#### DMCA

## Localization of Calderon Convolution in the Fourier Domain

### Citations

3203 |
A Wavelet Tour of Signal Processing
- Mallat
- 1998
(Show Context)
Citation Context ... D(k) T F (· − k). We also denote the shift-invariant space Vp(f1, . . . , fn) by Vp(F ) when F = (f1, . . . , fn) T . The finitely generated shift-invariant space Vp(F ) appears in wavelet analysis (=-=[5, 6, 10, 11]-=-), as well as in sampling theory ([1, 3, 4]). It is well known that the space of all band-limited functions in L 2 is a shift-invariant space generated by the sinc function sin πx πx . Let L p , 1 ≤ p... |

1036 |
An introduction to wavelets
- Chui
- 1992
(Show Context)
Citation Context ... D(k) T F (· − k). We also denote the shift-invariant space Vp(f1, . . . , fn) by Vp(F ) when F = (f1, . . . , fn) T . The finitely generated shift-invariant space Vp(F ) appears in wavelet analysis (=-=[5, 6, 10, 11]-=-), as well as in sampling theory ([1, 3, 4]). It is well known that the space of all band-limited functions in L 2 is a shift-invariant space generated by the sinc function sin πx πx . Let L p , 1 ≤ p... |

345 |
Ondelettes et Operateurs.
- Meyer
- 1990
(Show Context)
Citation Context ... D(k) T F (· − k). We also denote the shift-invariant space Vp(f1, . . . , fn) by Vp(F ) when F = (f1, . . . , fn) T . The finitely generated shift-invariant space Vp(F ) appears in wavelet analysis (=-=[5, 6, 10, 11]-=-), as well as in sampling theory ([1, 3, 4]). It is well known that the space of all band-limited functions in L 2 is a shift-invariant space generated by the sinc function sin πx πx . Let L p , 1 ≤ p... |

212 | Non-uniform sampling and reconstruction in shift-invariant spaces
- Aldroubi, Gröchenig
- 2001
(Show Context)
Citation Context ...riant space Vp(f1, . . . , fn) by Vp(F ) when F = (f1, . . . , fn) T . The finitely generated shift-invariant space Vp(F ) appears in wavelet analysis ([5, 6, 10, 11]), as well as in sampling theory (=-=[1, 3, 4]-=-). It is well known that the space of all band-limited functions in L 2 is a shift-invariant space generated by the sinc function sin πx πx . Let L p , 1 ≤ p ≤ ∞, be the space of all p-integrable func... |

188 | Using the refinement equations for the construction of pre-wavelets III: Elliptic splines
- Micchelli, Rabut, et al.
- 1991
(Show Context)
Citation Context ... ) be the space of all functions f so that ‖f‖ W (L p ,ℓ 1 ) := ∑ k∈Z d ‖f‖ L p (k+[0,1) d ) is finite. Clearly W (L p , ℓ 1 ) ⊂ L p for 1 ≤ p ≤ ∞. For any D ∈ (ℓ p ) (r) and F ∈ L p , it is shown in =-=[9]-=- that ‖F ∗ ′ D‖p ≤ ‖D‖ℓp‖F ‖Lp. (2.1) So in Theorem 2.1, we assume that the generator F of the shift-invariant space belongs to Lp for 1 ≤ p < ∞ and it belongs to W (L∞, ℓ1 ) for p = ∞. In that case V... |

45 | Wavelet analysis of refinement equations,” - Villemoes - 1994 |

30 |
Subdivision schemes in L p spaces
- Jia
- 1995
(Show Context)
Citation Context ...ere exist positive constants C and δ such that C −1 ‖D‖ℓp ≤ ‖F ∗′ D‖p ≤ C‖D‖ℓp ∀ D ∈ (ℓp B(ξ0,δ) )(r), (2.4) and that F has ℓ p stable shifts if (2.4) holds for all D ∈ (ℓ p ) (r) . (See for instance =-=[7, 8, 13, 15]-=-) and the references therein for applications of ℓ p stable shifts in the approximation by shift-invariant spaces, the regularity of scaling functions, and the convergence of cascade algorithms). The ... |

23 | Nonuniform average sampling and reconstruction of signals with finite rate of innovation
- Sun
- 2006
(Show Context)
Citation Context ...riant space Vp(f1, . . . , fn) by Vp(F ) when F = (f1, . . . , fn) T . The finitely generated shift-invariant space Vp(F ) appears in wavelet analysis ([5, 6, 10, 11]), as well as in sampling theory (=-=[1, 3, 4]-=-). It is well known that the space of all band-limited functions in L 2 is a shift-invariant space generated by the sinc function sin πx πx . Let L p , 1 ≤ p ≤ ∞, be the space of all p-integrable func... |

22 | Convolution, average sampling and a Calderon resolution of the identity for shift-invariant spaces
- Aldroubi, Sun, et al.
(Show Context)
Citation Context ...riant space Vp(f1, . . . , fn) by Vp(F ) when F = (f1, . . . , fn) T . The finitely generated shift-invariant space Vp(F ) appears in wavelet analysis ([5, 6, 10, 11]), as well as in sampling theory (=-=[1, 3, 4]-=-). It is well known that the space of all band-limited functions in L 2 is a shift-invariant space generated by the sinc function sin πx πx . Let L p , 1 ≤ p ≤ ∞, be the space of all p-integrable func... |

11 | Convergence of cascade algorithms and smoothness of refinable distributions
- Sun
- 2003
(Show Context)
Citation Context ...ere exist positive constants C and δ such that C −1 ‖D‖ℓp ≤ ‖F ∗′ D‖p ≤ C‖D‖ℓp ∀ D ∈ (ℓp B(ξ0,δ) )(r), (2.4) and that F has ℓ p stable shifts if (2.4) holds for all D ∈ (ℓ p ) (r) . (See for instance =-=[7, 8, 13, 15]-=-) and the references therein for applications of ℓ p stable shifts in the approximation by shift-invariant spaces, the regularity of scaling functions, and the convergence of cascade algorithms). The ... |

7 |
Stability of the shifts of globally supported distributions
- Sun
- 2000
(Show Context)
Citation Context ...p(F ) is a linear subspace of Lp by (2.1). We say that a function f is a C∞-function with ℓ1-decay if the partial derivative Dnf satisfies ∑ ‖D n f‖ L ∞ (k+[0,1) d ) < ∞ for any n ∈ (ZZ+) d k∈Zd (see =-=[12, 14]-=-). A Schwartz function is a C∞-function with ℓ1-decay, and so are linear combinations of the integer shifts of a Schwartz function using ℓ1 coefficients. In Theorem 2.1, the dual functions in the reco... |

5 |
p-frames and shift invariant spaces of L p
- Aldroubi, Sun, et al.
- 2001
(Show Context)
Citation Context ...s independent of ξ. Then G is a stable Calderon convolutor for Vp(F ) if and only if G is a stable Calderon convolutor for Vp(F ) at every frequency. For the function F in Theorem 3.1, it is shown in =-=[2, 14]-=- that the rank ( F (ξ+2kπ)) k∈Zd is independent of ξ if and only if the shift-invariant space Vp(F ) is closed in Lp . So applying Theorem 3.1, we see that under certain closedness assumption on the... |

5 |
A survey on L 2 -approximation order from shift-invariant spaces
- Jetter, Plonka
- 2001
(Show Context)
Citation Context ...ere exist positive constants C and δ such that C −1 ‖D‖ℓp ≤ ‖F ∗′ D‖p ≤ C‖D‖ℓp ∀ D ∈ (ℓp B(ξ0,δ) )(r), (2.4) and that F has ℓ p stable shifts if (2.4) holds for all D ∈ (ℓ p ) (r) . (See for instance =-=[7, 8, 13, 15]-=-) and the references therein for applications of ℓ p stable shifts in the approximation by shift-invariant spaces, the regularity of scaling functions, and the convergence of cascade algorithms). The ... |

1 | Localization of stability and p-frame in the Fourier domain
- Sun
(Show Context)
Citation Context ...established in [4]. In this paper, we introduce and study the localization of Calderon convolution for a finitely generated shift-invariant space in the Fourier domain, using similar techniques as in =-=[14]-=-, where semi-convolution and the frame operator in finitely generated shift-invariant spaces are localized in the Fourier domain. Define the Fourier transform ˆ f of an integrable function f by ˆ f(ξ)... |