## A Necessary And Sufficient Condition For The Linear Independence Of The Integer Translates Of A Compactly Supported Distribution (1987)

Venue: | Constr. Approx |

Citations: | 43 - 8 self |

### BibTeX

@ARTICLE{Ron87anecessary,

author = {Amos Ron and Beverly Sackler},

title = {A Necessary And Sufficient Condition For The Linear Independence Of The Integer Translates Of A Compactly Supported Distribution},

journal = {Constr. Approx},

year = {1987},

volume = {5},

pages = {297--308}

}

### Years of Citing Articles

### OpenURL

### Abstract

Given a multivariate compactly supported distribution OE, we derive here a necessary and sufficient condition for the global linear independence of its integer translates. This condition is based on the location of the zeros of b OE = The Fourier-Laplace transform of OE. The utility of the condition is demonstrated by several examples and applications, showing in particular, that previous results on box splines and exponential box splines can be derived from this condition by a simple combinatorial argument. July, 1987 1980 Mathematics Subject Classification (1985): Primary 41A63, 41A15. Key Words: Box Splines, Exponential Box Splines, Polynomial Box Splines, Integer Translates, Compactly Supported Function, Compactly Supported Distribution, Spectral Analysis, Global Linear Independence, Fourier Transform. 1. Introduction and Statement of Main Results Let E 0 (IR s ) be the space of all s-dimensional complex valued distributions of compact support. For each OE 2 E 0 (IR s ) ...

### Citations

468 | Functional Analysis - YOSIDA - 1980 |

242 |
Topological Vector Spaces, Distributions and Kernels
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- 1969
(Show Context)
Citation Context ...icient condition for the global linear independence of the integer translates of a OE 2 E 0 (IR s ). The methods here make an essential use of distribution theory. For background material we refer to =-=[T]-=-, which is also the source for some of the notations. The utility of this necessary and sufficient condition is demonstrated by several examples and applications. Given OE 2 E 0 (IR s ), it is well kn... |

163 | A Fourier analysis of the finite element variational method - Strang, Fix - 1971 |

60 | Translates of multivariate splines - Dahmen, Micchelli - 1983 |

59 |
B-splines from parallelepipeds
- Boor, HRollig
- 1982
(Show Context)
Citation Context ... k3 . To analyze Kφ let { 1 0 ≤ t ≤ 1, ρ(t) = 0 otherwise . 1∫ Then ρ(t) = e−itxdx and so ρ(t) = 0 if and only if t ∈ 2πZZ\0. Define 0 ψ(x1, x2) = ρ(x1) ∗ ρ(x2) ∗ ρ(x1 + x2) . It is well known (see =-=[BH1]-=-) that ψ(x1, x2) is the hat function (with support similar to that drawn in Fig. 3.1) and therefore Kψ = 0. But since ψ(x) = ρ(x1)ρ(x2)ρ(x1 + x2) we see from (3.2) that φ(x) = 0 ⇒ ψ(x) = 0 , a... |

27 |
Local linear independence of the translates of a box spline, Constructive Approximation 1
- Jia
- 1985
(Show Context)
Citation Context ... that the set of exponentials in K OE is identical to that of K/ . The second example generalizes the known results about the global linear independence of the translates of a polynomial box spline , =-=[J]-=-, and a real exponential box spline [R], [DM 3 ]. Theorem 1.4. Let X = fx 1 ; : : : ; x n g be a collection of non-trivial vectors in ZZ s , which span IR s . For every J = fi 1 ; : : : ; i s g ae f1;... |

20 |
Recent progress in multivariate splines, in Approximation Theory
- Dahmen, Micchelli
(Show Context)
Citation Context ... − λj) annihilates Q. Since dim Q = dim ker L = n we conclude Q = ker L. Remark 2.1. The above corollary shows that for univariate φ, Kφ∩ℓp = 0 for 1 ≤ p < ∞, thus contradicting Proposition 4.10.2 of =-=[DM2]-=-. Corollary 2.5. Assume s = 1, q ∈ Q and q is not an exponential polynomial (i.e., a linear combination of products of exponentials by polynomials). Then the linear span of {E α q}α∈ZZ is dense in Q. ... |

16 |
Bivariate box splines and smooth pp functions on a three-direction mesh
- Boor, Höllig
- 1983
(Show Context)
Citation Context ...t N(φ) = ∅ and application of Theorem 1.1(b) completes the proof. In our last example we prove the global linear independence of the integer translates of a bivariate function φ which was examined in =-=[BH2]-=-. Example 3.3. Let τ be the characteristic function of the triangle with vertices (0, 0), (1, 0), (1, 1). Let ψ be a three directional polynomial box spline, namely ∫ 1 ψ(x) = ( e −ix1t ∫ 1 k1 dt) ( ... |

9 |
Analyse Spectrale sur Zn
- Lefranc
- 1958
(Show Context)
Citation Context ...φ ∈ E ′ (IR s ) is a closed shift invariant subspace of Q. The result we need about shift invariant subspaces of Q is recorded in the following theorem which is due to Lefranc. Theorem 1.2. (Lefranc, =-=[Le]-=-). Every non-trivial closed shift invariant subspace of Q contains an exponential q(α) = z α . Here we used the standard notation z α = ∏ s j=1 zαj j . 1Given a closed shift invariant subspace Q of Q... |

4 |
Linear independence of the translates of an exponential box spline, preprint 1986, to appear
- Ron
- 1973
(Show Context)
Citation Context ...s identical to that of K/ . The second example generalizes the known results about the global linear independence of the translates of a polynomial box spline , [J], and a real exponential box spline =-=[R]-=-, [DM 3 ]. Theorem 1.4. Let X = fx 1 ; : : : ; x n g be a collection of non-trivial vectors in ZZ s , which span IR s . For every J = fi 1 ; : : : ; i s g ae f1; : : : ; ng denote by X J the matrix wi... |

3 |
Analyse spectrale sur
- Lefranc
- 1958
(Show Context)
Citation Context ...E 2 E 0 (IR s ) is a closed shift invariant subspace of Q. The result we need about shift invariant subspaces of Q is recorded in the following theorem which is due to Lefranc. Theorem 1.2. (Lefranc, =-=[Le]-=-). Every non-trivial closed shift invariant subspace of Q contains an exponential q(ff) = z ff . Here we used the standard notation z ff = Q s j=1 z ff j j . 1 Given a closed shift invariant subspace ... |

1 |
Multivariate E-splines, preprint
- Dahmen, Micchelli
- 1987
(Show Context)
Citation Context ...entical to that of Kψ. The second example generalizes the known results about the global linear independence of the translates of a polynomial box spline , [J], and a real exponential box spline [R], =-=[DM3]-=-. Theorem 1.4. Let X = {x 1 , . . . , x n } be a collection of non-trivial vectors in ZZ s , which span IR s . For every J = {i1, . . . , is} ⊂ {1, . . . , n} denote by XJ the matrix with columns x i1... |