Korobov Spaces with Respect to Digital Orthonormal Bases

 For an orthonormal basis (ONB) of we define classes of functions according to the order of decay of the Fourier coefficients with respect to the considered ONB . The rate is expressed in the real parameter α. We investigate the following problem: What is the order of decay, if any, when we consider with respect to another ONB ? If the function is expressable as an absolutely convergent Fourier series with respect to , we give bounds for the new order of decay, which we call . Special attention is given to digital orthonormal bases (dONBs) of which the Walsh and Haar systems are examples treated in the present paper. Bounding intervals and in several cases explicit values for are given for the case of dONBs. An application to quasi-Monte Carlo numerical integration is mentioned.     

Digital Expansions with Respect to Different Bases

We consider the g-ary expansion N=∑ _k b _k (N, g)g ^k of non-negative integers N and prove various results on the distribution and the mean value of the k-th digit b _k (N, g) if g varies in an interval of the form 2≤g≤N ^η. As an application we also consider the average value of the sum-of-digits function s(N, g)=∑ _k b _k (N, g). Received 5 November 2001 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Cesaro Summability with Respect to Two-Parameter Walsh Systems

 The one- and two-parameter Walsh system will be considered in the Paley as well as in the Kaczmarz rearrangement. We show that in the two-dimensional case the restricted maximal operator of the Walsh–Kaczmarz (C, 1)-means is bounded from the diagonal Hardy space H ^p to L ^p for every . To this end we consider the maximal operator T of a sequence of summations and show that the p-quasi-locality of T implies the same statement for its two-dimensional version T _α. Moreover, we prove that the assumption is essential. Applying known results on interpolation we get the boundedness of T _α as mapping from some Hardy–Lorentz spaces to Lorentz spaces. Furthermore, by standard arguments it will be shown that the usual two-parameter maximal operators of the (C, 1)-means are bounded from L ^p spaces to L ^p if . As a consequence, the a.e. convergence of the (C, 1)-means will be obtained for functions such that their hybrid maximal function is integrable. Of course, our theorems from the two-dimensional case can be extended to higher dimension in a simple way. (Received 20 April 2000; in revised form 25 September 2000)     

Isometric Shifts and Metric Spaces

 Let M be a complete metric space. If admits an isometric shift, then M is separable. (Received 2 February 2001; in revised form 9 April 2001)     

On Symmetric Schauder Bases in Non-Archimedean Fréchet Spaces

 It is proved that any infinite-dimensional non-archimedean Fréchet space with a symmetric basis is isomorphic to c ₀ or ?^ℕ. A similar result is shown for homogeneous bases. It is also proved that any infinite-dimensional nuclear non-archimedean Fréchet space with a subsymmetric basis is isomorphic to ?^ℕ. In fact, much stronger results are obtained. Received August 27, 2001; in revised form February 8, 2002     

Homomorphisms on Lipschitz Spaces

 Denote by the family of all real valued functions on a metric space which satisfy a Lipschitz condition on the compact (bounded) subsets of X. We prove that every homomorphism on is the evaluation at some point of X if and only if X is realcompact (every closed bounded subset of X is compact). (Received 4 November 1998; in revised form 31 May 1999)     

Critical Exponent of Short Even Filters andBurt-Adelson Biorthogonal Wavelets

 We determine the critical exponent of all positive filters having an even residual of degree two and present an extension to the case of degree four. We apply these results to Burt-Adelson filters, thus determining the critical exponent of all the biorthogonal wavelets they generate. After this, we consider the problem of smoothing the dual wavelets by considering longer dual filters: we first create new wavelets by imposing an extra zero at π on the new filters and study their regularity by determining all the critical exponents. Then we release this condition on the filters and present the results of a numerical simulation intended to maximize the Sobolev regularity. (Received 10 February 2000; in revised form 16 May 2000)     

Constant Mean Curvature Surfaces with Boundaryin Hyperbolic Space

 We study constant mean curvature compact surfaces immersed in hyperbolic space with non-empty boundary (=H-surfaces). We prove that the only H-surfaces with boundary circular and 0≤∣H∣≤1, are the umbilical examples. When the surface is embedded, conditions to be umbilical are given. Finally, we characterize umbilical surfaces bounded by a circle among all H-discs with small area. Received 27 March 1997; in final form 11 June 1998     

Haar Function Based Estimates of the Star-Discrepancy of Plane Digital Nets

 We apply the Haar function system to estimate the star-discrepancy of special digital (t,m,s)-nets in dimension . We use a basic technique based on discretization combined with an exact calculation of the discrete star-discrepancy. (Received 30 August 1999; in revised form 17 January 2000)     

Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces

 Let be a nilpotent connected and simply connected Lie group, and an analytic subgroup of G. Let , be a unitary character of H and let . Suppose that the multiplicities of all the irreducible components of τ are finite. Corwin and Greenleaf conjectured that the algebra of the differential operators on the Schwartz-space of τ which commute with τ is isomorphic to the algebra of H-invariant polynomials on the affine space . We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in . We prove also that if is an ideal of , then the finite multiplicities of τ is equivalent to the fact that the algebra is commutative. (Received 15 November 2000)     

Some characterizations of totally umbilical surfaces in three-dimensional warped product spaces

Surfaces with positive definite second fundamental form in a Riemannian, three-dimensional warped product space are considered. A formula expressing the Gaussian curvature with respect to this new metric on the surface in terms of the Gaussian and mean curvature of the first fundamental form is presented. This formula is then used to give some characterizations of compact, totally umbilical surfaces. Postdoctoral researcher of the F.W.O. Vlaanderen.     

On Hardy-Type Inequalities and Hankel Transforms

 For the real Hardy spaces , we shall show the Hardy type integral inequalities, and applying the inequalities we shall establish the Hardy’s inequalities with respect to Hankel transforms. Received 21 October 1997; in revised form 19 October 1998     

A variational problem for submanifolds in a sphere

Let be an n-dimensional submanifold in an (n + p)-dimensional unit sphere S ^{n + p} , M is called a Willmore submanifold (see [11], [16]) if it is a critical submanifold to the Willmore functional , where is the square of the length of the second fundamental form, H is the mean curvature of M. In [11], the second author proved an integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in S ^{n + p} . In this paper, we discover that a similar integral inequality of Simons’ type still holds for the critical submanifolds of the functional . Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation.     

Complete space-like hypersurfaces with constant scalar curvature

Let M ⁿ be a complete space-like hypersurface with constant normalized scalar curvature R in the de Sitter space S ^{n + 1} ₁ and denote . We prove that if the norm square of the second fundamental form of M ⁿ satisfies , then either and M ⁿ is a totally umbilical hypersurface; or , and, up to rigid motion, M ⁿ is a hyperbolic cylinder . Received: 8 February 2001 / Revised version: 27 April 2001     

On the area of constant mean curvature discs and annuli with circular boundaries

It is still an open question whether a constant mean curvature (CMC) disc which is bounded by a circle is necessarily a spherical cap or a flat disc. The authors together with López [1] recently showed that the only stable CMC discs which are bounded by a circle are spherical caps. In this paper we derive lower bounds for the area of constant mean curvature discs and annuli with circular boundaries in 3-dimensional space forms. Received November 8, 1999; in final form January 18, 2000 / Published online March 12, 2001     

Banach Spaces Admitting a Separating Polynomial and L P Spaces

 We prove that in a Banach space admitting a separating polynomial, each weakly null normalized sequence has a subsequence which is equivalent to the usual basis of some , p an even integer. We show that for each even integer p, the Schatten class admits a separating polynomial. For a space with a basis admitting a 4-homogeneous separating polynomial, we relate the unconditionality of the basis with the existence of certain type of polynomials defined in terms of infinite symmetric matrices. We find relations between the properties of the entries of these matrices and the geometrical structure of the space. Finally we study the existence of convex 4-homogeneous separating polynomials. Received 3 January 2001     

Hardy Spaces of Dirichlet Series and Their Composition Operators

 In [9], Hedenmalm, Lindqvist and Seip introduce the Hilbert space of Dirichlet series with square summable coefficients , and begin its study, with modern functional and harmonic analysis tools. The space is an analogue for Dirichlet series of the space for Fourier series. We continue their study by introducing , an analogue to the spaces . Thanks to Bohr’s vision of Dirichlet series, we identify with the Hardy space of the infinite polydisk . Next, we study a variant of the Poisson semigroup for Dirichlet series. We give a result similar to the one of Weissler ([25]) about the hypercontractivity of this semigroup on the spaces . Finally, following [8], we determine the composition operators on , and we compare some properties of such an operator and of its symbol. Received October 3, 2001; in revised form January 16, 2002 Published online July 12, 2002     

Complete Hypersurfaces with Constant Mean Curvature in a Unit Sphere

By investigating hypersurfaces M ⁿ in the unit sphere S ⁿ⁺¹(1) with constant mean curvature and with two distinct principal curvatures, we give a characterization of the torus S ¹(a) × , where . We extend recent results of Hasanis et al. [5] and Otsuki [10].     

The Existence of Good Extensible Polynomial Lattice Rules

Extensible (polynomial) lattice rules have been introduced recently and they are convenient tools for quasi-Monte Carlo integration. It is shown in this paper that for suitable infinite families of polynomial moduli there exist generating parameters for extensible rank-1 polynomial lattice rules such that for all these infinitely many moduli and all dimensions s the quantity R ^(s) and the star discrepancy are small. The case of Korobov-type polynomial lattice rules is also considered.Received April 30, 2002; in revised form August 21, 2002 Published online April 4, 2003     

A Moebius characterization of Veronese surfaces in $S^n$

Let be an umbilic-free submanifold in with I and II as the first and second fundamental forms. An important Moebius invariant for in Moebius differential geometry is the so-called Moebius form , defined by , where is a local basis of the tangent bundle with dual basis , is a local basis of the normal bundle, is the mean curvature vector and . In this paper we prove that if is an umbilics-free immersion of 2-sphere with vanishing Moebius form , then there exists a Moebius transformation and a 2k-equator with such that is the Veronese surface. Received August 12, 1999 / Published online March 12, 2001