Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ballB⊂C ⁿ with its relative logarithmic capacity inC ⁿ with respect to the same ballB. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace ofC ⁿ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity onC ⁿ as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂(C ⁿ . Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic functions. This work was partially supported by the programmes PARS MI 07 and AI.MA 180.     

Unramified cohomology of degree 3 and Noether’s problem

Let G be a finite group and W be a faithful representation of G over C. The group G acts on the field of rational functions C(W). The question whether the field of invariant functions C(W) ^G is purely transcendental over C goes back to Emmy Noether. Using the unramified cohomology group of degree 2 of this field as an invariant, Saltman gave the first examples for which C(W) ^G is not rational over C. Around 1986, Bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group G. In this paper, we prove a formula for the prime to 2 part of the unramified cohomology group of degree 3 of C(W) ^G . Specializing to the case where G is a central extension of an F _p -vector space by another, we get a method to construct nontrivial elements in this unramified cohomology group. In this way we get an example of a group G for which the field C(W) ^G is not rational although its unramified cohomology group of degree 2 is trivial. Dedicated to Jean-Louis Colliot-Thélène.     

Spherical Convergence of Double Fourier Integrals of Functions in Waterman Classes

The aim of the present paper is to obtain new results on the spherical convergence of double Fourier integrals of functions belonging to certain Waterman classes. A two-dimensional Waterman class of functions in L(R ²) is introduced in which the partial spherical Fourier integrals are uniformly bounded and converge at each point of continuity of the function in question, and which class is as large as possible. In addition, a one-dimensional Waterman class is established such that if the mean of a function belongs to this class, than its Fourier integral converges spherically at a given point, and this class is the largest possible in a certain sense.     

Branch point area methods in conformal mapping

The classical estimate of Bieberbach that ?a ₂?≤2 for a given univalent function ?(z)=z+a ₂ z ²+… in the classS leads to the best possible pointwise estimates of the ratio ?"(z)/?'(z) for ?∈S, first obtained by K?be and Bieberbach. For the corresponding class Σ of univalent functions in the exterior disk, Goluzin found in 1943 by variational methods the corresponding best possible pointwise estimates of ?"(z)/?'(z) for ψ∈Σ. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain an area-type theorem which has Goluzin's pointwise estimate as a corollary. This shows that Goluzin's estimate, like the K?be-Bieberbach estimate, is firmly rooted in areabased methods. The appearance of elliptic integrals finds a natural explanation: they arise because a certain associated covering surface of the Riemann sphere is a torus.     

The best <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> approximation of the Laplace operator by linear bounded operators in the classes of functions of two and three variables

Close two-sided estimates are obtained for the best approximation in the space L _p (? ^m ), m = 2 and 3, 1 ≤ p ≤ ∞, of the Laplace operator by linear bounded operators in the class of functions for which the second power of the Laplace operator belongs to the space L _p (? ^m ). We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class given with an error. We present an operator whose deviation from the Laplace operator is close to the best.     

First order absolute moment of Meyer-König and Zeller operators and their approximation for some absolutely continuous functions

A sharp estimate is given for the first order absolute moment of Meyer-König and Zeller operators M _n . This estimate is then used to prove convergence of approximation of a class of absolutely continuous functions by the operators M _n . The condition considered here is weaker than the condition considered in a previous paper and the rate of convergence we obtain is asymptotically the best possible.     

Non-oscillating Paley-Wiener functions

A non-oscillating Paley-Wiener function is a real entire functionf of exponential type belonging toL ₂(R) and such that each derivativef ⁽ⁿ⁾,n=0, 1, 2,…, has only a finite number of real zeros. It is established that the class of such functions is non-empty and contains functions of arbitrarily fast decay onR allowed by the convergence of the logarithmic integral. It is shown that the Fourier transform of a non-oscillating Paley-Wiener function must be infinitely differentiable outside the origin. We also give close to best possible asymptotic (asn→∞) estimates of the number of real zeros of then-th derivative of a functionf of the class and the size of the smallest interval containing these zeros.     

Decay parameter and related properties of 2-type branching processes

We consider the decay parameter, invariant measures/vectors and quasi-stationary dis- tributions for 2-type Markov branching processes. Investigating such properties is crucial in realizing life period of branching models. In this paper, some important properties of the generating functions for 2-type Markov branching q-matrix are firstly investigated in detail. The exact value of the decay parameter λC of such model is given for the communicating class C = Z+2 \ 0. It is shown that this λC can be directly ...     

Irrationality of linear combinations of eigenvectors

A givenn ×n matrix of rational numbers acts onC ^π and onQ ^π. We assume that its characteristic polynomial is irreducible and compare a basis of eigenvectors forC ^π with the standard basis forQ ^π. Subject to a hypothesis on the Galois group we prove that vectors from these two bases are as independent of each other as possible.     

Associating curves of low genus to infinite nilpotent groups via the zeta function

It is known from work of du Sautoy and Grunewald in [duSG1] that the zeta functions counting subgroups of finite index in infinite nilpotent groups depend upon the behaviour of some associated system of algebraic varieties on reduction modp. Further to this, in [duS1, duS2] du Sautoy constructed a group whose local zeta function was determined by the number of points on the elliptic curveE:Y ²=X ³−X. In this work we generalise du Sautoy’s construction to define a class of groups whose local zeta functions are dependent upon the number of points on the reduction of a given elliptic curve with a rational point. We also construct a class of groups that behave the same way in relation to any curve of genus 2 with a rational point. We end with a discussion of problems arising from this work.     

Comonotone Adaptive Interpolating Splines

For any given data we propose the construction of an interpolating spline of class C ¹, which is either a quadratic polynomial or a linear/linear rational function between the knots, and preserves the monotonicity of the data on the sections of rational intervals. We prove the uniqueness and existence of this spline. Numerical tests show good approximation properties and flexibility due to the non-coincidence of the given data arguments and the spline knots which can be chosen freely.     

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.     

Inequalities for Rational Functions

A new hyperbolic area estimate for the level sets of finite Blaschke products is presented. The following inversion of the usual Sobolev embedding theorem is proved:

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Hereris a rational function of degreenwith poles outside . This estimate implies a new inverse theorem for rational approximation of analytic functions with respect to areaL^pnorm.     

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class of all rational functions of order (n,m) are considered. Let ρ _n,m = ρ _n,m (f;E) be the distance of f in the uniform metric on E from the class . We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation { ρ _n,m(n) } _n=0 ^∞ , m(n)/n → θ ∈ (0,1] as n → ∞ . In particular, we give an upper estimate for the liminf _{n →∞} ρ _n,m(n) ^1/(n+m(n)) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators. June 16, 1997. Date revised: December 1, 1997. Date accepted: December 1, 1997. Communicated by Ronald A. DeVore.     

Some geometric aspects of the calculus of fractions

We give necessary and sufficient conditions on class S of maps of a category C so that a good calculus of fractions is possible in C[S ^-1], and a geometric characterization of the communitative diagrams in the category of fractions. These conditions are also described simply in terms of Grothendiek topologies.We characterize the categories which arise in this manner by the fact that the functor C C[S ^-1] is uniformly flat and give some applications of this result.     

Interpolation in the unit ball ofC n

A necessary and sufficient condition is given for a discrete multiplicity variety in the unit ballB _n ofC ⁿ to be an interpolating variety for weighted spaces of holomorphic functions inB _n . Partially supported by NSF Grant DMS-9706376.     

Improving bounds on the minimum Euclidean distance for block codes by inner distance measure optimization

The minimum Euclidean distance is a fundamental quantity for block coded phase shift keying (PSK). In this paper we improve the bounds for this quantity that are explicit functions of the alphabet size q, block length n and code size |C|. For q=8, we improve previous results by introducing a general inner distance measure allowing different shapes of a neighborhood for a codeword. By optimizing the parameters of this inner distance measure, we find sharper bounds for the outer distance measure, which is Euclidean.The proof is built upon the Elias critical sphere argument, which localizes the optimization problem to one neighborhood. We remark that any code with q=8 that fulfills the bound with equality is best possible in terms of the minimum Euclidean distance, for given parameters n and |C|. This is true for many multilevel codes.     

Roots of Dehn twists

Margalit and Schleimer found examples of roots of the Dehn twist t _C about a nonseparating curve C in a closed orientable surface, that is, homeomorphisms h such that h ⁿ   =  t _C in the mapping class group. Our main theorem gives elementary number-theoretic conditions that describe the n for which an n ^th root of t _C exists, given the genus of the surface. Among its applications, we show that n must be odd, that the Margalit-Schleimer roots achieve the maximum value of n among the roots for a given genus, and that for a given odd n, n ^th roots exist for all genera greater than (n − 2)(n − 1)/2. We also describe all n ^th roots having n greater than or equal to the genus.     

Global convergence of the method of successive approximations on S1

The class of iterating functions of C(S¹, S¹) for which the method of successive approximations converges for any starting point is characterized; such characterization is given by (i) the existence of a fixed point; (ii) the non-existence of periodic points of an even period.     

Error estimation of a kind of rational spline

This paper deals with the approximation properties of a kind of rational spline with linear denominator when the function being interpolated is C³$C^3$ in an interpolating interval. Error estimate expressions of interpolating functions are derived, convergence is established, the optimal error coefficient, _ci$c_i$, is proved to be symmetric about the parameters of the rational interpolation and it is bounded. Finally, the precise jump measurements of the second derivatives of the interpolating function at the knots are given.