Canadian Mathematical Society
Canadian Mathematical Society
  location:  Publicationsjournals
Search results

Search: All articles in the CMB digital archive with keyword isometry

  Expand all        Collapse all Results 1 - 6 of 6

1. CMB Online first

Chen, Wengu; Ge, Huanmin
A sharp bound on RIC in generalized orthogonal matching pursuit
Generalized orthogonal matching pursuit (gOMP) algorithm has received much attention in recent years as a natural extension of orthogonal matching pursuit (OMP). It is used to recover sparse signals in compressive sensing. In this paper, a new bound is obtained for the exact reconstruction of every $K$-sparse signal via the gOMP algorithm in the noiseless case. That is, if the restricted isometry constant (RIC) $\delta_{NK+1}$ of the sensing matrix $A$ satisfies $ \delta_{NK+1}\lt \frac{1}{\sqrt{\frac{K}{N}+1}}$, then the gOMP can perfectly recover every $K$-sparse signal $x$ from $y=Ax$. Furthermore, the bound is proved to be sharp. In the noisy case, the above bound on RIC combining with an extra condition on the minimum magnitude of the nonzero components of $K$-sparse signals can guarantee that the gOMP selects all of support indices of the $K$-sparse signals.

Keywords:sensing matrix, generalized orthogonal matching pursuit, restricted isometry constant, sparse signal
Categories:65D15, 65J22, 68W40

2. CMB 2016 (vol 60 pp. 350)

Ma, Yumei
Isometry on Linear $n$-G-quasi Normed Spaces
This paper generalizes the Aleksandrov problem: the Mazur-Ulam theorem on $n$-G-quasi normed spaces. It proves that a one-$n$-distance preserving mapping is an $n$-isometry if and only if it has the zero-$n$-G-quasi preserving property, and two kinds of $n$-isometries on $n$-G-quasi normed space are equivalent; we generalize the Benz theorem to n-normed spaces with no restrictions on the dimension of spaces.

Keywords:$n$-G-quasi norm, Mazur-Ulam theorem, Aleksandrov problem, $n$-isometry, $n$-0-distance
Categories:46B20, 46B04, 51K05

3. CMB 2016 (vol 59 pp. 234)

Beardon, Alan F.
Non-discrete Frieze Groups
The classification of Euclidean frieze groups into seven conjugacy classes is well known, and many articles on recreational mathematics contain frieze patterns that illustrate these classes. However, it is only possible to draw these patterns because the subgroup of translations that leave the pattern invariant is (by definition) cyclic, and hence discrete. In this paper we classify the conjugacy classes of frieze groups that contain a non-discrete subgroup of translations, and clearly these groups cannot be represented pictorially in any practical way. In addition, this discussion sheds light on why there are only seven conjugacy classes in the classical case.

Keywords:frieze groups, isometry groups
Categories:51M04, 51N30, 20E45

4. CMB 2014 (vol 58 pp. 9)

Chavan, Sameer
Irreducible Tuples Without the Boundary Property
We examine spectral behavior of irreducible tuples which do not admit boundary property. In particular, we prove under some mild assumption that the spectral radius of such an $m$-tuple $(T_1, \dots, T_m)$ must be the operator norm of $T^*_1T_1 + \cdots + T^*_mT_m$. We use this simple observation to ensure boundary property for an irreducible, essentially normal joint $q$-isometry provided it is not a joint isometry. We further exhibit a family of reproducing Hilbert $\mathbb{C}[z_1, \dots, z_m]$-modules (of which the Drury-Arveson Hilbert module is a prototype) with the property that any two nested unitarily equivalent submodules are indeed equal.

Keywords:boundary representations, subnormal, joint p-isometry
Categories:47A13, 46E22

5. CMB 2011 (vol 54 pp. 680)

Jiménez-Vargas, A.; Villegas-Vallecillos, Moisés
$2$-Local Isometries on Spaces of Lipschitz Functions
Let $(X,d)$ be a metric space, and let $\mathop{\textrm{Lip}}(X)$ denote the Banach space of all scalar-valued bounded Lipschitz functions $f$ on $X$ endowed with one of the natural norms $ \| f\| =\max \{\| f\| _\infty ,L(f)\}$ or $\|f\| =\| f\| _\infty +L(f), $ where $L(f)$ is the Lipschitz constant of $f.$ It is said that the isometry group of $\mathop{\textrm{Lip}}(X)$ is canonical if every surjective linear isometry of $\mathop{\textrm{Lip}}(X) $ is induced by a surjective isometry of $X$. In this paper we prove that if $X$ is bounded separable and the isometry group of $\mathop{\textrm{Lip}}(X)$ is canonical, then every $2$-local isometry of $\mathop{\textrm{Lip}}(X)$ is a surjective linear isometry. Furthermore, we give a complete description of all $2$-local isometries of $\mathop{\textrm{Lip}}(X)$ when $X$ is bounded.

Keywords:isometry, local isometry, Lipschitz function
Categories:46B04, 46J10, 46E15

6. CMB 2010 (vol 53 pp. 398)

Botelho, Fernanda; Jamison, James
Projections in the Convex Hull of Surjective Isometries
We characterize those linear projections represented as a convex combination of two surjective isometries on standard Banach spaces of continuous functions with values in a strictly convex Banach space.

Keywords:isometry, convex combination of isometries, generalized bi-circular projections
Categories:47A65, 47B15, 47B37

© Canadian Mathematical Society, 2017 :