Cardinal translation invariant Tchebycheffian B-splines

Cardinal Tchebycheffian B-splines, defined by weight functions of the form with , are investigated. It is shown that these B-splines are translation invariant, have a geometric representation and satisfy a generalized Hermite-Genocchi formula. For pure exponential weight functions the above results lead to a product type expression for the Fourier transform of the cardinal exponential B-splines, showing that these functions are convolutions of lower order ones. Similar conclusions are obtained for the corresponding Greens' functions.     

Partially translation invariant linear systems

The subject of this paper is the analysis of abstract linear systems in several variables, i.e. continuous linear mappings from Banach space valued test functions as inputs to Banach space valued distributions as outputs, N being partially translation invariant in the sense that it commute with translations from a given closed subgroup G of the additive group. The classical convolution representation of the invariant case generalizes to a Fourier series representation whose coefficients N_k are linear systems commuting with translations from the vector subgroup lin G generated by G. Thus N is essentially a "Fourier superposition" of classical systems. It is shown that causality of N with respect to a subsemigroup P of implies causality of the N_k which in turn corresponds to carrier conditions on the N_k.Dedicated to Professor Heinz König on the occasion of his 50th birthday.     

Monotone translation invariant valuations on convex bodies

Archiv der Mathematik -     

Wavelet subspaces invariant under groups of translation operators

We study the action of translation operators on wavelet subspaces. This action gives rise to an equivalence relation on the set of all wavelets. We show by explicit construction that each of the associated equivalence classes is non-empty.     

FFT-based homogenization on periodic anisotropic translation invariant spaces

In this paper we derive a discretisation of the equation of quasi-static elasticity in homogenization in form of a variational formulation and the so-called Lippmann–Schwinger equation, in anisotropic spaces of translates of periodic functions. We unify and extend the truncated Fourier series approach, the constant finite element ansatz and the anisotropic lattice derivation. The resulting formulation of the Lippmann–Schwinger equation in anisotropic translation invariant spaces unifies and analyses for the first time both the Fourier methods and finite element approaches in a common mathematical framework. We further define and characterize the resulting periodised Green operator. This operator coincides in case of a Dirichlet kernel corresponding to a diagonal matrix with the operator derived for the Galerkin projection stemming from the truncated Fourier series approach and to the anisotropic lattice derivation for all other Dirichlet kernels. Additionally, we proof the boundedness of the periodised Green operator. The operator further constitutes a projection if and only if the space of translates is generated by a Dirichlet kernel. Numerical examples for both de la Vallée Poussin means and Box splines illustrate the flexibility of this framework.     

On the invariant translation approximation property for discrete groups

Recently J.Roe considered the question of whether for a discrete group the reduced group -algebra is the fixed point algebra of Ad acting on the uniform Roe algebra . is said to have the invariant translation approximation property in this case. We consider a slight generalization of this property which, for exact , is equivalent to the operator space approximation property of . We also give a new characterization of exactness and a short proof of the equivalence of exactness of and exactness of for discrete groups.

     

Harmonic spaces associated with non-symmetric translation invariant Dirichlet forms

Continuing earlier work on construction of harmonic spaces from translation invariant Dirichlet spaces defined on locally compact abelian groups, it is shown that the potential kernel for a non-symmetric translation invariant Dirichlet form on a locally compact abelian group under the extra assumptions that

On translation invariant operators which preserve the B-spline recurrence

It was observed in [4] that the Hilbert transform of the univariate B-spline preserves the B-spline recurrence. Motivated by this observation, we characterize translation invariant operators that preserve the multivariate B-spline recurrence and analogous results are also provided for the multivariate cube spline. Charles A. Micchelli was supported in part by the US National Science of Foundation under grant CCR-0407476. Yuesheng Xu was supported in part by the US National Science Foundation under grant CCR-0407476, by the Natural Science Foundation of China under grant 10371122, by the Chinese Academy of Sciences under the program “One Hundred Distinguished Chinese Scientists” and by Ministry of Education, People’s Republic of China, under the Changjian Scholarship through Zhongshan University.     

A compactness criterion for translation invariant Banach spaces of functions

Установлено, что для и нвариантных относит ельно сдвига банаховых про странствВ (классов) измеримых фу нкций на локально ком пактной группе, удовлетворяю щих некоторым дополнительным усло виям, справедлив след ующий критерий компактнос ти. Замкнутое подмножествоМ?В яв ляется компактным вВ тогда и только тогда, к огда оно удовлетворя ет условиям:

1. М ограничено вВ;
2. для каждого ?>0 сущест вуетkε?(G) такое, что ∥k*f-f∥_BfεМ;
3. для каждого ?0 существ уетh??(G) такое, что ∥hf - f ∥_BfεМ.

     

Flat translation invariant surfaces in the 3-dimensional Heisenberg group

Flat translation invariant surfaces in 3-dimensional Heisenberg group are classified.     

Dimension of the space of unitary equivariant translation invariant tensor valuations

Following the work of Semyon Alesker in the scalar valued case and of Thomas Wannerer in the vector valued case, the dimensions of the spaces of continuous translation invariant and unitary equivariant tensor valuations are computed. In addition, a basis in the vector valued case is presented.     

On the ground state energy of the translation invariant Pauli-Fierz model

In this note, we determine the ground state energy of the translation invariant Pauli-Fierz model to subleading order with respect to powers of the finestructure constant and prove rigorous error bounds of order . A main objective of our argument is its brevity.