Moves towards Authentic Freedom. Church and State in Switzerland, and Beyond

Many of the Swiss Cantons have regulated the relations between church and state by establishing, in their public law, corporations at the levels of the municipality and of the canton. The role and the rights of these corporations, especially obligatory membership in them, is the object of ongoing political and legal debate. Both on the side of the courts and of the church, the present system has come under scrutiny, while the corporation representatives and also a majority of the population seem intent on maintaining it. This paper explains and examines the presently valid church-state relations, focusing on the Canton of Zurich, and looks at the suggestions for reform elaborated by an experts’ commission instituted by the Conference of Swiss Bishops. In conclusion, it presents some more general reflections on the challenges to individual and corporative religious freedom today, in Switzerland and beyond

From Frazier-Jawerth characterizations of Besov spaces to Wavelets and Decomposition spaces

This article describes how the ideas promoted by the fundamental papers published by M. Frazier and B. Jawerth in the eighties have influenced subsequent developments related to the theory of atomic decompositions and Banach frames for function spaces such as the modulation spaces and Besov-Triebel-Lizorkin spaces. Both of these classes of spaces arise as special cases of two different, general constructions of function spaces: coorbit spaces and decomposition spaces. Coorbit spaces are defined by imposing certain decay conditions on the so-called voice transform of the function/distribution under consideration. As a concrete example, one might think of the wavelet transform, leading to the theory of Besov-Triebel-Lizorkin spaces. Decomposition spaces, on the other hand, are defined using certain decompositions in the Fourier domain. For Besov-Triebel-Lizorkin spaces, one uses a dyadic decomposition, while a uniform decomposition yields modulation spaces. Only recently, the second author has established a fruitful connection between modern variants of wavelet theory with respect to general dilation groups (which can be treated in the context of coorbit theory) and a particular family of decomposition spaces. In this way, optimal inclusion results and invariance properties for a variety of smoothness spaces can be established. We will present an outline of these connections and comment on the basic results arising in this context

The inner kernel theorem for a certain Segal algebra

The Segal algebra ${\textbf{S}}_{0}(G)$ is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups $G$. Despite the fact that it is a Banach space it is possible to derive a kernel theorem similar to the Schwartz kernel theorem, of course without making use of the Schwartz kernel theorem. First we characterize the bounded linear operators from ${\textbf{S}}_{0}(G_1)$ to ${\textbf{S}}_{0}'(G_2)$ by distributions in ${\textbf{S}}_{0}'(G_1 \times G_2)$. We call this the "outer kernel theorem". The "inner kernel theorem" is concerned with the characterization of those linear operators which have kernels in the subspace ${\textbf{S}}_{0}(G_1 \times G_2)$, the main subject of this manuscript. We provide a description of such operators as regularizing operators in our context, mapping ${\textbf{S}}_{0}'(G_1)$ into test functions in ${\textbf{S}}_{0}(G_2)$, in a $w^{*}$-to norm continuous manner. The presentation provides a detailed functional analytic treatment of the situation and applies to the case of general LCA groups, without recurrence to the use of so-called Wilson bases, which have been used for the case of elementary LCA groups. The approach is then used in order to describe natural laws of composition which imitate the composition of linear mappings via matrix multiplications, now in a continuous setting. We use here that in a suitable (weak) form these operators approximate general operators. We also provide an explanation and mathematical justification used by engineers explaining in which sense pure frequencies "integrate" to a Dirac delta distribution

Compactness Criteria in Function Spaces

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for L^2(R^d), and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin spaces, modulation spaces and Bargmann-Fock spaces

Flexible Gabor-wavelet atomic decompositions for L2-Sobolev spaces

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