization for some finite set of elements (B _k ) in . The corresponding result is not true for the subalgebra . We identify states on which do not extend to states on . It follows from a result of Powers (and Arveson) that such states on cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on , and this property is not in general shared by representations generated by states defined only on the subalgebra .Work supported in part by the NSF     

On the Spectrum of Degenerate Operator Equations

We study the distribution in the complex plane of the spectrum of the operator , generated by the closure in of the operation originally defined on smooth functions with values in a Hilbert space satisfying the Dirichlet conditions . Here and A is a model operator acting in . Criterial conditions on the parameter for the eigenfunctions of the operator to form a complete and minimal system as well as a Riesz basis in the Hilbert space H are given.     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.