Some properties of frames of subspaces obtained by operator theory methods

We study the relationship among operators, orthonormal basis of subspaces and frames of subspaces (also called fusion frames) for a separable Hilbert space H. We get sufficient conditions on an orthonormal basis of subspaces E={Ei}_i∈I of a Hilbert space K and a surjective T∈L(K,H) in order that {T(Ei)}_i∈I is a frame of subspaces with respect to a computable sequence of weights. We also obtain generalizations of results in [J.A. Antezana, G. Corach, M. Ruiz, D. Stojanoff, Oblique projections and frames, Proc. Amer. Math. Soc. 134 (2006) 1031-1037], which relate frames of subspaces (including the computation of their weights) and oblique projections. The notion of refinement of a fusion frame is defined and used to obtain results about the excess of such frames. We study the set of admissible weights for a generating sequence of subspaces. Several examples are given.     

On structural decompositions of finite frames

A frame in an n-dimensional Hilbert space H _n is a possibly redundant collection of vectors {f _i } _i∈I that span the space. A tight frame is a generalization of an orthonormal basis. A frame {f _i } _i∈I is said to be scalable if there exist nonnegative scalars {c _i } _i∈I such that {c _i f _i } _i∈I is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame {f _i } _i∈I to be a collection of subsets of I ordered by inclusion so that nonempty J?I is in the factor poset iff {f _j } _j∈J is a tight frame for H _n . We study various properties of factor posets and address the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset P. We then turn our attention to scalable frames and present partial results regarding when a frame can be scaled to have a given factor poset; in doing so we present a bridge between erasure resilience (as studied via prime tight frames) and scalability.     

Frames and bases of subspaces in Hilbert spaces

In this paper we develop the frame theory of subspaces for separable Hilbert spaces. We will show that for every Parseval frame of subspaces {Wi}_i∈I for a Hilbert space H, there exists a Hilbert space K⊇H and an orthonormal basis of subspaces {Ni}_i∈I for K such that Wi=P(Ni), where P is the orthogonal projection of K onto H. We introduce a new definition of atomic resolution of the identity in Hilbert spaces. In particular, we define an atomic resolution operator for an atomic resolution of the identity, which even yield a reconstruction formula.     

Redundancy for localized frames

Redundancy is the qualitative property which makes Hilbert space frames so useful in practice. However, developing a meaningful quantitative notion of redundancy for infinite frames has proven elusive. Though quantitative candidates for redundancy exist, the main open problem is whether a frame with redundancy greater than one contains a subframe with redundancy arbitrarily close to one. We will answer this question in the affirmative for ℓ ¹-localized frames. We then specialize our results to Gabor multi-frames with generators in M ¹(R ^d ), and Gabor molecules with envelopes in W(C, l ¹). As a main tool in this work, we show there is a universal function g(x) so that, for every ε =s> 0, every Parseval frame {f _i } _i=1 ^M for an N-dimensional Hilbert space H _N has a subset of fewer than (1+ε)N elements which is a frame for H _N with lower frame bound g(ε/(2M/N − 1)). This work provides the first meaningful quantative notion of redundancy for a large class of infinite frames. In addition, the results give compelling new evidence in support of a general definition of redundancy given in [5].     

A note on uniqueness problem for meromorphic mappings with 2N + 3 hyperplanes

In this paper,a uniqueness theorem for meromorphic mappings partially sharing 2N+3 hyperplanes is proved.For a meromorphic mapping f and a hyperplane H,set E(H,f) = {z|ν(f,H)(z) 0}.Let f and g be two linearly non-degenerate meromorphic mappings and {Hj}j2=N1+ 3be 2N + 3 hyperplanes in general position such that dim f-1(Hi) ∩ f-1(Hj) n-2 for i = j.Assume that E(Hj,f) E(Hj,g) for each j with 1 j 2N +3 and f = g on j2=N1+ 3f-1(Hj).If liminfr→+∞ 2j=N1+ 3N(1f,Hj)(r) j2=N1+ 3N(1g,Hj)(r) NN+1,then f ≡ g.     

Critical Pairs of Sequences of a Mixed Frame Potential

The classical frame potential in a finite-dimensional Hilbert space has been introduced by Benedetto and Fickus, who showed that all finite unit-norm tight frames can be characterized as the minimizers of this energy functional. This was the starting point of a series of new results in frame theory, related to finding tight frames with determined lengths. The frame potential has been studied in the traditional setting as well as in the finite-dimensional fusion frame context. In this work we introduce the concept of mixed frame potential, which generalizes the notion of the Benedetto-Fickus frame potential. We study properties of this new potential, and give the structure of its critical pairs of sequences on a suitable restricted domain. For a given sequence {α _m } _{m=1,…, N} in K, where K is ? or ?, we obtain necessary and sufficient conditions in order to have a dual pair of frames {f _m } _{m=1,…, N} , {g _m } _{m=1,…, N} such that ? f _m , g _m  ? = α _m for all m = 1,…, N.     

Simultaneous approximation by greedy algorithms

We study nonlinear m-term approximation with regard to a redundant dictionary $\mathcal {D}$ in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f∈H and any dictionary $\mathcal {D}$ an expansion into a series $$f=\sum_{j=1}^{\infty}c_{j}(f)\varphi_{j}(f),\quad\varphi_{j}(f)\in \mathcal {D},\ j=1,2,\ldots,$$ with the Parseval property: ‖f‖²=∑_j|c _j(f)|². Following the paper of A. Lutoborski and the second author we study analogs of the above expansions for a given finite number of functions f ¹,.?.?.,f ^N with a requirement that the dictionary elements φ_j of these expansions are the same for all f ⁱ, i=1,.?.?.,N. We study convergence and rate of convergence of such expansions which we call simultaneous expansions.     

A generalized moment problem for self-adjoint operators

Let the sequence {λ _i } (i≧0) satisfy condition (1.1) and let {A _n} (n≧0) be a sequence of bounded self-adjoint operators over a complex Hilbert spaceH. We give a necessary and sufficient condition in order that {A _n} (n≧0) should possess the representation (1.2).     

On the Duality Principle by Casazza, Kutyniok, and?Lammers

The R-dual sequences of a frame {f _i } _i∈I , introduced by Casazza, Kutyniok and Lammers in (J. Fourier Anal. Appl. 10(4):383–408, 2004), provide a powerful tool in the analysis of duality relations in general frame theory. In this paper we derive conditions for a sequence {ω _j } _j∈I to be an R-dual of a given frame {f _i } _i∈I . In particular we show that the R-duals {ω _j } _j∈I can be characterized in terms of frame properties of an associated sequence {n _i } _i∈I . We also derive the duality results obtained for tight Gabor frames in (Casazza et al. in J. Fourier Anal. Appl. 10(4):383–408, 2004) as a special case of a general statement for R-duals of frames in Hilbert spaces. Finally we consider a relaxation of the R-dual setup of independent interest. Several examples illustrate the results.     

Nonvanishing of Rankin–Selberg L-functions for Hilbert modular forms

Let F be a totally real number field of degree n over $\mathbb{Q}$ with ring of integers $\mathcal{O}$ and narrow class number one. Let S _2k (Γ) be the vector space of cuspidal Hilbert modular forms of parallel weight 2k for $\varGamma=\mathrm{SL}_{2}(\mathcal{O})$ , and let B _2k be an orthogonal Hecke eigenbasis for this space. For any fixed Hecke eigenform f∈S _2k (Γ) and any ε>0, we prove that $$\# \biggl\{ g \in B_{2k}: L \biggl(f \times g, \frac{1}{2} \biggr) \ne0 \biggr\} \gg k^{n- \varepsilon}, $$ where L(f×g,s) is the Rankin–Selberg L–function of f and g.     

On super edge-magic decomposable graphs

Let G be any graph and let {H _i } _i??I be a family of graphs such that $E\left( {H_i } \right) \cap E\left( {H_j } \right) = \not 0$ when i ?? j, ?? _i??I E(H _i ) = E(G) and $E\left( {H_i } \right) \ne \not 0$ for all i ?? I. In this paper we introduce the concept of {H _i } _i??I -super edge-magic decomposable graphs and {H _i } _i??I -super edge-magic labelings. We say that G is {H _i } _i??I -super edge-magic decomposable if there is a bijection ??: V(G) ?? {1,2,..., |V(G)|} such that for each i ?? I the subgraph H _i meets the following two requirements: ??(V(H _i )) = {1,2,..., |V(H _i )|} and {??(a) +??(b): ab ?? E(H _i )} is a set of consecutive integers. Such function ?? is called an {H _i } _i??I -super edge-magic labeling of G. We characterize the set of cycles C _n which are {H ₁, H ₂}-super edge-magic decomposable when both, H ₁ and H ₂ are isomorphic to (n/2)K ₂. New lines of research are also suggested.     

Inequalities for martingale transforms and related characterizations of Hilbert spaces

Suppose that f is a martingale taking values in a Banach space B and g is its transform by a deterministic sequence of numbers in {−1,1}, such that sup_n‖g_n‖≥1 almost surely. We show that a certain family of Φ-estimates for f holds true if and only B is a Hilbert space.     

Asymptotic properties of plug-in level set estimators for right censored data

We assume T1,...,Tn are i.i.d.data sampled from distribution function F with density function f and C1,...,Cn are i.i.d.data sampled from distribution function G.Observed data consists of pairs(Xi,δi),i=1,...,n,where Xi=min{Ti,Ci},δi=I(Ti Ci),I(A)denotes the indicator function of the set A.Based on the right censored data{Xi,δi},i=1,...,n,we consider the problem of estimating the level set{f c}of an unknown one-dimensional density function f and study the asymptotic behavior of the plug-in level set estimators.Under some regularity conditions,we establish the asymptotic normality and the exact convergence rate of theλg-measure of the symmetric difference between the level set{f c}and its plug-in estimator{fn c},where f is the density function of F,and fn is a kernel-type density estimator of f.Simulation studies demonstrate that the proposed method is feasible.Illustration with a real data example is also provided.     

Approximately reducing subspaces for unbounded linear operators

We give sufficient conditions for generation of strongly continuous contraction semigroups of linear operators on Hilbert or Banach space. Let L be a dissipative (unbounded) linear operator in a Hilbert space $\text{H}$ and let {P_n} be an increasing sequence of self-adjoint projections converging weakly to the identity projection. We show that if there is a positive integer k such that for all n the range of P_n is contained in the domain of L and mapped by L into the range of P_{n + k}, and if the sequence {LP_n ? P_nLP_n} is dominated in norm (∥LP_n ? P_nLP_n∥ ? a_n) by some {a_n} ? $\text{R}$₊ with ∑_{n = 1}^∞a_n^?1 = ∞, then the closure of the restriction of L to ∪_{n = 1}^∞ range (P_n) is the infinitesimal generator of a strongly continuous contraction semigroup on $\text{H}$. Applications to an important class of finite perturbations, properly larger than the finite Kato perturbations, are given.We also give sufficient conditions for generation of contraction semigroups when {P_γ} (indexed by a directed set) is a set of bounded self-adjoint operators converging weakly to the identity and each having range contained in D(L). In the latter theorem, and in an analogous theorem for dissipative linear operators L in a Banach space, we do not assume that L interchanges at most finitely many of the approximately reducing operators P_γ.     

A Characterization of Hardy Spaces Associated with Certain Schrödinger Operators

Let {K _t } _t>0 be the semigroup of linear operators generated by a Schrödinger operator ?L = Δ ? V (x) on ? ^d , d ≥ 3, where V (x) ≥ 0 satisfies Δ ^?1 V ∈ L ^∞. We say that an L ¹-function f belongs to the Hardy space \({H^{1}_{L}}\) if the maximal function ? _L f (x) = sup _t>0 |K _t f (x)| belongs to L ¹ (? ^d ). We prove that the operator (?Δ)^1/2 L ^?1/2 is an isomorphism of the space \({H^{1}_{L}}\) with the classical Hardy space H ¹(? ^d ) whose inverse is L ^1/2(?Δ)^?1/2. As a corollary we obtain that the space \({H^{1}_{L}}\) is characterized by the Riesz transforms \(R_{j}=\frac {\partial }{\partial x_{j}}L^{-1\slash 2}\) .     

Ultrapowers ofL 1(μ) and the subsequence splitting principle

We study the ultrapowers $L_1 (\mu )_\mathfrak{U} $ of aL ₁(μ) space, by describing the components of the well-known representation $L_1 (\mu )_\mathfrak{U} = L_1 (\mu _\mathfrak{U} ) \oplus _1 L_1 (\nu _\mathfrak{U} )$ , and we give a representation of the projection from $L_1 (\mu )_\mathfrak{U} $ onto $L_1 (\mu _\mathfrak{U} )$ . Moreover, the subsequence splitting principle forL ₁(μ) motivates the following question: if $\mathfrak{V}$ is an ultrafilter on ? and $[f_i ] \in L_1 (\mu )_\mathfrak{V} $ , is it possible to find a weakly convergent sequence (g _i ) ?L ₁(μ) following $\mathfrak{V}$ and a disjoint sequence (h _i ) ?L ₁(μ) such that [f _i ]=[g _i ]+[h _i ]? If $\mathfrak{V}$ is a selective ultrafilter, we find a positive answer by showing that $f = [f_i ] \in L_1 (\mu )_\mathfrak{V} $ belongs to $L_1 (\mu _{_\mathfrak{V} } )$ if and only if its representatives {f _i } are weakly convergent following $\mathfrak{V}$ and $f \in L_1 (\nu _\mathfrak{V} )$ if and only if it admits a representative consisting of pairwise disjoint functions. As a consequence, we obtain a new proof of the subsequence splitting principle. If $\mathfrak{V}$ is not a p-point then the above characterizations of $L_1 (\nu _{_\mathfrak{V} } )$ and $L_1 (\nu _{_\mathfrak{V} } )$ fail and the answer to the question is negative.     

Greedy expansions in Hilbert spaces

We study the rate of convergence of expansions of elements in a Hilbert space H into series with regard to a given dictionary D. The primary goal of this paper is to study representations of an element f ∈ H by a series f ～ ∑ _j=1 ^∞ c _j (f)g _j (f), $g_j \left( f \right) \in \mathcal{D}$ . Such a representation involves two sequences: {g _j (f)} _j=1 ^∞ and {c _j (f) _j=1 ^∞ . In this paper the construction of {g _j (f)} _j=1 ^∞ is based on ideas used in greedy-type nonlinear approximation, hence the use of the term greedy expansion. An interesting open problem questions, “What is the best possible rate of convergence of greedy expansions for f ∈ A ₁(D)?” Previously it was believed that the rate of convergence was slower than $m^{ - \tfrac{1} {4}}$ . The qualitative result of this paper is that the best possible rate of convergence of greedy expansions for $f \in A_1 \left( \mathcal{D} \right)$ is faster than $m^{ - \tfrac{1} {4}}$ . In fact, we prove it is faster than $m^{ - \tfrac{2} {7}}$ .     

Periodic-type functionals and their extensions

Let Ω denote a simply connected domain in the complex plane and let $\text{K[Ω]}$ be the collection of all entire functions of exponential type whose Laplace transforms are analytic on Ω′, the complement of Ω with respect to the sphere. Define a sequence of functionals $\text{{L}{}_{\mathrm{n}}\text{}}\text{on}\text{K[Ω]}$ by $\text{L}{}_{\mathrm{n}}\text{(f) =}\text{1}\text{2πi}\text{∝}{}_{\mathrm{\Gamma }}\text{g}{}_{\mathrm{n}}\text{(ζ) F(ζ) dζ}$, where F denotes the Laplace transform of f, Γ ? Ω is a simple closed contour chosen so that F is analytic outside and on Ω, and g_n is analytic on Ω. The specific functionals considered by this paper are patterned after the Lidstone functions, L_2n(f) = f⁽²ⁿ⁾(0) and L_{2n + 1}(f) = f⁽²ⁿ⁾(1), in that their sequence of generating functions {g_n} are “periodic.” Set g_{pn + k}(ζ) = h_k(ζ) ζ^pn, where p is a positive integer and each h_k (k = 0, 1,…, p ? 1) is analytic on Ω. We find necessary and sufficient conditions for $\text{f ∈ k[Ω]}\text{with}\text{L}{}_{\mathrm{n}}\text{(f) = 0 (n = 0, 1,…)}$. DeMar previously was able to find necessary conditions [7]. Next, we generalize {L_n} in several ways and find corresponding necessary and sufficient conditions.     

Methods of summation and best approximation

Let λ={λ _k ⁿ } be a triangular method of summation,f ε L_p (1 ≤ p ≤ ∞), $$U_n (f,x,\lambda ) = \frac{{a_0 }}{2} + \sum _{k = 1}^n \lambda _k^n (a_k \cos kx + b_k \sin kx).$$ Consideration is given to the problem of estimating the deviations ∥f ? U_n (f, λ) ∥ L_p in terms of a best approximation E_n (f) L_p in abstract form (for a sequence of projectors in a Banach space). Various generalizations of known inequalities are obtained.     

Some results on non-linear recurrence

We study the behavior of measure-preserving systems with continuous time along sequences of the form {n ^α}n∈#x2115;} where α is a positive real number¹. Let {S ^t } _t∈? be an ergodic continuous measure preserving flow on a probability Lebesgue space (X, β, μ). Among other results we show that:

1. For all but countably many α (in particular, for all α∈???) one can find anL ^∞-functionf for which the averagesA _N (f)(1/N)=Σ _n=1 ^N f(S ^nα x) fail to converge almost everywhere (the convergence in norm holds for any α!).
2. For any non-integer and pairwise distinct numbers α₁, α₂,..., α _k ∈(0, 1) and anyL ^∞-functionsf ₁,f ₂, ...,f _k , one has $$\mathop {lim}\limits_{N \to \infty } \left\| {\frac{1}{N}\sum\limits_{n - 1}^N {\prod\limits_{i - 1}^k {f_i (S^{n^{\alpha _i } } x) - \prod\limits_{i - 1}^k {\int_X {f_i d\mu } } } } } \right\|_{L^2 } = 0$$

We also show that Furstenberg’s correspondence principle fails for ?-actions by demonstrating that for all but a countably many α>0 there exists a setE?? having densityd(E)=1/2 such that, for alln∈?, $$d(E \cap (E - n^\alpha )) = 0$$ .