The unitary extension principle for locally compact abelian groups with co-compact subgroups

An explicit method for the construction of a tight wavelet frame generated by the Walsh polynomials with the help of extension principles was presented by Shah (Shah, 2013). In this article, we extend the notion of wavelet frames to periodic wavelet frames generated by the Walsh polynomials on R+ by using extension principles. We first show that under some mild conditions, the periodization of any wavelet frame constructed by the unitary extension principle is still a periodic wavelet frame on R + . Then, we construct a pair of dual periodic wavelet frames generated by the Walsh polynomials on R + using the machinery of the mixed extension principle and Walsh–Fourier transforms.

A refined method for calculating wind load combinations on open-framed structures

The theory of compressed sensing asserts that one can recover signals in $$\mathbb {R}^n$$ from far fewer samples or measurements, if the signal has a sparse representation in some orthonormal basis; from non-adaptive linear measurements by solving a $$\mathbb {L}_1$$ norm minimisation problem. The non-adaptive measurements have the character of random linear combinations of the basis or frame elements. However, for large-scale 2D image signals, the randomized sensing matrix consumes enormous computational resources that makes it impractical. The problem has been addressed in the paper as a block compressed sensing (BCS) with sparsity normalization in the transformed domain in the preprocessing stage. The blocks obtained are converted to non-adaptive measurements using identically independent weighted Gaussian random matrices. The feasibility of reconstruction is verified using orthogonal matching pursuit. Simulation results show that better reconstruction performance can be achieved by the proposed technique in comparison with the existing BCS approaches.

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.

The recently introduced and characterized scalable frames can be considered as those frames which allow for perfect preconditioning in the sense that the frame vectors can be rescaled to yield a tight frame. In this paper we define m−scalability, a refinement of scalability based on the number of non-zero weights used in the rescaling process. We enlighten a close connection between this notion and elements from convex geometry. Another focus lies in the topology of scalable frames. In particular, we prove that the set of scalable frames with “usual” redundancy is nowhere dense in the set of frames.

The Kazakov-Migdal model, if considered as a functional of external fields, can always be represented as an expansion over characters of the GL group. The integration over “matter fields” can be interpreted as going over the model (the space of all highest weight representations) of GL. In the case of compact unitary groups the integrals should be substituted by discrete sums over the weight lattice. The D=0 version of the model is the generalized Kontsevich integral, which in the above-mentioned unitary (discrete) situation coincides with the partition function of 2D Yang-Mills theory with the target space of genus g=0 and m=0, 1, 2 holes. This particular quantity is always a bilinear combination of characters and appears to be a Toda lattice τ function. (This is a generalization of the classical statement that individual GL characters are always singular KP τ functions.) The corresponding element of the universal Grassmannian is very simple and somewhat similar to the one arising in investigations of the c=1 string models. However, in certain circumstances the formal sum over representations should be evaluated by the steepest descent method, and this procedure leads to some more-complicated elements of the Grassmannian. This “Kontsevich phase,” as opposed to the simple “character phase,” deserves further investigation.

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

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 this contribution we introduce the concept of multiresolution analysis (MRA) and give an explanation of the relationship between MRA and orthonormal wavelet basis. The construction of the orthonormal B-spline wavelet bases is described in detail. We extend the B-splines to `non-integral order' cases and thus obtain a new family of orthonormal wavelet bases for the space L2(R). Some good properties of the new wavelets are demonstrated. The new wavelet family gives satisfactory performances in our research projects including seismic signal compression and gravity tide data processing.

Studies the autocorrelation and cross-correlation structure of the scaling and wavelet functions associated with compactly supported orthonormal wavelet bases. These correlation structures play an important role in both wavelet-based interpolation and in answering the question of existence of scale-limited signals. It is shown how to efficiently compute the correlation functions approximately at the M-adic rationals. Furthermore, when the tight frame is, in particular, an orthonormal wavelet basis, it is shown that the approximations involved in the computation of the samples of the correlations cancel in a manner to make the computations exact. For the case of orthonormal wavelet bases, an attempt is made to give a complete description of the zeros at the M-adic rationals. An interesting fact that arises from the analysis is that all the correlation functions possible have infinitely many zeros in their support. >

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.

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 this paper, we propose an orthonormal wavelet basis having a dilation factor (R + 1)/ R with an arbitrary real number R > 0. This orthonormal wavelet basis requires an infinite number of wavelet shapes when the dilation factor (R + 1)/ R is ir-rational, and (R + 1)/ R can be taken an arbitrary real number greater than 1 with the appropriate real number R > 0.

We study the existence of many nonclosed pure subgroups of nondiscrete locally compact abelian groups. It is shown that every nondiscrete locally compact abelian group has uncountably many nonclosed pure subgroups. This in particular solves a question of Armacost. It is also shown that, if G G is a nondiscrete locally compact abelian group and if either G G is a compact group or the torsion part t ( G ) t\left ( G \right ) of G G is nonopen, then G G has 2 c {2^c} proper dense pure subgroups, where c c denotes the power of the continuum. This in particular gives a partial answer to another question of Armacost.

In this note we point out two properties (Theorems 1 and 2) of the group algebra of the direct product of a compact group and a locally compact abelian group and indicate briefly how these properties enable one to extend to these group algebras theorems which are known in the compact and abelian cases. Our interest in these properties is two-fold; they establish the applicability of known results in [4]1 to the group algebras in question, and they provide an interesting example of a type of B-algebra, a sort of Kroliecker product algebra, which may very well deserve independenlt systematic study. The reader is referred to results of Kaplansky [1 ] and Segal [2 ] of which this note is essentially a continuation. Let C be a compact group, A be a locally compact abelian group, and G = CXA be their direct product group. We are interested in the group algebras L1(C), L1(A), and L1(G). In particular, we ask how the latter is related to the first two. We will show (1) how the B-algebra L1(G) is formed out of the B-algebras L1(C) and L1(A) and (2) how the structure space of L1(G) is formed from the structure spaces of L1(C) and L1(A). DEFINITION 1. The complex algebra R is the Kronecker product of the complex algebras R1 and R2 (written R = R1 XR2) if there exists a map (xl, X2)>X1 X X2 of (R1, R2) into R such that