On local oscillatory integrals with variable Calderón-Zygmund kernels

Letp(1, ). In this paper, the authors investigate the uniformL ^p ( ⁿ ) in of the oscillatory singular integral operatorT defined by

Asymptotics of polynomials orthogonal with respect to varying measures

Letd be a finite positive Borel measure on the interval [0, 2] such that >0 almost everywhere; andW _n be a sequence of polynomials, degW _n =n, whose zeros (w _n ,1,,w _n,n lie in [|z|1]. Let d _n <> for each_nN, whered _n =d/|W _n (e ⁱ )|². We consider the table of polynomials _n,m such that for each fixednN the system _n,m,mN, is orthonormal with respect tod _n . If

On the relationship between differentiability conditions and existence of a strong gradient

It is proved that for any n 2 there exists continuous function f : ⁿ which is differentiable almost everywhere, but has no strong gradient almost everywhere.     

Subadditive functions on modules and subgroups of abelian groups

Thepositive half A ₊ of an ordered abelian groupA is the set {x Ax 0} andM A ₊ is amodule if for allx, y M alsox + y, |x – y| M. If A ₊ \M thenM() is the module generated byM and. S M isunbounded inM if(x M)(y S)(x y) and isdense inM if (x₁, x₂ M)(y S) (x₁ <>₂ x₁ y x₂). IfM is a module, or a subgroup of any abelian group, a real-valuedg: M R issubadditive ifg(x + y) g(x) + g(y) for allx, y M. The following hold:

An extension of Knight's theorem on Cauchy distribution

If is a probability on , the set of distributions ofa+pX wherea,p>0 andX has distribution is called the type of . F. B. Knight has shown that if a type has no atom and if it is invariant byi:x–1/x, the type must be the Cauchy one. We show here thati can be replaced by any Cayley nonaffine function.

Parabolische Kollineationen,die eine ebene konvexe Menge invariant lassen

Collineations ₁, ₂ of PG(2, ) leaving invariant a compact convex setK ² are called parabolic if |K Fix _i|=1. Conditions are stated under which the existence of ₁, ₂ imply that K is an ellipse.

Herrn Helmut R. Salzmann zum 65. Geburtstag gewidmet     

An Elementary Approach to Quasi-Isometries of Tree x ℝ n

We prove by elementary means a regularity theorem for quasi-isometries of T x ⁿ (where T denotes an infinite tree), and of many other metric spaces with similar combinatorial properties, e.g. Cayley graphs of Baumslag–Solitar groups. For quasi-isometries of T x ⁿ, it states that the image of {x} x ⁿ (xT) is uniformly close to {y} x ⁿ for some yT, and there is a well-defined surjection . Even stronger, the image of a quasi-isometric embedding of ⁿ⁺¹ in T x ⁿ is close to (a geodesic in T)T)x ⁿ.     

Total oscillation diminishing property for scalar conservation laws

Summary. We prove a BV estimate for scalar conservation laws that generalizes the classical Total Variation Diminishing property. In fact, for any Lipschitz continuous monotone :, we have that |(u)|_TV() is nonincreasing in time. We call this property Total Oscillation Diminishing because it is in contradiction with the oscillations observed recently in some numerical computations based on TVD schemes. We also show that semi-discrete Total Variation Diminishing finite volume schemes are TOD and that the fully discrete Godunov scheme is TOD.Mathematics Subject Classification (2000): 35L65, 35K55, 65M20     

Constrained normalization of Hamiltonian systems and perturbed Keplerian motion

Summary Consider a Hamiltonian system (H, _2n ,). LetM be a symplectic submanifold of (_2n ,). The system (H, _2n ,) constrained toM is (HM, M, M). In this paper we give an algorithm which normalizes the system on _2n in such a way that restricted toM we have normalized the constrained system. This procedure is then applied to perturbed Kepler systems such as the lunar problem and the main problem of artificial satellite theory.

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Zusammenfassung Wir betrachten ein Hamiltonisches System (H, _2n ,). SeiMein symplectisches Submanifold von (_2n ,). Das System (H, _2n ,), aufM beschränkt, ist (HM,M,M). In der vorliegenden Arbeit wird ein Algorithmus vorgeschlagen, der dieses System so auf _2n normalisiert, daß das aufM beschränkte System auch normalisiert ist. Dieser Algorithmus wird dann auf gestörte Keplersysteme, wie z. B. das Hill-sche Mondproblem und das Hauptproblem der Theorie der künstlichen Satelliten, angewendet.

     

A characterization of multivariate quasi-interpolation formulas and its applications

Summary Let be a compactly supported function on ^s andS () the linear space withgenerator ; that is,S () is the linear span of the multiinteger translates of . It is well known that corresponding to a generator there are infinitely many quasi-interpolation formulas. A characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasi-interpolants. We introduce a generalization of interpolation called -interpolation and a notion of higher order quasi-interpolation called -approximation. A characterization of -approximants similar to that of quasi-interpolants is obtained with similar applications. Among these applications are estimating least-squares approximants without matrix inversion, surface fitting to incomplete or semi-scattered discrete data, and constructing generators with one-point quasi-interpolation formulas. It will be seen that the exact values of the generator at the multi-integers ^s facilitates the above study. An algorithm to yield this information for box splines is discussed.Supported by the National Science Foundation and the U.S. Army Research Office     

Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh

Summary We study here the discretisation of the nonlinear hyperbolic equationu _t +div(vf(u))=0 in ³ × ₊, with given initial conditionu(.,0)=u ₀(.) in ², wherev is a function from ² × ₊ to ² such that divv=0 andf is a given nondecreasing function from to . An explicit Euler scheme is used for the time discretisation of the equation, and a triangular mesh for the spatial discretisation. Under a usual stability condition, we prove the convergence of the solution given by an upstream finite volume scheme towards the unique entropy weak solution to the equation.     

On local oscillatory integrals with variable Calderón-Zygmund kernels, II

Let , be a real analytic function or a real-C function on ⁿ andk be a variable Calderón-Zygmund kernel. Define the oscillatory singular integral operatorT by

Fibred links and a construction of real singularities via complex geometry

This note gives a method for constructing real analytic maps from ²ⁿ into ², with an isolated critical point at 0 ²ⁿ , for alln>1. This provides infinite families of real singularities which fiber a la Milnor.Research partially supported by CONACYT, Mexico, grant 1206-E92103.     

On the uniqueness of canonical points in the Hobby-Rice theorem

According to the Hobby-Rice theorem for anyn-dimensional subspaceU _n ofL ₁([a, b], ) ( positive, finite, nonatomic) there exist points =s ₀x ₁x _m+1=b, where 0mn, such that

Conditionally positive functions andp-norm distance matrices

In [4], deep results were obtained concerning the invertibility of matrices arising from radial basis function interpolation. In particular, the Euclidean distance matrix was shown to be invertible for distinct data. In this paper, we investigate the invertibility of distance matrices generated byp-norms. In particular, we show that, for anyp(1, 2), and for distinct pointsx ¹,,x ^{n d} , wheren andd may be any positive integers, with the proviso thatn2, the matrixA ^n×n defined by

Note on initial topologies on rational vector spaces induced by realvalued linear mappings

LetG be a vector space over the field of rational numbers andf, g:G -linear mappings. equipped with the usual norm topology. Denote by _f , _g the initial topologies onG induced byf respectivelyg.Then the following result holds: If there is a nonvoid open setU whose complement contains at least one inner point such thatf ^–1 U _g , then there is ac withf=cg. In particular, iff0, the topologies coincide.Furthermore, a -linear mappingh: (G, _f )(G, _g ) is continuous if and only if there is a real constantc withg ^o h=cf.Dedicated to Professor János Aczél on his 60th birthday     

Convex polynomial and spline approximation in C[−1, 1]

We prove that a convex functionf C[–1, 1] can be approximated by convex polynomialsp _n of degreen at the rate of ₃(f, 1/n). We show this by proving that the error in approximatingf by C² convex cubic splines withn knots is bounded by ₃(f, 1/n) and that such a spline approximant has anL third derivative which is bounded by n³₃(f, 1/n). Also we prove that iff C²[–1, 1], then it is approximable at the rate ofn ^–2 (f, 1/n) and the two estimates yield the desired result.Communicated by Ronald A. DeVore.     

A test of convergence of Fourier series with respect to multiplicative systems,analogous to the Jordan test

, {p _n} _n=0 (p₀=1, _n2 n2). : x f(t) V(G)