Finite extensions of generalized Bessel sequences to generalized frames

The objective of this paper is to investigate the question of modifying a given generalized Bessel sequence to yield a generalized frame or a tight generalized frame by finite extension. Some necessary and sufficient conditions for the finite extensions of generalized Bessel sequences to generalized frames or tight generalized frames are provided, and every result is illustrated by the corresponding example.     

On properties of generalized quadratic operators

In this paper, we investigate the set ω(P) of generalized quadratic operators A satisfying the equation A²=αA+βP for all complex numbers α and β and for an idempotent operator P such that AP=PA=A. Furthermore, the close relationship between the operator A∈ω(P) and the idempotent operator P are established and expressions for the inverse, the Moore-Penrose inverse and the Drazin inverse of A∈ω(P) are given. Some related results are also obtained.     

On dimensional properties of the generalized Smirnov's spaces

We show that the transfinite inductive dimensions modulo PP-trind and P-trInd introduced in M.G. Charalambous (1997) [2] differ by simple spaces, where P is the absolutely additive Borel class A(α) or the absolutely multiplicative Borel class M(α), 0?α<ω₁.     

System bandwidth and the existence of generalized shift-invariant frames

We consider the question whether, given a countable family of lattices ${({\mathrm{\Gamma }}_j)}_{j\in J}$ in a locally compact abelian group G, there exist functions ${(g_j)}_{j\in J}$ such that the resulting generalized shift-invariant system ${(g_j(??\gamma ))}_{j\in J,\gamma \in {\mathrm{\Gamma }}_j}$ is a tight frame of $L^2(G)$. This paper develops a new approach to the study of generalized shift-invariant system via almost periodic functions, based on a novel unconditional convergence property. From this theory, we derive characterizing relations for tight and dual frame generators, we introduce the system bandwidth as a measure of the total bandwidth a generalized shift-invariant system can carry, and we show that the so-called Calderón sum is uniformly bounded from below for generalized shift-invariant frames. Without the unconditional convergence property, we show, counter intuitively, that even orthonormal bases can have arbitrary small system bandwidth. Our results show that the question of existence of frame generators for a general lattice system is rather subtle and depends on analytical and algebraic properties of the lattice system.     

On local properties of spatial generalized quasi-isometries

Mathematical Notes - An upper bound for the measure of the image of a ball under mappings of a certain class generalizing the class of branched spatial quasi-isometries is determined. As a...     

On the existence of frames

This paper deals with two topics, namely, frames and pairwise balanced designs (PBD's). Frames, which were introduced by W.D. Wallis for the construction of (skew) Room squares, are shown to exist for most orders congruent to 1 (mod 4). This result relies heavily on the existence of PBD's since the set F = {v | there is a frame of order v] is shown to be PBD-closed. By employing a generalization of the usual recursive construction for PBD's, it is shown that $\text{B}${5, 9, 13, 17}?$\text{B}${5, 9, 13}∪{69, 77, 97, 137, 237, 277, 317, 377, 569}?{n | n  1 (mod 4), n>0}?{29, 33, 49, 57, 93, 129, 133}, where $\text{B}$(K) denotes the set of orders of PBD's of index one having block-sizes from the set K. Frames of orders 5, 9, 13 and 17 are exhibited which immediately implies that F?$\text{B}${5, 9, 13, 17}. D.R. Stinson and W.D. Wallis have shown that {29, 49}?F. Thus there is a frame of order υ for every positive integer υ congruent to 1 (mod 4) with the possible exceptions of υ ? {33, 57, 93, 133}.     

On generalized weak subdifferentials and some properties

In this article, some characterizations for gw-subdifferentiability of functions from ? ⁿ to ? ^m are stated. Some criteria for gw-subdifferentiability of generalized lower locally Lipschitz functions and positively homogeneous functions are given. Furthermore, it is proved that every Lipschitz function is gw-subdifferentiable at any point in its domain. Finally, the relationship between directional derivative and gw-subdifferential is given and a convexity criteria for Fréchet differentiable function is given by using gw-subdifferential.     

On some properties of functions of generalized variation

We show that the spaceV ^* generated by a -variation (or -variation of Riesz, or -variation of Waterman) forms a commutative Banach algebra with respect to the pointwise multiplication under the appropriate choice of norms.     

Distributive residuated frames and generalized bunched implication algebras

We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions, we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames.     

On some geometric properties of the generalized CES production functions

In this paper we prove that the generalized CES production function has constant return to scale if and only if the corresponding hypersurface is developable. Moreover, we establish that this production function has decreasing/increasing return to scale if and only if the corresponding hypersurface has positive/negative Gaussian curvature. These results are a generalization of some recent results concerning the generalized Cobb-Douglas production functions [G.E. Vîlcu, A geometric perspective on the generalized Cobb-Douglas production functions, Appl. Math. Lett. 24 (2011) 777-783].     

ON THE STABILITY OF FUSION FRAMES （FRAMES OF SUBSPACES）

A frame is an orthonormal basis-like collection of vectors in a Hilbert space, but need not be a basis or orthonormal. A fusion frame (frame of subspaces) is a frame-like collection of subspaces in a Hilbert space, thereby constructing a frame for the whole space by joining sequences of frames for subspaces. Moreover the notion of fusion frames provide a framework for applications and providing efficient and robust information processing algorithms.In this paper we study the conditions under which removing an element from a fusion frame, again we obtain another fusion frame. We give another proof of [5, Corollary 3.3(iii)] with extra information about the bounds.     

On the duality of fusion frames

The fusion frames were considered recently by P.G. Casazza, G. Kutyniok and S. Li in connection with distributed processing and are related to the construction of global frames from local frames. In this paper we give new results on the duality of fusion frames in Hilbert spaces.     

On the number of harmonic frames

There is a finite number $h_{n,d}$ of tight frames of n distinct vectors for ${\mathbb{C}}^d$ which are the orbit of a vector under a unitary action of the cyclic group ${\mathbb{Z}}_n$. These cyclic harmonic frames (or geometrically uniform tight frames) are used in signal analysis and quantum information theory, and provide many tight frames of particular interest. Here we investigate the conjecture that $h_{n,d}$ grows like $n^{d-1}$. By using a result of Laurent which describes the set of solutions of algebraic equations in roots of unity, we prove the asymptotic estimate$h_{n,d}\approx \frac{n^d}{\phi (n)}\ge n^{d-1},n\to \infty .$ By using a group theoretic approach, we also give some exact formulas for $h_{n,d}$, and estimate the number of cyclic harmonic frames up to projective unitary equivalence.     

On epi-extensions of frames

A characterization of finitely generated epi-extensions of frames is presented.Dedicated to the memory of Alan DayPresented by J. Sichler.     

On properties of the generalized elliptic pseudo-differential operators on closed manifolds

It is shown that two classes of generalized elliptic pseudo-differential operators, GEL(X) and REL(X), selected by the author from the class of classical linear pseudo-differential operators coincide. It is also shown that for any operators A, B ∈GEL(X) their composition AB and their global parametrices P_A, P_B belong to GEL(X). An operator A belongs to GEL(X) independently of the choice of a basis in E and of the weighted order of A. Some properties of the classes EFL(U) and REL(U) arising in microalocal analysis of generalized elliptic operators are studied. Bibliography: 12 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 215–269. Translated by N. A. Karazeeva.     

On constrained generalized inverses of matrices and their properties

A matrix G is called a generalized inverse (g-invserse) of matrix A if AGA = A and is denoted by G = A ⁻. Constrained g-inverses of A are defined through some matrix expressions like E(AE)⁻, (FA)⁻ F and E(FAE)⁻ F. In this paper, we derive a variety of properties of these constrained g-inverses by making use of the matrix rank method. As applications, we give some results on g-inverses of block matrices, and weighted least-squares estimators for the general linear model.     

On fine fractal properties of generalized infinite Bernoulli convolutions

The paper is devoted to the investigation of generalized infinite Bernoulli convolutions, i.e., the distributions μξ of the following random variables: where ak are terms of a given positive convergent series; ξk are independent random variables taking values 0 and 1 with probabilities p0k and p1k correspondingly.We give (without any restriction on {an}) necessary and sufficient conditions for the topological support of ξ to be a nowhere dense set. Fractal properties of the topological support of ξ and fine fractal properties of the corresponding probability measure μξ itself are studied in details for the case where ak?rk:=ak+1+ak+2+? (i.e., rk−1?2rk) for all sufficiently large k. The family of minimal dimensional (in the sense of the Hausdorff–Besicovitch dimension) supports of μξ for the above mentioned case is also studied in details. We describe a series of sets (with additional structural properties) which play the role of minimal dimensional supports of generalized Bernoulli convolutions. We also show how a generalization of M. Cooper's dimensional results on symmetric Bernoulli convolutions can easily be derived from our results.     

On some properties of generalized quasiisometries with unbounded characteristic

We consider a family of open discrete mappings f:D ?[`(\mathbb R<sup>n</sup>)] f:D \to \overline {{{\mathbb R}^n}} that distort, in a special way, the p-modulus of a family of curves that connect the plates of a spherical condenser in a domain D in \mathbb R<sup>n</sup> {{\mathbb R}^n} ; p > n-1; p < n; and bypass a set of positive p-capacity. We establish that this family is normal if a certain real-valued function that controls the considered distortion of the family of curves has finite mean oscillation at every point or only logarithmic singularities of order not higher than n - 1: We show that, under these conditions, an isolated singularity x <sub>0</sub> ∈ D of a mapping f:D\{ x<sub>0</sub> } ?[`(\mathbb Rⁿ)] f:D\backslash \left\{ {{x_0}} \right\} \to \overline {{{\mathbb R}^n}} is removable, and, moreover, the extended mapping is open and discrete. As applications, we obtain analogs of the known Liouville and Sokhotskii–Weierstrass theorems.