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.     

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.     

Effective intervals and regular Dirichlet subspaces

It is shown in Li and Ying (2019) that a regular and local Dirichlet form on an interval can be represented by so-called effective intervals with scale functions. This paper focuses on how to operate on effective intervals to obtain regular Dirichlet subspaces. The first result is a complete characterization for a Dirichlet form to be a regular subspace of such a Dirichlet form in terms of effective intervals. Then we give an explicit road map how to obtain all regular Dirichlet subspaces from a local and regular Dirichlet form on an interval, by a series of intuitive operations on the effective intervals in the representation above. Finally applying previous results, we shall prove that every regular and local Dirichlet form has a special standard core generated by a continuous and strictly increasing function.     

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.     

由给定的子空间构造新的子空间

本给出并证明了若干个子空间的并以及两个子空间的基构成子空间的充要条件，从而本质地揭示了除子空间的交与和是构造新的予空间的方法外，集合的其它运算不能构造新的子空间，最后分析了子空间直和的两种不同定义的优缺点，指出了张禾瑞教材中子空间直和定义推广时应注意的一个问题。     

The Grassmannian of affine subspaces

Foundations of Computational Mathematics - The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being 0-dimensional affine subspaces, and the usual...     

Reconstruction of functions in spline subspaces from local averages

In this paper, we study the reconstruction of functions in spline subspaces from local averages. We present an average sampling theorem for shift invariant subspaces generated by cardinal B-splines and give the optimal upper bound for the support length of averaging functions. Our result generalizes an earlier result by Aldroubi and Gröchenig.

     

Dense subspaces of quasinormable spaces

Dense linear subspaces of quasinormable Fréchet spaces need not be quasinormable, as an example due to J. Bonet and S. Dierolf proved. A characterization of the quasinormability of dense linear subspaces of quasinormable locally convex spaces and several consequences are given. Moreover, an example of a dense linear subspace of a countable direct sum of Banach spaces, which is not quasinormable, is provided. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The finite-dimensional decomposition property in non-Archimedean Banach spaces

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property（OFDDP） if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces.This property has an influence in the non-Archimedean Grothendieck＇s approximation theory,where an open problem is the following： Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E.Does D have the OFDDP？ In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP.Next we prove that,however,for certain classes of Banach spaces of countable type,the OFDDP is preserved by taking finite-codimensional subspaces.     

Planar and affine spaces

In this note, we characterize finite three-dimensional affine spaces as the only linear spaces endowed with set Ω of proper subspaces having the properties (1) every line contains a constant number of points, say n, with n>2; (2) every triple of noncollinear points is contained in a unique member of Ω; (3) disjoint or coincide is an equivalence relation in Ω with the additional property that every equivalence class covers all points. We also take a look at the case n=2 (in which case we have a complete graph endowed with a set Ω of proper complete subgraphs) and classify these objects: besides the affine 3-space of order 2, two small additional examples turn up. Furthermore, we generalize our result in the case of dimension greater than three to obtain a characterization of all finite affine spaces of dimension at least 3 with lines of size at least 3.     

Extremal subspaces and their submanifolds

It was proved in the paper [KM1] that the properties of almost all points of being not very well (multiplicatively) approximable are inherited by nondegenerate in (read: not contained in a proper affine subspace) smooth submanifolds. In this paper we consider submanifolds which are contained in proper a.ne subspaces, and prove that the aforementioned Diophantine properties pass from a subspace to its nondegenerate submanifold. The proofs are based on a correspondence between multidimensional Diophantine approximation and dynamics of lattices in Euclidean spaces. Submitted: March 2002; Revision: August 2002     

Affiliated subspaces and infiniteness of von Neumann algebras

We show that the structural properties of von Neumann algebra s are connected with the metric and order theoretic properties of various classes of affiliated subspaces. Among others we show that properly infinite von Neumann algebra s always admit an affiliated subspace for which (1) closed and orthogonally closed affiliated subspaces are different; (2) splitting and quasi‐splitting affiliated subspaces do not coincide. We provide an involved construction showing that concepts of splitting and quasi‐splitting subspaces are non‐equivalent in any GNS representation space arising from a faithful normal state on a Type I factor. We are putting together the theory of quasi‐splitting subspaces developed for inner product spaces on one side and the modular theory of von Neumann algebra s on the other side.     

Shift-type invariant subspaces of contractions

Using the Sz.-Nagy-Foias functional model it was shown in [L. Kérchy, Injection of unilateral shifts into contractions with non-vanishing unitary asymptotes, Acta Sci. Math. (Szeged) 61 (1995) 443-476] that under certain conditions on a contraction T the natural embedding of a Hardy space of vector-valued functions into the corresponding L² space can be factored into the product of two transformations, intertwining T with a unilateral shift and with an absolutely continuous unitary operator, respectively. The norm estimates in the Factorization Theorem of this paper are sharpened to their best possible form by essential improvements in the proof. As a consequence we obtain that if the residual set of a contraction covers the whole unit circle then those invariant subspaces, where the restriction is similar to the unilateral shift with a similarity constant arbitrarily close to 1, span the whole space. Furthermore, the hyperinvariant subspace problem for asymptotically non-vanishing contractions is reduced to these special circumstances.     

The Structure of Finitely Generated Shift-Invariant Subspaces in Super Hilbert Spaces

We study the structure of finitely generated shift-invariant subspaces with generators from the super Hilbert space L ²(? ^d )^(N). We give a characterization for these subspaces. Moreover, we show that every finitely generated shift-invariant subspace possesses a tight frame. We also give a necessary and sufficient condition for such a space to be principal. Our results generalize similar ones for which generators are from L ²(? ^d ).     

Zero-sets of complex homogeneous polynomials

In this paper we study the zero-sets of continuous n-homogeneous polynomials on complex nonseparable Banach spaces. We prove that the zero-set of any complex n-homogeneous polynomial P is a subspace if, and only if, there is a functional ? such that P(x)=? (x)ⁿ for every x. We give sufficient conditions on the Banach space to ensure that every continuous 2-homogeneous polynomial is identically zero on a nonseparable subspace. Also, we prove that, in the 2-homogeneous case, one of the following three properties holds: P ^?1(0) is a subspace; P ^?1(0) is the union of two different subspaces; and P ^?1(0) is the union of infinitely many different subspaces.     

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.      

RICE CONDITION NUMBERS OF CERTAIN CHARACTERISTIC SUBSPACES

This paper proposes the Rice condition numbers for invariant subspace, singular sub-spaces of a matrix and deflating subspaces of a regular matrix pair. The first-order perturbation estimations for these subspaces are derived by applying perturbation expansions of orthogonal projection operators.     

Wavelet frames and shift-invariant subspaces of periodic functions

A general approach based on polyphase splines, with analysis in the frequency domain, is developed for studying wavelet frames of periodic functions of one or higher dimensions. Characterizations of frames for shift-invariant subspaces of periodic functions and results on the structure of these subspaces are obtained. Starting from any multiresolution analysis, a constructive proof is provided for the existence of a normalized tight wavelet frame. The construction gives the minimum number of wavelets required. As an illustration of the approach developed, the one-dimensional dyadic case is further discussed in detail, concluding with a concrete example of trigonometric polynomial wavelet frames.     

A generalized affine isoperimetric inequality

A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry.     

Principal subspaces for the quantum affine vertex algebra in type A1(1)

By using the ideas of Feigin and Stoyanovsky and Calinescu, Lepowsky and Milas we introduce and study the principal subspaces associated with the Etingof–Kazhdan quantum affine vertex algebra of integer level $k?1$ and type $A_1^{(1)}$. We show that the principal subspaces possess the quantum vertex algebra structure, which turns to the usual vertex algebra structure of the principal subspaces of generalized Verma and standard modules at the classical limit. Moreover, we find their topological quasi-particle bases which correspond to the sum sides of certain Rogers–Ramanujan-type identities.