Finite Normalized Tight Frames

Frames are interesting because they provide decompositions in applications where bases could be a liability. Tight frames are valuable to ensure fast convergence of such decompositions. Normalized frames guarantee control of the frame elements. Finite frames avoid the subtle and omnipresent approximation problems associated with the truncation of infinite frames. In this paper the theory of finite normalized tight frames (FNTFs) is developed. The main theorem is the characterization of all FNTFs in terms of the minima of a potential energy function, which was designed to measure the total orthogonality of a Bessel sequence. Examples of FNTFs abound, e.g., in R ³ the vertices of the Platonic solids and of a soccer ball are FNTFs.     

On the construction of Wold decomposition for multivariate stationary processes

A method to construct the Wold decomposition for multivariate stationary stochastic processes x_k, k Z, is presented. The method is based on orthogonal decompositions for x_k, k Z, obtained by forming orthogonal projections of x_k, k Z, onto its component processes , k Z, j = 1, …, q. The method does not give a complete solution to the Wold decomposition problem.     

d-Cube decompositions ofK n/K m

Necessary conditions onn, m andd are given for the existence of an edge-disjoint decomposition ofK _n/K _m into copies of the graph of ad-dimensional cube. Sufficiency is shown whend=3 and, in some cases, whend=2^t. We settle the problem of embedding 3-cube decompositions ofK _m into 3-cube decompositions ofK _n, wherenm.Research of P.A. and D.E.B. supported by Australian Research Council grant A49532750Research of D.E.B. supported by Australian Research Council grant ARCPDF015GResearch of S.I.E. and C.V.E. supported by Illinois State University Research Office     

Componentwise perturbation bounds for some matrix decompositions

Componentwise perturbation bounds for the Cholesky,LDL ^H ,QR andLU decompositions are derived. The bounds improve known results of the same type and reveal the structural characteristics of the perturbations.This subject was supported by the Institute of Information Processing of the University of Umeå and the Swedish Natural Science Research Council.     

Continuous spectral decompositions of Abelian group actions on -algebras

Let G be a locally compact Abelian group. Following Ruy Exel, we view Fell bundles over the Pontrjagin dual group of G as continuous spectral decompositions of G-actions on C^*-algebras. We classify such spectral decompositions using certain dense subspaces related to Marc Rieffel's theory of square-integrability. There is a unique continuous spectral decomposition if the group acts properly on the primitive ideal space of the C^*-algebra. But there are also examples of group actions without or with several inequivalent spectral decompositions.     

Proper actions on imprimitivity bimodules and decompositions of Morita equivalences

We consider a class of proper actions of locally compact groups on imprimitivity bimodules over C^*-algebras which behave like the proper actions on C^*-algebras introduced by Rieffel in 1988. We prove that every such action gives rise to a Morita equivalence between a crossed product and a generalized fixed-point algebra, and in doing so make several innovations which improve the applicability of Rieffel's theory. We then show how our construction can be used to obtain canonical tensor-product decompositions of important Morita equivalences. Our results show, for example, that the different proofs of the symmetric imprimitivity theorem for actions on graph algebras yield isomorphic equivalences, and this gives new information about the amenability of actions on graph algebras.     

Hodge–Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media

We study in detail Hodge–Helmholtz decompositions in nonsmooth exterior domains Ω??^N filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms of rank q belonging to the weighted L²‐space L_s^{2, q}(Ω), s∈?, into irrotational and solenoidal q‐forms. These decompositions are essential tools, for example, in electro‐magnetic theory for exterior domains. To the best of our knowledge, these decompositions in exterior domains with nonsmooth boundaries and inhomogeneous and anisotropic media are fully new results. In the Appendix, we translate our results to the classical framework of vector analysis N=3 and q=1, 2. Copyright © 2008 John Wiley & Sons, Ltd.     

Perturbations of Banach Frames and Atomic Decompositions

Banach frames and atomic decompositions are sequences that have basis-like properties but which need not be bases. In particular, they allow elements of a Banach space to be written as linear combinations of the frame or atomic decomposition elements in a stable manner. In this paper we prove several functional — analytic properties of these decompositions, and show how these properties apply to Gabor and wavelet systems. We first prove that frames and atomic decompositions are stable under small perturbations. This is inspired by corresponding classical perturbation results for bases, including the Paley — Wiener basis stability criteria and the perturbation theorem el kato. We introduce new and weaker conditions which ensure the desired stability. We then prove quality properties of atomic decompositions and consider some consequences for Hilbert frames. Finally, we demonstrate how our results apply in the practical case of Gabor systems in weighted L² spaces. Such systems can form atomic decompositions for L²_w(IR), but cannot form Hilbert frames but L²_w(IR) unless the weight is trivial.     

On tactical decompositions of class number 2

In this article we consider tactical decompositions of class number 2 of symmetric designs. Our main result says that if the orders are prime, then the only decompositions are of affine type. Moreover, we study symmetric decompositions of finite projective planes and show that, except in some cases, they are related to Baer subplanes, unitals, or 2 - ((m ² - m + 1)m, m, 1)designs.     

Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x ²ⁿ + y ²ⁿ , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.     

Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality

In this paper we study how prime filtrations and squarefree Stanley decompositions of squarefree modules over the polynomial ring and over the exterior algebra behave with respect to Alexander duality. The results which we obtained suggest a lower bound for the regularity of a \mathbb Zⁿ{\mathbb {Z}^n}-graded module in terms of its Stanley decompositions. For squarefree modules this conjectured bound is a direct consequence of Stanley’s conjecture on Stanley decompositions. We show that for pretty clean rings of the form R/I, where I is a monomial ideal, and for monomial ideals with linear quotient our conjecture holds.     

Jordan decompositions in alternative loop rings

It is observed that the additive as well as multiplicative Jordan decompositions hold in alternative loop algebras of finiteRA loops and theRA loops for which the additive Jordan decomposition holds in the integral loop ring are characterized. Multiplicative Jordan decomposition (MJD) inZL, whereL is a finiteRA loop with cyclic centre is analysed, besides settling MJD for integral loop rings of allRA loops of order ≤32. It is also shown that for any finiteRA loopL,U (ZL) is an almost splittable Moufang loop. Research of the second author is supported by CSIR.     

The Number of Negative Entries in a Sign Pattern Allowing Orthogonality

A ± sign pattern is a matrix whose entries are in the set {+,–}. An n×n ± sign pattern A allows orthogonality if there exists a real orthogonal matrix B in the qualitative class of A. In this paper, we prove that for n3 there is an n×n ± sign pattern A allowing orthogonality with exactly k negative entries if and only if n–1kn²–n+1.Research supported by Shanxi Natural Science Foundation 20011006, 20041010Final version received: October 22, 2003     

Telescoping partial fractions decompositions, the little q-Jacobi functions of complex order, and the nonterminating q-Saalschütz sum

We use telescoping partial fractions decompositions to give new proofs of the orthogonality property and the normalization relation for the little q-Jacobi polynomials, and the q-Saalschütz sum. In [20], we followed the development [19] of Schur functions for partitions with complex parts, and we showed that there exist natural little q-Jacobi functions of complex order which satisfy extensions of the orthogonality property and normalization relation of the little q-Jacobi polynomials, and that these two results follow from and together imply the nonterminating form of the q-Saalschütz sum. Writing the q-Pochhammer symbol of complex order as a ratio of infinite products in the usual way, we obtain new telescoping partial fractions decomposition proofs of our results [20] for the little q-Jacobi functions of complex order. We give several new proofs of the q-Saalschütz sum and its nonterminating form. For our friends Dick and Liz 2000 Mathematics Subject Classification Primary—42C05; Secondary—33C45, 33C47     

Decomposing complete equipartite graphs into odd square‐length cycles: number of parts odd

In this article, we introduce a new technique for obtaining cycle decompositions of complete equipartite graphs from cycle decompositions of related multigraphs. We use this technique to prove that if n, m and λ are positive integers with n ≥ 3, λ≥ 3 and n and λ both odd, then the complete equipartite graph having n parts of size m admits a decomposition into cycles of length λ² whenever nm ≥ λ² and λ divides m. As a corollary, we obtain necessary and sufficient conditions for the decomposition of any complete equipartite graph into cycles of length p², where p is prime. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:401‐414, 2010     

Polar decompositions and related classes of operators in spaces ∏κ

Polar decompositions with respect to an indefinite inner product are studied for bounded linear operators acting on a space. Criteria are given for existence of various forms of the polar decompositions, under the conditions that the range of a given operatorX is closed and that zero is not an irregular critical point of the selfadjoint operatorX ^[*]X. Both real and complex spaces are considered. Relevant classes of operators having a selfadjoint (in the sense of the indefinite inner product) square root, or a selfadjoint logarithm, are characterized.The work of this author was partially supported by INdAM-GNCS and MURSTThe work of this author was partially supported by NSF grant DMS-9988579.     

A New Characterization of p-Smoothable Spaces

In this paper atomic decompositions for Banach space valued martingales are given. In the other direction, p-smoothable Banach spaces are characterized in terms of atomic decompositions for Banach space valued martingales. This revised version was published online in June 2006 with corrections to the Cover Date.     

Wold decomposition,prediction and parameterization of stationary processes with infinite variance

Summary A discrete time stochastic process {_t} is said to be a p-stationary process (1<p2)if , for all integers n1, t ₁,...t _n,h and scalars b ₁,...b _n.The class of p-stationary processes includes the class of second-order weakly stationary stochastic processes, harmonizable stable processes of order (1<2), and p ^thorder strictly stationary processes. For any nondeterministic process in this class a finite Wold decomposition (moving average representation) and a finite predictive decomposition (autoregressive representation) are given without alluding to any notion of covariance or spectrum. These decompositions produce two unique (interrelated) sequences of scalar which are used as parameters of the process {_t}. It is shown that the finite Wold and predictive decomposition are all that one needs in developing a Kolmogorov-Wiener type prediction theory for such processes.     

Vilenkin systems and generalized triangular truncation operator

We study actions of the Vilenkin group ^{k=0 m(k)} onL _p -spaces associated with a semi-finite von Neumann algebra , via a generalized triangular truncation operator. The systems of eigenspaces that arise contain the classical unbounded Vilenkin systems and we show that such systems with the inverse lexicographic enumeration form Schauder decompositions in all reflexive non-commutativeL _p -spaces. This is a non-commutative analogue of a theorem of Paley for unbounded Vilenkin systems, which in the classical setting is due to W.S. Young, F. Schipp and P. Simon.Research is partially supported by the NSF Grant DMS 99-71587Research is partially supported by the Australian Research Council (ARC)