SHIFT INVARIANT SPACES FOR LOCAL FIELDS

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

We investigate several issues surrounding the general question of when a function in a finitely generated shift invariant subspace of L2(R) can be determined by certain of its sample values just as a function band limited to (-1/2, ») can be expressed in terms of its integer samples. The main theme here is how answers to this question depend on general properties of the generators of the shift invariant space, such as orthogonality properties, scaling relations, smoothness and so forth. One of the main issues that we address is the question of how to control aliasing error.© (2000) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

In this paper the structure of all open semigroups in any compact topological group is determined. We can then extend this result to a classification of the angular semigroups of any abelian topological group H which contains a compact open subgroup K. Coupling this with the results for Euclidean n-space, we obtain an essentially complete description of the angular semigroups of an arbitrary locally compact group. For compact groups, the result is remarkably simple.

The canonical mapping on the product of a LCA group with its dual is shown to extend uniquely in a homomorphic and analytic way to the product of the respective complexifications.

The concept of super‐wavelet was introduced by Balan, and Han and Larson over the field of real numbers which has many applications not only in engineering branches but also in different areas of mathematics. To develop this notion on local fields having positive characteristic we obtain characterizations of super‐wavelets of finite length as well as Parseval frame multiwavelet sets of finite order in this setup. Using the group theoretical approach based on coset representatives, further we establish Shannon type multiwavelet in this perspective while providing examples of Parseval frame (multi)wavelets and (Parseval frame) super‐wavelets. In addition, we obtain necessary conditions for decomposable and extendable Parseval frame wavelets associated to Parseval frame super‐wavelets.

LetG be a locally compact abelian topological group containing a suitable sequence of compact open subgroups and let Γ be its dual group. LetK (α,p, q; G) andK (α,p, q; Γ) denote the so-called Herz spaces onG and Γ, respectively. In this paper we shall prove that for 1<p≤2 and 0≤α<1/p′=1−1/p, the Fourier transform mapsK (α,p, p; G) continuously intoK (−α,p′, 2; Γ). The proof requires two results that are of independent interest: an extension of the Hausdorff-Young inequality to certain weightedL p-spaces onG and a Littlewood-Paley theorem for certain weightedL p-spaces onG.

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

For a connected reductive group G over a nonarchimedean local field F of positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter ${\mathcal {L}}^{ss}(\pi )$ to each irreducible representation $\pi $ . Our first result shows that the Genestier-Lafforgue parameter of a tempered $\pi $ can be uniquely refined to a tempered L-parameter ${\mathcal {L}}(\pi )$ , thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of ${\mathcal {L}}^{ss}(\pi )$ for unramified G and supercuspidal $\pi $ constructed by induction from an open compact (modulo center) subgroup. If ${\mathcal {L}}^{ss}(\pi )$ is pure in an appropriate sense, we show that ${\mathcal {L}}^{ss}(\pi )$ is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show $\mathcal {L}^{ss}(\pi )$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is ${\mathbb {P}}^1$ and a simple application of Deligne’s Weil II.

We study Pontryagin reflexivity in the class of precompact topological Abelian groups. We find reflexive groups among precompact not pseudocompact and among pseudocompact not compact groups. Making use of Martin's axiom we give an example of a reflexive countably compact not compact Abelian group. We also prove that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

Given a compact group G and r ∈ N, let sr(G) denote the least upper bound of real ϵ > 0 such that for every measurable r-coloring of G, there exists a monochrome symmetric subset of measure ≥ ϵ. A subset A ⊆ G is symmetric if there exists g ∈ G such that gA −1 g = A. We give a general picture of asymptotic behaviour of the function sr(G) for compact Abelian groups.

In Pontrjagin's theory of duality for compact abelian groups, the following theorem is known : Let G be a compact abelian group, G* the dual group. Then the topological dimension of G, in the sense of Lebesgue, is equal to the rank of discrete abelian group G*. It was Prof. T. Tannaka who has called my attention to the lack of corresponding theorem in non-commutative case. I intend to give, in this note, a theorem of this kind in the following form: THEOREM A. Let G be an arbitrary compact group, G the aggregate of continuous finite dimensional representstions of G, C[ G ] the algebra over the complex numbers C generated by the coefficients of representations in G, i.e., the representative ring of G in the sense of C. Chevalley^. Then the topological dimension of G, in the sense of Lebesgue, is equal to the transcendental degree of C[G] over C.

In general, a two-dimensional display is defined by two orthogonal unit vectors. In developing the display, discriminant analysis has the shortcoming that, in general, the extracted axes are not orthogonal. In order to overcome this shortcoming, the authors propose a discriminant analysis which provides an orthonormal system in the transformed space. The transformation preserves the discriminatory ability in terms of the Fisher ratio. The authors present a necessary and sufficient condition under which discriminant analysis in the original space provides an orthonormal system. Relationships between orthogonal discriminant analysis and the Karhunen-Loeve expansion in the original space are investigated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We study lattices in a product$G=G_{1}\times \cdots \times G_{n}$of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor isquasi just-non-compact, meaning that$G_{i}$is non-compact and every closed normal subgroup of$G_{i}$is discrete or cocompact (e.g. $G_{i}$is topologically simple). We show that the set of discrete subgroups of$G$containing a fixed cocompact lattice$\unicode[STIX]{x1D6E4}$with dense projections is finite. The same result holds if$\unicode[STIX]{x1D6E4}$is non-uniform, provided$G$has Kazhdan’s property (T). We show that for any compact subset$K\subset G$, the collection of discrete subgroups$\unicode[STIX]{x1D6E4}\leqslant G$with$G=\unicode[STIX]{x1D6E4}K$and dense projections is uniformly discrete and hence of covolume bounded away from $0$. When the ambient group$G$is compactly presented, we show in addition that the collection of those lattices falls into finitely many$\operatorname{Aut}(G)$-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group$G$is a Chabauty limit of discrete subgroups, then some compact open subgroup of$G$is an infinitely generated pro-$p$group for some prime$p$. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.

Embeddable quantum homogeneous spaces

Let be an arbitrary locally compact Abelian group and let be the space of all continuous complex-valued functions on . A closed linear subspace is referred to as an invariant subspace if it is invariant with respect to the shifts , . By definition, an invariant subspace admits strict spectral synthesis if coincides with the closure in of the linear span of all characters of belonging to . We say that strict spectral synthesis holds in the space on if every invariant subspace admits strict spectral synthesis. An element of a topological group is said to be compact if is contained in some compact subgroup of . A group is said to be element-wise compact if all elements of are compact. The main result of the paper is the proof of the fact that strict spectral synthesis holds in for a locally compact Abelian group if and only if is element-wise compact. Bibliography: 14 titles.