Wavelet frames for (not necessarily reducing) affine subspaces II: The structure of affine subspaces

This is a continuation of the investigation into the theory of wavelet frames for general affine subspaces. The main focus of this paper is on the structural properties of affine subspaces. We show that every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces, while every reducing subspace (with respect to the dilation and translation operators) is the orthogonal direct sum of two purely non-reducing ones. This result is obtained through considering the basic question as to when the orthogonal complement of an affine subspace in another one is still affine. Motivated by the fundamental question as to whether every affine subspace is singly-generated, and by a recent result that every singly generated purely non-reducing subspace admits a singly generated wavelet frame, we prove that every affine subspace can be decomposed into the direct sum of a singly generated affine subspace and some space of “small size”. As a consequence we establish a connection between the above mentioned two questions.     

关于L2(Rd)中仿射子空间小波标架的一个注记

研究了L2(Rd)的有限生成仿射子空间中小波标架的构造.证明了任意有限生成仿射子空间都容许一个具有有限多个生成元的Parseval小波标架,并且得到了仿射子空间是约化子空间的一个充分条件.对其傅里叶变换是一个特征函数的单个函数生成的仿射子空间,得到了与小波标架构造相关的投影算子在傅里叶域上的明确表达式,同时也给出了一些例子.     

On decomposability of affine partial linear spaces

In the paper we characterize normal subspaces of an affine partial linear space and characterize affine partial linear spaces which can not be represented as the Segre product of some affine partial linear spaces.     

A magic decomposition

A lively example to use in a first course in linear algebra to clarify vector space notions is the space of square matrices of fixed order with its subspaces of affine, coaffine, doubly affine, and magic squares. In this note, the projection theorem is illustrated by explicitly constructing the orthogonal projections (in closed forms) of any matrix U onto these subspaces. The results follow directly from a canonical decomposition of U.     

On the linear convergence of the circumcentered-reflection method

In order to accelerate the Douglas–Rachford method we recently developed the circumcentered-reflection method, which provides the closest iterate to the solution among all points relying on successive reflections, for the best approximation problem related to two affine subspaces. We now prove that this is still the case when considering a family of finitely many affine subspaces. This property yields linear convergence and incites embedding of circumcenters within classical reflection and projection based methods for more general feasibility problems.     

Gearhart–Koshy acceleration for affine subspaces

The method of cyclic projections finds nearest points in the intersection of finitely many affine subspaces. To accelerate convergence, Gearhart & Koshy proposed a modification which, in each iteration, performs an exact line search based on minimising the distance to the solution. When the subspaces are linear, the procedure can be made explicit using feasibility of the zero vector. This work studies an alternative approach which does not rely on this fact, thus providing an efficient implementation in the affine setting.     

中约化子空间上的仿射与伪仿射对偶小波标架

本文讨论中约化子空间上的仿射(伪仿射)对偶小波标架.我们建立了仿射系与伪仿射系之间的一个标架 和对偶标架保持定理,并且在没有任何衰减性假设的条件下获得了仿射(伪仿射)对偶小波标架在傅立叶域上的一个刻画.进一步, 我们也给出了仿射Parseval标架在傅立叶域上的刻画.       

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Pseudoframes for subspaces have been recently introduced by Li and Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence. In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of ℝ ^d by employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time–frequency shifts of an approximation by elements from the pseudoframe. The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.      

Extrapolation algorithm for affine-convex feasibility problems

The convex feasibility problem under consideration is to find a common point of a countable family of closed affine subspaces and convex sets in a Hilbert space. To solve such problems, we propose a general parallel block-iterative algorithmic framework in which the affine subspaces are exploited to introduce extrapolated over-relaxations. This framework encompasses a wide range of projection, subgradient projection, proximal, and fixed point methods encountered in various branches of applied mathematics. The asymptotic behavior of the method is investigated and numerical experiments are provided to illustrate the benefits of the extrapolations.     

Subspaces and Parallelity in semiaffine partial linear spaces

In the paper we characterize subspaces and strong subspaces of a semiaffine partial linear space and consider definability of projective and semiaffine planes, affine lines and parallelity in terms of projective lines. We also give some construction of a wide class of semiaffine partial linear spaces.     

Affine pseudoframes for subspaces of L(R) associated with a generalized multiresolution structure

In this paper, the notion of an m-band generalized multiresolution structure (GMS) of L²(R) is introduced. We give the definition and the characterization of affine pseudoframes for subspaces. The construction of a GMS of Paley–Wiener subspaces of L²(R) is investigated. The pyramid decomposition scheme is derived based on such a GMS. As a major new contribution the construction of affine frames for L²(R) based on a GMS is presented.     

Convergence of minimum norm elements of projections and intersections of nested affine spaces in Hilbert space

We consider a Hilbert space, an orthogonal projection onto a closed subspace and a sequence of downwardly directed affine spaces. We give sufficient conditions for the projection of the intersection of the affine spaces into the closed subspace to be equal to the intersection of their projections. Under a closure assumption, one such (necessary and) sufficient condition is that summation and intersection commute between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces. Another sufficient condition is that the cosines of the angles between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces, be bounded away from one. Our results are then applied to a general infinite horizon, positive semi-definite, linear quadratic mathematical programming problem. Specifically, under suitable conditions, we show that optimal solutions exist and, modulo those feasible solutions with zero objective value, they are limits of optimal solutions to finite-dimensional truncations of the original problem.     

Lyapunov exponents and stationary solutions for affine stochastic delay equations

A certain class of affine delay equations is considered. Two cases for the forcingfunction M are treated: M locally integrable deterministic, and M a random process with stationaryincrements. The Lyapunov spectrum of the homogeneous equation is used to decompose the state spaceinto finite-dimensional and finite-codimensional subspaces. Using a suitable variation of constants representation, formulas for the projection of the trajectories onto the above subspaces are obtained. If the homogeneous equation is hyperbolic and M has stationary increments, existence and uniqueness of a stationary solution for the affine stochastic delay equation is proved. The existence of Lyapunov exponents for the affine equation and their dependence on initial conditions is als studied.     

Diophantine exponents and the Khintchine Groshev theorem

We prove the convergence case of the Khintchine–Groshev theorem for affine subspaces and their nondegenerate submanifolds, answering a conjecture of D. Kleinbock.     

Diophantine exponents and the Khintchine Groshev theorem

We prove the convergence case of the Khintchine–Groshev theorem for affine subspaces and their nondegenerate submanifolds, answering a conjecture of D. Kleinbock.     

The product of affine orthogonal projections

In the case of a finite number of subspaces in a given Hilbert space, by a theorem of J. von Neumann, the iteration of the product of projectors is always convergent. In a finite dimensional Hilbert space, this theorem has been generalized for affine subspaces. In this paper we construct an example which shows that this result does not hold in the infinite dimensional case.     

A note on wavelet subspaces

The wavelet subspaces of the space of square integrable functions on the affine group with respect to the left invariant Haar measure dν are studied using the techniques from Vasilevski (Integral Equ. Operator Theory 33:471–488, 1999) with respect to wavelets whose Fourier transforms are related to Laguerre polynomials. The orthogonal projections onto each of these wavelet subspaces are described and explicit forms of reproducing kernels are established. Isomorphisms between wavelet subspaces are given.     

Principal subspaces for the affine Lie algebras in types D,E and F

Journal of Algebraic Combinatorics - We consider the principal subspaces of certain level $$k\geqslant 1$$ integrable highest weight modules and generalized Verma modules for the untwisted affine...     

Representation of relatively uniform and order convergence topologies by an inductive limit

Using the concept of a topological affine space, it is proved that a partially ordered topological linear spaces associated with relatively uniform and order convergence can be represented by an inductive limit of its subspaces.     

Diophantine exponents of affine subspaces: The simultaneous approximation case

We apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of Rn and their nondegenerate submanifolds.