Subspaces Of L2 Research Articles

Completeness in the set of wavelets

We study the completeness properties of the set of wavelets in L 2 ( R ) L^{2}(\mathbb {R}) . It is well-known that this set is not closed in the unit ball of L 2 ( R ) L^{2}(\mathbb {R}) . However, if one considers the metric inherited as a subspace (in the Fourier transform side) of L 2 ( R , d ξ ) ∩ L 2 ( R ∗ , d ξ | ξ | ) L^{2}(\mathbb {R},d\xi ) \cap L^{2}(\mathbb {R}_{*},{\frac {{d\xi }}{{|\xi |}}}) , we do obtain a complete metric space.

L 1 (μ, X) as a constrained subspace of its bidual

In this note we consider the property of being constrained in the bidual, for the space of Bochner integrable functions. For a Banach spaceX having the Radon-Nikodym property and constrained in its bidual and forY ⊂ X, under a natural assumption onY, we show thatL 1 (μ, X/Y) is constrained in its bidual andL 1 (μ, Y) is a proximinal subspace ofL 1(μ, X). As an application of these results, we show that, ifL 1(μ, X) admits generalized centers for finite sets and ifY ⊂ X is reflexive, thenL 1 μ, X/Y) also admits generalized centers for finite sets.

A Hardy space for Fourier integral operators

We introduce a new function space, denoted by H FIO 1 (ℝn), which is preserved by the algebra of Fourier integral operators of order 0 associated to canonical transformations. A subspace of L1 (ℝn), this space in many aspects resembles the real Hardy space of Fefferman-Stein. In particular, we obtain an atomic characterization of H FIO 1 (ℝn). In contrast to the standard Hardy space, these atoms are localized in frequency space as well as in real space.

Compact Toeplitz operators on Bergman spaces

Let Bn denote the open unit ball in Cn. We write V to denote Lebesgue volume measure on Bn normalized so that V(Bn)=1. Fix −1<γ<∞ and let Vγ denote the measure given by dVγ(z)=cγ (1−[mid ]z[mid ]2)γdV(z), for z∈Bn, where cγ=Γ(n+γ+1)/ (n!Γ(γ+1)); then Vγ(Bn)=1. The weighted Bergman space A2,γ(Bn) is the space of all analytic functions in L2(Bn, dVγ). This is a closed linear subspace of L2(Bn, dVγ). Let Pγ denote the orthogonal projection of L2(Bn, dVγ) onto A2,γ(Bn). For a function f∈L∞(Bn) the Toeplitz operator Tf is defined on A2,γ(Bn) by Tfh=Pγ(fh), for h∈A2,γ(Bn). It is clear that Tf is bounded on A2,γ(Bn) with ∥Tf∥[les ]∥f∥∞. In this paper we will consider the question for which f∈L∞(Bn) the operator Tf is compact on A2,γ(Bn). Although a complete answer has been given by the author and D. Zheng (see the next section), the condition for compactness is somewhat unnatural. In this article we will give a more natural description for compactness of Toeplitz operators with sufficiently nice symbols. We will describe compactness in terms of behaviour of the so-called Berezin transform of the symbol, which has been useful in characterizing compactness of Toeplitz operators with positive symbols (see [5, 9]). Before we can define this Berezin transform we need to introduce more notation.

Aϕ-Invariant Subspaces on the Torus

AbstractGeneralizing the notion of invariant subspaces on the 2-dimensional torus T2, we study the structure of Aϕ-invariant subspaces of L2(T2). A complete description is given of Aϕ-invariant subspaces that satisfy conditions similar to those studied by Mandrekar, Nakazi, and Takahashi.

The Segal–Bargmann Transform for Path-Groups

LetKbe a connected Lie group of compact type and letW(K) denote the set of continuous paths inK, starting at the identity and with time-interval [0, 1]. ThenW(K) forms an infinite-dimensional group under the operation of pointwise multiplication. Letρdenote the Wiener measure onW(K). We construct an analog of the Segal–Bargmann transform forW(K). LetKCbe the complexification ofK,W(KC) the set of continuous paths inKCstarting at the identity, andμthe Wiener measure onW(KC). Our transform is a unitary map ofL2(W(K),ρ) onto the “holomorphic” subspace ofL2(W(KC),μ). By analogy with the classical transform, our transform is given by convolution with the Wiener measure, followed by analytic continuation. We prove that the transform forW(K) is nicely related by means of the Itô map to the classical Segal–Bargmann transform for the path-space in the Lie algebra ofK.

Discrete Series Representations for the Orbit Spaces Arising from Two Involutions of Real Reductive Lie Groups

LetH⊂Gbe real reductive Lie groups. A discrete series representation for a homogeneous spaceG/His an irreducible representation ofGrealized as a closedG-invariant subspace ofL2(G/H). The condition for the existence of discrete series representations forG/Hwas not known in general except for reductive symmetric spaces. This paper offers a sufficient condition for the existence of discrete series representations forG/Hin the setting thatG/His a homogeneous submanifold of a symmetric space G/HwhereG⊂G⊃H. We prove that discrete series representations are non-empty for a number of non-symmetric homogeneous spaces such asSp(2n,R)/Sp(n0,C)×GL(n1,C)×…×GL(nk,C) (∑nj=n) andO(4m,n)/U(2m,j) (0⩽2j⩽n).

Selfadjoint commutators and invariant subspaces on the torus II

In a previous paper, the authors determined the invariant subspaces ofL2(T2) on which a certain commutator is selfadjoint. In this paper, we give its generalization.

Admissible Wavelets Associated with the Affine Automorphism Group of the Siegel Upper Half-Plane

LetP=NAMbe the minimal parabolic subgroup ofSU(n+1,1), which can be regarded as the affine automorphism group of the Siegel upper half-planeUn+1,Palso acts on the Heisenberg groupHn, the boundary ofUn+1. ThereforePhas a natural representationUonL2(Hn). We decomposeL2(Hn) into the direct sum of the irreducible invariant closed subspaces underU. The restrictions ofUon these subspaces are square-integrable. We give the characterization of the admissible condition in terms of the Fourier transform and define the wavelet transform with respect to admissible wavelets. The wavelet transform gives isometric operators from the irreducible invariant closed subspaces ofL2(Hn) toL2(P).

Stability of spectra of Hodge-de Rham laplacians

In this paper we investigate the stability of eigenspaces of the Laplace operator acting on differential forms satisfying relative or absolute boundary conditions on a compact, oriented, Riemannian manifold with boundary (this includes, in particular, both Neumann and Dirichlet conditions for the Laplace-Beltrami operator on functions). More precisely, our main result is that the gap between corresponding eigenspaces (precise definition will be recalled below) measured using the L∞ norm, converges to zero when smooth metrics g converge to g0 in the C 1 topology. It is quite well known (cf. [3] or [14]) that the eigenvalues of the Laplacian vary continuously under C 0-continuous perturbations of the metric. It is perhaps less well known, but implicit in the work of Cheeger [3], that eigenspaces vary continuously as subspaces of L2 when the metric is perturbed C 0-continuously. We reprove this C 0 L2 stability in Sect. 4 for completeness and in order to be able to use certain notation, conventions and partial results in the proof of C 1 L∞ stability in Sect. 5. The second section of the paper contains a review of the Hodge theory for the Laplace operator with absolute and relative boundary conditions. We also state here the Sobolev embedding theorems and the basic a priori estimates for the square root d + δ of the Laplacian ∆. We need to work with d + δ rather than ∆ = (d + δ)2 since the coefficients of ∆ depend on the second derivatives of the metric tensor and we allow only C 1-continuous perturbations of the metric and do not assume any bounds on the second derivatives. In the third section we review following Kato [12] and Osborn [16] general results from functional analysis concerning perturbation theory for compact operators on Banach spaces, that reduce proving convergence of eigenvalues and eigenspaces

𝐿²-homology over traced *-algebras

Given a unital complex *-algebra A A , a tracial positive linear functional τ \tau on A A that factors through a *-representation of A A on Hilbert space, and an A A -module M M possessing a resolution by finitely generated projective A A -modules, we construct homology spaces H k ( A , τ , M ) H_k(A,\tau ,M) for k = 0 , 1 , … k = 0, 1, \ldots . Each is a Hilbert space equipped with a *-representation of A A , independent (up to unitary equivalence) of the given resolution of M M . A short exact sequence of A A -modules gives rise to a long weakly exact sequence of homology spaces. There is a Künneth formula for tensor products. The von Neumann dimension which is defined for A A -invariant subspaces of L 2 ( A , τ ) n L^2(A,\tau )^n gives well-behaved Betti numbers and an Euler characteristic for M M with respect to A A and τ \tau .

Minimal Projections onto Two Dimensional Subspaces ofl(4)∞

LetYbe a two-dimensional subspace ofl(4)∞. A formula for a minimal projection froml(4)∞ontoYwill be presented. Also, a complete characterization of the unicity of this minimal projection will be given

Wavelet transform of the dilation equation

AbstractIn this article we study the dilation equation f(x) = ∑h ch f (2x − h) in ℒ2(R) using a wavelet approach. We see that the structure of Multiresolution Analysis adapts very well to the study of scaling functions. The equation is reduced to an equation in a subspace of ℒ2(R) of much lower resolution. This simpler equation is then “wavelet transformed” to obtain a discrete dilation equation. In particular we study the case of compactly supported solutions and we see that conditions for the existence of solutions are given by convergence of infinite products of matrices. These matrices are of the type obtained by Daubechies, and, when the analyzing wavelet is the Haar wavelet, they are exactly the same.

On subspaces of L1 which embed into 1.

On subspaces of L1 which embed into 1.

Separable Banach space theory needs strong set existence axioms

We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π 1 1 \Pi ^1_1 comprehension, is needed to prove such basic facts as the existence of the weak- ∗ * closure of any norm-closed subspace of ℓ 1 = c 0 ∗ \ell _1=c_0^* . This is in contrast to earlier work in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for Π 2 0 \Pi ^0_2 sentences. En route to our main results, we prove the Krein-Šmulian theorem in A C A 0 \mathsf {ACA}_0 , and we give a new, elementary proof of a result of McGehee on weak- ∗ * sequential closure ordinals.

Order of linear approximation from shift-invariant spaces

A Fourier analysis approach is taken to investigate the approximation order of scaled versions of certain linear operators into shift-invariant subspaces ofL2(Rd). Quasi-interpolants and cardinal interpolants are special operators of this type, and we give a complete characterization of the order in terms of some type of ellipticity condition for a related function. We apply these results by showing that theL2-approximation order of a closed shift-invariant subspace can often be realized by such an operator.

Sobolev regularity of the weighted bergman projections and estimates for minimal solutions to the -equation

Let Ω be a C∞ bounded strongly pseudoconvex domain in C n. For be the space of functions on Ωthat are square integrable with respect to the measure |ρ|v dm. Let be the subspace of L2 (|ρ|v dm) consisting of the holomorphic functions. We consider the weighted Bergman projection . We prove that Pv admits both local and global regularity estimates in Sobolev norms, and that the weighted kernel function Kv (z,w) is smooth up to the boundary off the boundary diagonal. As a consequence we prove estimates for the solution to the -equation which is minimal in the L2 (|ρ|v dm)-norm.

Generalized Sectional Convergence and Multipliers

We define a generalized sectional convergence scheme (gscs) as a sequence (Tn) of finitely non-zero matrices which converges coordinatewise to the identity matrix. If Tnx converges to x for each x in a topological sequence space S, then we say that S has AK (Tn), a generalization of sectional convergence (AK). We prove that a generalization of the Dieudonné Weak Basis Theorem is valid in this new context. A gscs (Tn) together with a dense subspace Λ of l1 determines a matrix space M = Λ(Tn). If Λ is barrelled and each member of M is a matrix representation of a linear operator T which maps the locally convex K space S into itself (denote this by MS ⊂ S), then we can draw conclusions about topological and approximation properties of S. For an appropriate type of (Tn) we will have the following: If S is an FK space with AD and MS ⊂ S, then S has the generalized sectional convergence associated with (Tn). If S is a sequence space, which has AD in the βφ topology and MS ⊂ S, then S is barrelled in the βφ topology. The many dense barrelled subspaces of l1 are examples of dense βφ subspaces Λ of l1, and the latter still support our conclusions even though Λ and the corresponding multiplier space M may be very small. This reduction of M is novel even in the context of ordinary sectional convergence.

A generalization of de Boor’s stability result and symmetric preconditioning

We provide a symmetric preconditioning method based on weighted divided differences which can be applied in order to solve certain ill-conditioned scattered-data interpolation problems in a stable way; more precisely, our method applies to cases where the ill-conditioning comes from the fact that the basis function is slowly growing, and the number of interpolation points is large. Concerning the theoretical background, an a priori unbounded operator on l2(ℤ) is preconditioned so as to get a bounded and coercive operator. The method has another and probably even more interesting interpretation in terms of constructing certain Riesz bases of appropriate closed subspaces ofL 2(ℝ). In extending Mallat’s multiresolution analysis to the scattered data case, we construct nested sequences of spaces giving rise to orthogonal decompositions of functions inL 2(ℝ); in this way the idea of wavelet decompositions is (theoretically) carried over to scattered-data methods.

On the fourier coefficients of functions concentrated on a subset of the circle

We consider\(\mathbb{T} = \{ z \in \mathbb{C} :{\mathbf{ }}|z|{\mathbf{ }} = 1\} ,{\mathbf{ }}E{\mathbf{ }} \subset {\mathbf{ }}\mathbb{T}\),mE > 0,G(E) is a certain subspace of L1(E) consisting of functions concentrated on E and integrable, and {dk}, (k ∈ ℤ) in a summable sequence of positive numbers. It is proved that if G(E)=Lp(E), p≥2, then there exists f∈G(E) such that |f(n)|≥dn,\(\hat f(n)|{\mathbf{ }} \geqslant {\mathbf{ }}d_n ,{\mathbf{ }}n \in \mathbb{Z}\) (one of the questions involved in the majorization problem). Sufficient conditions are obtained for certain other function classes G(E). We study the question of partial majorization. Bibliography: 2 titles.