Die Verteilung der schlichten Funktionen in einem Funktionenraum

We consider the set of regular functions . We construct a Borel measure and a class of outer measures _h onH. With these and _h we show that: (HS)=0 and _h (HS)=0, (S is the set of normed univalent functions). From _h (HS)=0 follows—forh=t —that the Hausdorff—Billingsley-dimension ofHS is zero.     

Small into isomorphisms onL ∞ spaces

IfT is an isomorphism ofL (A, ) intoL (B, ) which satisfies the condition T T ^–11+, where (A, ) is a -finite measure space, thenT/T is close to an isometry with an error less than 4.     

Disintegration with respect to L p-density functions and singular measures

Summary Let A be a real or complex commutative ordered algebra with identity and involution. Let denote the set of positive multiplicative linear functionals on A. Equip with the topology of simple convergence. For a fixed non-negative probability measure on the set _p of linear functionals f on A which admit an integral representation of the form with FL _p () (1p) is biuniquely identified with L _p () via the map tfF. The norm on _p under which this map becomes an isometry is characterized and a formula for approximating F is derived. The linear functionals which admit representation of the form with are also characterized and appropriately normed. The theory is applied to solve abstract versions of trigonometric and n-dimensional moment problems as well as provide an alternate point of view to the theory of L _p-spaces. New proofs of classical theorems are offered.Research for this paper was sponsored in part by the Danish Natural Science Research Council (Grant No.511-10302) and in part by the National Science Foundation (Grant No. MCS78-03397)The results contained herein include the proofs of theorems announced in [15]     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.     

Best approximations of continuous functions by spline functions

An investigation of the approximation on [0, 1] of functionsf (x) by spline functions s(f,; x) of degree 2r-1 and of deficiency r (r>1) depending on the vector function = ₁ (x),..., _r-1(x) and interpolatingf (x) at fixed points. For the optimal choice of the vector ₀, exact estimates are obtained of the norms f(x)-s (f, ₀; x)_C[0,1] and f (x)-s (f, ₀; x)_{L[0, 1]} on the function classes H Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 41–46, July, 1970.In conclusion we would like to thank N. P. Korneichuk for suggesting this problem and for his valuable advice.     

An Extension of the Roots Separation Theorem

Let T _n be an n×n unreduced symmetric tridiagonal matrix with eigenvalues ₁<₂<< _n and W _k is an (n–1)×(n–1) submatrix by deleting the kth row and the kth column from T _n , k=1,2,...,n. Let _{12 n–1} be the eigenvalues of W _k . It is proved that if W _k has no multiple eigenvalue, then ₁<₁<₂<₂<< _n–1< _n–1< _n ; otherwise if _i = _i+1 is a multiple eigenvalue of W _k , then the above relationship still holds except that the inequality _i < _i+1< _i+1 is replaced by _i = _i+1= _i+1.     

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾     

On a Property of Subexponential Distributions

Let F be a distribution function (d.f.) on [0, ) with finite first moment m >0. We define the integrated tail distribution function F ₁ of F by F ₁(t)=m^-1 ₀ ^t (1- F(u))du, t0. In this paper, we obtain sufficient conditions under which implications FSF ₁S and F ₁S FS hold, where S is the class of subexponential distributions.     

A Favard theorem for orthogonal rational functions on the unit circle

We consider forn=0, 1,... the nested spaces _n of rational functions of degreen at most with given poles . Given a finite measure supported on the unit circle, we associate with it a nested orthogonal basis of rational functions ₀,..., _n for _n ,n=0, 1,.... These _n satisfy a recurrence relation that generalizes the recurrence for Szeg polynomials.In this paper we shall prove a Favard type theorem which says that if one has a sequence of rational functions _{n n} which are generated by such a recurrence, then there will be a measure supported on the unit circle to which they are orthogonal. We shall give a sufficient condition for the uniqueness of this measure.     

Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups

Let ( _t ) _t0 be a -semistable convolution semigroup of probability measures on a Lie groupG whose idempotent ₀ is the Haar measure on some compact subgroupK. Then all the measures ₁ are supported by theK-contraction groupC _K() of the topological automorphism ofG. We prove here the structure theoremC _K()=C()K, whereC() is the contraction group of . Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.     

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.     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

Covering a convex body by smaller homothetic copies

Let a convex bodyAE ⁿ be covered bys smaller homothetic copies with coefficients ₁, ..., _s , respectively. It is conjectured that ₁ + ...+ _s n. This conjecture is confirmed in two cases:n is arbitrary ands=n+1;s is arbitrary andn=2.     

Fractional integrodifferentiation in Hölder classes of arbitrary order

Hölder classes of variable order (x) are introduced and it is shown that the fractional integralI ₀₊ has Hölder order (x)+ (0 < , ₊, +₊ < 1, ₊ = sup (x)).     

Graphs of Constant Mean Curvature in Hyperbolic Space

We study the problem of finding constant mean curvature graphsover a domain of a totally geodesic hyperplane andan equidistant hypersurface Q of hyperbolic space. We findthe existence of graphs of constant mean curvature H overmean convex domains Q and with boundary for –H < H |h|, where H > 0 is the mean curvature of the boundary . Here h is the mean curvature respectively of the geodesic hyperplane (h= 0) and of the equidistant hypersurface (0 < |h|< 1). The lower bound on H is optimal.     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

Tritronquée Solutions of Perturbed First Painlevé Equations

We consider solutions of the class of ODEs y=6y ²–x , which contains the first Painlevé equation (PI) for =1. It is well known that PI has a unique real solution (called a tritronquée solution) asymptotic to and decaying monotonically on the positive real line. We prove the existence and uniqueness of a corresponding solution for each real nonnegative 1.     

Measurable Functionals and Finitely Absolutely Continuous Measures on Banach Spaces

We consider the structure of orthogonal polynomials in the space L ₂(B, ) for a probability measure on a Banach space B. These polynomials are described in terms of Hilbert–Schmidt kernels on the space of square-integrable linear functionals. We study the properties of functionals of this sort. Certain probability measures are regarded as generalized functionals on the space (B, ).