For which reproducing kernel Hilbert spaces is Pick's theorem true?

Pick's theorem tells us that there exists a function inH , which is bounded by 1 and takes given values at given points, if and only if a certain matrix is positive.H is the space of multipliers ofH ², and this theorem has a natural generalisation whenH is replaced by the space of multipliers of a general reproducing kernel Hilbert spaceH(K) (whereK is the reproducing kernel). J. Agler has shown that this generalised theorem is true whenH(K) is a certain Sobolev space or the Dirichlet space, so it is natural to ask for which reproducing kernel Hilbert spaces this generalised theorem is true. This paper widens Agler's approach to cover reproducing kernel Hilbert spaces in general, replacing Agler's use of the deep theory of co-analytic models by a relatively elementary, and more general, matrix argument. The resulting theorem gives sufficient (and usable) conditions on the kernelK, for the generalised Pick's theorem to be true forH(K), and these are then used to prove Pick's theorem for certain weighted Hardy and Sobolev spaces and for a functional Hilbert space introduced by Saitoh.     

The gauge theorem for a class of additive functionals of zero energy

Summary In earlier works, the gauge theorem was proved for additive functionals of Brownian motion of the form ₀ ^t q(B _s )ds, whereq is a function in the Kato class. Subsequently, the theorem was extended to additive functionals with Revuz measures in the Kato class. We prove that the gauge theorem holds for a large class of additive functionals of zero energy which are, in general, of unbounded variation. These additive functionals may not be semi-martingales, but correspond to a collection of distributions that belong to the Kato class in a suitable sense. Our gauge theorem generalizes the earlier versions of the gauge theorem.Research supported in part by NSA grant MDA-92-H-30324     

Derivation of Vector-Valued Boolean Functions

The notion of Jacobi matrix is introduced for B ^m B ⁿ mappings. The chain rule, the inverse function theorem and the implicit function theorem are proved. The meaning of functional dependence in the context of Boolean functions is discussed. Parallel is drawn with usual calculus.     

The Central Approximation Theorems for the Method of Left Gamma Quasi-Interpolants in L p spaces

The optimal degree of approximation of the method of Gammaoperators G _n in L _p spaces is O(n ^-1). In order to obtain much faster convergence, quasi-interpolants G _n ^(k) of G _n in the sense of Sablonnière are considered. We show that for fixed k the operator-norms G _n ^(k) _p are uniformly bounded in n. In addition to this, for the first time in the theory of quasi-interpolants, all central problems for approximation methods (direct theorem, inverse theorem, equivalence theorem) could be solved completely for the L _p metric. Left Gamma quasi-interpolants turn out to be as powerful as linear combinations of Gammaoperators [6].     

The Diametric Theorem in Hamming Spaces—Optimal Anticodes

For a Hamming space (ⁿ_α, d_H), the set ofn-length words over the alphabet _α = {0, 1,…,α − 1} endowed with the distanced_H, which for two wordsxⁿ = (x₁,…,x_n),yⁿ = (y₁,…,y_n) ∈ ⁿ_αcounts the number of different components, we determine the maximal cardinality of subsets with a prescribed diameterdor, in another language, anticodes with distanced. We refer to the result as the diametric theorem.In a sense anticodes are dual to codes, which have a prescribedlowerbound on the pairwise distance. It is a hopeless task to determine their maximal sizes exactly.We find it remarkable that the diametric theorem (for arbitrary α) can be derived from our recent complete intersection theorem, which can be viewed as a diametric theorem (for α = 2) in the restricted case, where alln-length words considered have exactlykones.     

On a theorem about projectivities

The main result of this paper is a theorem about projectivities in then-dimensional complex projective spaceP ⁿ (n 2). Puttingn = 2, we showed in [3] that the theorem of Desargues inP ⁿ is a special case of this theorem. And not only the theorem of Desargues but also the converse of the theorem of Pascal, the theorem of Pappus-Pascal, the theorem of Miquel, the Newton line, the Brocard points and a lot of lesser known results in the projective, the affine and the Euchdean plane were obtained from this theorem as special cases without any further proof. Many extensions of classical theorems in the projective, affine and Euclidean plane to higher dimensions can be found in the literature and probably some of these are special cases of this theorem inP ⁿ. We only give a few examples and leave it as an open problem which other special cases could be found.     

An implicit function theorem

Suppose thatF:DR ⁿ×R^mRⁿ, withF(x ⁰,y ⁰)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ₁ F(x ⁰,y ⁰) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.     

A Note on *-paranormal Operators

In this note we show that if either T or T^* is totally *-paranormal then Weyls theorem holds for f(T) for every f , and also a-Weyls theorem holds for f(T) if T is totally *-paranormal. We prove that if either T or T^* is *-paranormal then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.     

On Nevanlinna's second main theorem in projective space

Summary We first prove a theorem concerning higher order logarithmic partial derivatives for meromorphic functions of several complex variables. Then we show the best nature of the second main theorem in Nevanlinna theory under two different assumptions of non-degeneracy of meromorphic mappingsf : ^{n n} for arbitrary positive integersn andm. Moreover, we derive a upper bound of the error term in the second main theorem for meromorphic mappings of finite order. Finally, we demonstrate the sharpness of all upper bounds in our main theorems.Oblatum 28-IX-1994 & 29-V-1995     

Eine Übertragung des Satzes von Bonnet auf Regelflächen im einfach isotropen Raum

J ₃ ⁽¹⁾ . Bonnet's classical theorem about ruled surfaces in the three-dimensional Euclidean space does not hold inJ ₃ ⁽¹⁾ . To get an isotropic version of this theorem the terms geodetic line and isogonal-trajectory of the generators are replaced by new, the isotropic space adapted properties of curves on ruled surfaces.     

Dini-Verbände mit fast gut frisiertem Dualkegel

Summary We prove a theorem which is a generalisation of a theorem due to Goullet de Rugy: LetE be a locally convex vector lattice such that the structure spaces of the principal ideals ofE are universally measurable in their Stone-ech-compactifications respectively. Then, ifE is infrabarrelled or its topology is finer than the topology of compact convergence,E is a Dini lattice iffE ₊ is nearly well-capped.     

The Darling-Erdős theorem for sums of I.I.D. random variables

Summary We obtain a general Darling-Erds type theorem for the maximum of appropriately normalized sums of i.i.d. mean zero r.v.'s with finite variances. We infer that the Darling-Erds theorem holds in its classical formulation if and only ifE[X ² 1 {|X|t}]=o((loglogt)^-1) ast. Our method is based on an extension of the truncation techniques of Feller (1946) to non-symmetric r.v.'s. As a by-product we are able to reprove fundamental results of Feller (1946) dealing with lower and upper classes in the Hartman-Wintner LIL.     

A Formal Chevalley Restriction Theorem for Kac–Moody Groups

Let G be a symmetrizable Kac–Moody group over a field of characteristic zero, let T be a split maximal torus of G. By using a completion of the algebra of strongly regular functions on G, and its restriction on T, we give a formal Chevalley restriction theorem. Specializing to the affine case, and to the field of complex numbers, we obtain a convergent Chevalley restriction theorem, by choosing the formal functions, which are convergent on the semi-groups of trace class elements G ^trG resp. T ^trT.     

Unification and extensions of some stability theorems in mathematical programming

In this paper, we provide the stability theorem for the program: inf{f(x, t)|xH(t)}, using the uniformlyN-type functions (also called -chainable functions^[10]). This theorem generalizes the results of Dantzig^[1], Hogan^[2], Greenberg^[3], Ying Mei-qian^[4] et al.Project supported by the Science Foundation of the Chinese Academy of Science.     

Sukzessive Minima mit und ohne Nebenbedingungen

Given semi-normsf andg on ⁿ and a real number >0. Then the successive minima off under the constraintg are defined by _j : = inf {: there existj linear independent vectors inZ ⁿ withf andg}. The main theorem of this paper (Lagrange multiplier theorem) states that the successive minima of a certainnorm h on ⁿ (without constraints) coincide with the _j 's up to bounded factors. Moreover, this norm is constructed explicitly. Using Minkowski's wellknown theorem on successive minima and our result certain inequalities on simultaneous Diophantine approximations are derived.     

Groupoid C *-Algebras and Index Theory on Manifolds with Singularities

The simplest case of a manifold with singularities is a manifold M with boundary, together with an identification M M × P, where P is a fixed manifold. The associated singular space is obtained by collapsing P to a point. When P = Z/k or S ¹, we show how to attach to such a space a noncommutative C ^*-algebra that captures the extra structure. We then use this C ^*-algebra to give a new proof of the Freed–Melrose Z/k-index theorem and a proof of an index theorem for manifolds with S ¹ singularities. Our proofs apply to the real as well as to the complex case. Applications are given to the study of metrics of positive scalar curvature.     

On Compactness with Respect to Countable Extensions of Ideals and the Generalized Banach Category Theorem

We extend a theorem of Hamlett and Jankovi by proving that if a topological space (X, ) is compact with respect to the countable extension of I, then the local function A ^*(I) of every subset A of X with respect to and I is a compact subspace with respect to the extension in A ^* (I). We also give a generalized version of the Banach category theorem.     

Abstract,weighted, and multidimensional Adamjan-Arov-Krein theorems,and the singular numbers of Sarason commutants

The classical Adamjan-Arov-Krein (A-A-K) theorem relating the singular numbers of Hankel operators to best approximations of their symbols by rational functions is given an abstract version. This provides results for Hankel operators acting in weightedH ²(T; ), as well as inH ²(T ^d ), and an A-A-K type extension of Sarason's interpolation theorem. In particular, it is shown that all compact Hankel operators inH ²(T ^d ) are zero.Author partially supported by NSF grant DMS89-11717.     

A “from scratch” proof of a theorem of Rockafellar and Fulkerson

The theorem of this paper is of the same general class as Farkas' Lemma, Stiemke's Theorem, and the Kuhn—Fourier Theorem in the theory of linear inequalities. LetV be a vector subspace ofR ⁿ , and let intervalsI ¹,, I ⁿ of real numbers be prescribed. A necessary and sufficient condition is given for existence of a vector (x ¹ ,, x ⁿ ) inV such thatx ⁱ I ⁱ (i = 1, ,n); this condition involves the elementary vectors (nonzero vectors with minimal support) ofV . The proof of the theorem uses only elementary linear algebra.The author at present holds a Senior Scientist Award of the Alexander von Humboldt Stiftung.     

Stability in Aleksandrov's problem for a convex body,one of whose projections is a ball

A stability theorem is proved corresponding to this uniqueness theorem forn5,k3, and under the hypothesis that one of the projections of the convex body on a hyperplane ofR ⁿ is a ball. In particular the stability theorem holds in the case when the body is a solid of revolution.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 50–62.