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     

Global Representatives of Colombeau Holomorphic Generalized Functions

It is proved that a holomorphic generalized function in the sense of Colombeau has a representative consisting of a net of holomorphic functions. More generally, the existence of global representatives of Colombeau solutions to elliptic partial differential equations is proved. Further, it is shown that a generalized holomorphic function on Ω, which is equal to zero in an open set , is equal to zero. First author partially supported by FWF (Austria), grants P16820 and Y237. Third author partially supported by MNZŽ (Serbia), project 144016.     

Fréchet-Valued Real Analytic Functions on Fréchet Spaces

The aim of this paper is to establish the equivalence between the real and topologically real analyticity of Fréchet-valued functions on Fréchet spaces. This is an extension of recent results of Bonet and Domanski [1, 2] to infinite dimension.Received February 19, 2002; in revised form June 25, 2002 Published online May 16, 2003     

Measures on the Product of Compact Spaces

 If K is an uncountable metrizable compact space, we prove a “factorization” result for a wide variety of vector valued Borel measures μ defined on K ⁿ . This result essentially says that for every such measure μ there exists a measure μ′ defined on K such that the measure μ of a product A ₁ × ⋯ × A _n of Borel sets of K equals the measure μ′ of the intersection A ₁′∩⋯∩A _n ′, where the A _i ′’s are certain transforms of the A _i ’s. Partially supported by DGICYT grant PB97-0240. Received August 23, 2001; in revised form March 21, 2002     

A Discontinuous Colombeau Differential Calculus

Starting from the Colombeau Generalized Functions, the sharp topologies and the notion of generalized points, we introduce a new kind of differential calculus (for functions between totally disconnected spaces). We also define here the notions of holomorphic generalized functions (in this new framework) and generalized manifold. Finally we give an answer to a question raised in [6].Research partially supported by CNPq (Proc 300652/95-0).     

On the Dimension of Coinvariants of Permutation Representations

Let be a univariate, separable polynomial of degree n with roots x ₁,…,x _n in some algebraic closure of the ground field . It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on by permutation of the coordinates and each defines a relation P − P(x ₁,…,x _n ) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an H-basis of G-invariant relations if and only if the algebra of coinvariants has dimension ‖G‖ over . To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite group G has dimension ‖G‖ if and only if G = Σ _n acting via the tautological permutation representation.     

Bounds in Polynomial Rings over Artinian Local Rings

Let R be a (mixed characteristic) Artinian local ring of length l and let X be an n-tuple of variables. We prove that several algebraic constructions in the ring R[X] admit uniform bounds on the degrees of their output in terms of l, n and the degrees of the input. For instance, if I is an ideal in R[X] generated by polynomials g _i of degree at most d and if f is a polynomial of degree at most d belonging to I, then f = q ₁ f ₁ + ··· + q _s f _s , for some q _i of degree bounded in terms of d, l and n only. Similarly, the module of syzygies of I is generated by tuples all of whose entries have degree bounded in terms of d, l and n only.     

Equiframed Curves – A Generalization of Radon Curves

Equiframed curves are centrally symmetric convex closed planar curves that are touched at each of their points by some circumscribed parallelogram of smallest area. These curves and their higher-dimensional analogues were introduced by Peczynski and Szarek (1991, Math Proc Cambridge Philos Soc 109: 125–148). Radon curves form a proper subclass of this class of curves. Our main result is a construction of an arbitrary equiframed curve by appropriately modifying a Radon curve. We give characterizations of each type of curve to highlight the subtle difference between equiframed and Radon curves and show that, in some sense, equiframed curves behave dually to Radon curves.Research supported by a grant from a cooperation between the Deutsche Forschungsgemeinschaft in Germany and the National Research Foundation in South Africa. Parts of this paper were written during a visit to the Department of Mathematics, Applied Mathematics and Astronomy of the University of South Africa.     

Distributional Fourier Transform and Convolution Associated to Chébli-Trimèche Hypergroups

 In this paper we investigate the convolution and the generalized Fourier transform related to Chébli-Trimèche hypergroups on new spaces of distributions. Boundedness, smoothness, uniqueness, and inversion theorems are established for this transform, as well as the main properties of the convolution. The theory developed is used in solving a differential equation involving a singular differential operator. (Received 26 January 2000; in final form 24 July 2001)     

Zeros of Generalized Holomorphic Functions

Let f be a generalized holomorphic function on a connected open set . It is proved that f equals zero if and only if there exists a smooth curve and a set A of positive (one-dimensional) measure such that f takes zero value on A. Also, a holomorphic generalized function different from zero on the disc, which takes zero values on a dense G _δ-set of the disc, is constructed. The generalized zero set of a holomorphic function is introduced and studied in an analogous way.     

New spaces of functions and hyperfunctions for Hankel transforms and convolutions

In this paper we study the Hankel transformation and convolution on certain spaces of entire functions and its dual that is a space of hyperfunctions and contains the (even)-Schwartz space S _e ′. We prove that the Hankel transform is an automorphism of . Also the Hankel convolutors of are investigated. Authors’ addresses: Jorge J. Betancor, Claudio Jerez and Lourdes Rodríguez-Mesa, Departamento de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez, s/n, 38271 La Laguna (Sta. Cruz de Tenerife), Espa?a; Sandra M. Molina, Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350 (7600), Mar del Plata, Argentina     

A counterexample on non-archimedean regularity

A non-regular inductive sequence of non-archimedean reflexive Fréchet spaces is constructed. On the other hand, it is proved that every inductive sequence of reflexive Banach spaces over a spherically complete field is regular. Also, some applications are given.     

On metrizability of compactoid sets in non-archimedean locally convex spaces

In 2003, N. De Grande-De Kimpe, J. Kąkol and C. Perez-Garcia using t-frames and some machinery concerning tensor products proved that compactoid sets in non-archimedean (LM)-spaces (i.e. the inductive limits of a sequence of non-archimedean metrizable locally convex spaces) are metrizable. In this paper we show a similar result for a large class of non-archimedean locally convex space with a £-base, i.e. a decreasing base (U_α)_αN^N of neighbourhoods of zero. This extends the first mentioned result since every non-archimedean (LM)-space has a £-base. We also prove that compactoid sets in non-archimedean (DF)-spaces are metrizable.     

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.     

Unbounded Disjointness Preserving Linear Functionals

Let X be a locally compact Hausdorff space and C ₀(X) the Banach space of continuous functions on X vanishing at infinity. In this paper, we shall study unbounded disjointness preserving linear functionals on C ₀(X). They arise from prime ideals of C ₀(X), and we translate it into the cozero set ideal setting. In particular, every unbounded disjointness preserving linear functional of c ₀ can be constructed explicitly through an ultrafilter on complementary to a cozero set ideal. This ultrafilter method can be extended to produce many, but in general not all, such functionals on C ₀(X) for arbitrary X. We also make some remarks where C ₀(X) is replaced by a non-commutative C*-algebra.     

A bounded N-tuplewise independent and identically distributed counterexample to the CLT

Summary. A sequence of random variables X ₁,X ₂,X ₃,… is said to be N-tuplewise independent if X i ₁,X i ₂,…,X i _N are independent whenever (i ₁,i ₂,…,i _N ) is an N-tuple of distinct positive integers. For any fixed N∈ℤ⁺, we construct a sequence of bounded identically distributed N-tuplewise independent random variables which fail to satisfy the central limit theorem. Received: 17 May 1996 / In revised form: 28 January 1998     

Orlicz-Pettis Polynomials on Banach Spaces

 We introduce the class of Orlicz-Pettis polynomials between Banach spaces, defined by their action on weakly unconditionally Cauchy series. We give a number of equivalent definitions, examples and counterexamples which highlight the differences between these polynomials and the corresponding linear operators. (Received 17 May 1999; in revised form 6 October 1999)     

Real analytic generalized functions

Real analytic generalized functions are introduced and investigated. The analytic singular support and analytic wave front of a generalized function in are introduced and described. Authors’ addresses: S. Pilipović, Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovića 4, 21000 Novi Sad, Serbia; D. Scarpalezos, U.F.R. de Mathématiques, Université Paris 7, 2 Place Jussieu, Paris 5^e, 75005, France; V. Valmorin, Université des Antilles et de la Guyane, Département Math-Info, Campus de Fouillole, 97159 Pointe á Pitre Cedex, France