Asymmetric multi-channel sampling in shift invariant spaces

We develop an asymmetric multi-channel sampling on a shift invariant space V(?) with a Riesz generator ?(t) in L²(R), where each channeled signal is assigned a uniform but distinct sampling rate. We use Fourier duality between V(?) and L²[0,2π] to find conditions under which there is a stable asymmetric multi-channel sampling formula on V(?).     

Beurling's theorem and invariant subspaces for the shift on Hardy spaces

Let G be a bounded open subset in the complex plane and let H~2(G) denote the Hardy space on G. We call a bounded simply connected domain W perfectly connected if the boundary value function of the inverse of the Riemann map from W onto the unit disk D is almost 1-1 with respect to the Lebesgue measure on D and if the Riemann map belongs to the weak-star closure of the polynomials in H~∞(W). Our main theorem states: in order that for each M∈Lat (M_z), there exist u∈H~∞(G) such that M=∨{uH~2(G)}, it is necessary and sufficient that the following hold: (1) each component of G is a perfectly connected domain; (2) the harmonic measures of the components of G are mutually singular; (3) the set of polynomials is weak-star dense in H~∞(G). Moreover, if G satisfies these conditions, then every M∈Lat (M_z) is of the form uH~2(G), where u∈H~∞(G) and the restriction of u to each of the components of G is either an inner function or zero.     

The structure of shift invariant spaces on a locally compact abelian group

We investigate shift invariant subspaces of L²(G), where G is a locally compact abelian group. We show, among other things, that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame.     

On principal invariant subspaces

Let F and G be closed subspaces of the complex Hilbert space H, and U and V be closed subspaces of F and G, respectively. In this paper, using the technique of operator block, we present the necessary and sufficient conditions under which (U, V) is a pair of (strictly, non-degenerate) principal invariant subspaces for (F, G).     

Sums of Hilbert space frames

We give simple necessary and sufficient conditions on Bessel sequences {fi} and {gi} and operators L₁, L₂ on a Hilbert space H so that {L₁fi+L₂gi} is a frame for H. This allows us to construct a large number of new Hilbert space frames from existing frames.     

Generalized average sampling in shift invariant spaces

Average sampling is motivated by realistic needs, e.g., physical limitation of acquisition devices or characteristics of sampling procedure. As an extension of the average sampling, we study generalized average sampling in shift invariant spaces with frame generators, in which averages are taken from suitable channeled version of signals. Illustrative examples are given.     

Univalent mappings and invariant subspaces of the Bergman and Hardy spaces

In both the Bergman space and the Hardy space , the problem of determining which bounded univalent mappings of the unit disk have the wandering property is addressed. Generally, a function in has the wandering property in , where denotes either or , provided that every -invariant subspace of is generated by the orthocomplement of within . It is known that essentially every function which has the wandering property in either space is the composition of a univalent mapping with a classical inner function, and that the identity mapping has this property in both spaces. Consequently, weak-star generators of also have the wandering property in both settings. The present paper gives a partial converse to this, and shows that in both settings there is a large class of bounded univalent mappings which fail to have the wandering property.

     

Rank-one cross commutators on backward shift invariant subspaces on the bidisk

For a backward shift invariant subspace N in H^2（Г^2）, the operators Sz and Sw on N are defined by Sz = PNTz｜N and Sw, = PNTw｜N, where PN is the orthogonal projection from L^2（Г^2） onto N. We give a characterization of N satisfying rank [Sz, Sw^＊] = 1.     

Bessel sequences,wavelet frames,duals and extensions

From the perspectives of duality and extensions, Gabor frames and wavelet frames have contrasting behaviour. Our chief concern here is about duality. Canonical duals of wavelet frames may not be wavelet frames, whereas canonical duals of Gabor frames are Gabor frames. Keeping these in view, we give several constructions of wavelet frames with wavelet canonical duals. For this, a simple characterisation of Bessel sequences and a general commutativity result are given, the former also leading naturally to some extension results.     

Multishifts in Hilbert spaces

We introduce and study a multishift structure in a Hilbert space. This structure is a noncommutative analog of the (simple one-sided) shift operator, well known in function theory and functional analysis. Subspaces invariant under the multishift are described. A theorem on the factorization into an inner and an outer factor is established for operators commuting with the multishift.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 69–81, 2005Original Russian Text Copyright © by P. A. TerekhinSupported by the RFBR under grant No. 03-01-00390, the program Leading Scientific Schools of the Russian Federation under grant No. NSh-1295.2003.1, and the INTAS under grant No. 99-00089.Translated by V. E. Nazaikinskii     

Approximation by polynomials and rational functions in weighted rearrangement invariant spaces

Let Γ be a Dini-smooth curve in the complex plane, and let G:=IntΓ. We prove some direct and inverse theorems of approximation theory by algebraic polynomials and rational functions in the weighted rearrangement invariant Smirnov spaces EX(G,ω) defined on G.     

Error estimation for non-uniform sampling in shift invariant space

To reconstruct a function from its sampling value is not always exact, error may arise due to a lot of reasons, therefore error estimation is useful in reconstruction. For non-uniform sampling in shift invariant space, three kinds of errors of the reconstruction formula are discussed in this article. For every kind of error, we give an estimation. We find the accuracy of the reconstruction formula mainly depends on the decay property of the generator and the sampling function.     

Duality problem for disjointly homogeneous rearrangement invariant spaces

Let $1\le p<\infty$. A Banach lattice E is said to be disjointly homogeneous (resp. p-disjointly homogeneous) if two arbitrary normalized disjoint sequences from E contain equivalent in E subsequences (resp. every normalized disjoint sequence contains a subsequence equivalent in E to the unit vector basis of $l_p$). Answering a question raised in the paper [11], for each $1<p<\infty$, we construct a reflexive p-disjointly homogeneous rearrangement invariant space on $[0,1]$ whose dual is not disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to $l_p$, $1\le p<\infty$, or $c_0$.     

The stability of Banach frames in Banach spaces

In this paper, we give some equivalent conditions on a Banach frame for a Banach space by using the pseudoinverse operator. We also consider the stability of a Banach frame for a Banach space X with respect to Xd or an Xd-frame for a Banach space X under perturbation. These results generalize and improve the related works of Balan, Casazza, Christensen, Stoeva and Jian et al.     

关于次可分解算子的不变子空间

In this paper,we prove the Mohebi-Radjabalipour Conjecture under an ad-ditional condition,and obtain an invariant subspace theorem on subdecomposableoperators.     

Symmetric approximation of frames and bases in Hilbert spaces

We introduce the symmetric approximation of frames by normalized tight frames extending the concept of the symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces. We prove existence and uniqueness results for the symmetric approximation of frames by normalized tight frames. Even in the case of the symmetric orthogonalization of bases, our techniques and results are new. A crucial role is played by whether or not a certain operator related to the initial frame or basis is Hilbert-Schmidt.

     

The continuity of superposition operators on some sequence spaces defined by moduli

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x _k ): (ϕk(|x _k |)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψ_k), we characterize the continuity of the superposition operators P _f from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type. This research was supported by Estonian Science Foundation Grant 5376.     

Time-frequency partitions and characterizations of modulation spaces with localization operators

We study families of time-frequency localization operators and derive a new characterization of modulation spaces. This characterization relates the size of the localization operators to the global time-frequency distribution. As a by-product, we obtain a new proof for the existence of multi-window Gabor frames and extend the structure theory of Gabor frames.