#### DMCA

## C.Karanikas, Truncation Error on Wavelet sampling Expansions

Venue: | J. Computational Anal. and Applications |

Citations: | 2 - 1 self |

### Citations

82 |
Wavelets and Other Orthogonal Systems With Applications
- Walter
- 1994
(Show Context)
Citation Context ...φ(.−n), n ∈ Z} is an orthonormal basis on V ; it is clear that V is a shift invariant space 1 and its reproducing kernel K(x, t) is given by: K(x, t) = ∑ n∈Z φ(x− n)φ(t− n), (x, t) ∈ R2, (see [9] and =-=[10]-=-). Throughout this work, we assume that the function φ satisfies the following conditions: (i) |φ(x)| ≤ (cons.) |B(x)||x|1+ε , where ε ≥ 0 and |B(x)| is bounded and 1-periodic function on R. (ii) ∑ n∈... |

82 |
A sampling theorem for wavelet subspaces,”
- Walter
- 1992
(Show Context)
Citation Context ...ros on [−pi, pi]. It is known that conditions (i) and (ii) imply that {K(x, n), n ∈ Z} is a Riesz basis on V , with a unique biorthogonal Riesz basis {S(x− n), n ∈ Z}, (see [10] Proposition (9.1) and =-=[11]-=-); moreover, the function S is determined by its Fourier transform Ŝ(γ) = φ̂(γ)∑ n∈Z φ(n) e−inγ , γ ∈ R. It is also well known that for each f ∈ V we may write f(x) = ∑ n∈Z < f, K(. , n) > S(x− n) an... |

35 |
General sampling theorems for functions in reproducing kernel Hilbert spaces,”
- Nashed, Walter
- 1991
(Show Context)
Citation Context ...h that {φ(.−n), n ∈ Z} is an orthonormal basis on V ; it is clear that V is a shift invariant space 1 and its reproducing kernel K(x, t) is given by: K(x, t) = ∑ n∈Z φ(x− n)φ(t− n), (x, t) ∈ R2, (see =-=[9]-=- and [10]). Throughout this work, we assume that the function φ satisfies the following conditions: (i) |φ(x)| ≤ (cons.) |B(x)||x|1+ε , where ε ≥ 0 and |B(x)| is bounded and 1-periodic function on R. ... |

27 |
Applications of reproducing kernel Hilbert spaces—Bandlimited signal models,”
- Yao
- 1967
(Show Context)
Citation Context ...ampling function of V ; other properties of sampling functions can be found in [3] and [10]. The idea to derive sampling series from reproducing kernels on Hilbert spaces is not new; see [6], [9] and =-=[13]-=-. Moreover from the theory of frames (a notion more general than the notion of a basis in a Hilbert space), one can obtain general sampling series involving regular as well irregular sample points; se... |

21 |
Irregular sampling and the theory of frames,”
- Benedetto, Heller
- 1990
(Show Context)
Citation Context ... from the theory of frames (a notion more general than the notion of a basis in a Hilbert space), one can obtain general sampling series involving regular as well irregular sample points; see [1] and =-=[2]-=-. Notice that in case where the space V is generated by a Riesz basis (which is a special case of frames), one can apply the orthonormalization trick to obtain an orthonormal basis (see [5] p. 139-140... |

14 |
Bounds for truncation error of the sampling expansion,”
- Jagerman
- 1966
(Show Context)
Citation Context ..., have been studied rather extensively in engineering literature. For the case of band-limited functions f (i.e. f̂(γ) = 0 outside a bounded interval), this error has been estimated by D. Jagerman in =-=[8]-=-. By imposing some extra conditions on f besides being band-limited, the truncation error can be decreased; see [4], [7], [8], [14] and [15] for an overview of the results concerning this type of erro... |

10 |
A Sampling Theorem for Irregularly Spaced Sample Points
- Higgins
- 1976
(Show Context)
Citation Context ...s S(x) is a sampling function of V ; other properties of sampling functions can be found in [3] and [10]. The idea to derive sampling series from reproducing kernels on Hilbert spaces is not new; see =-=[6]-=-, [9] and [13]. Moreover from the theory of frames (a notion more general than the notion of a basis in a Hilbert space), one can obtain general sampling series involving regular as well irregular sam... |

6 |
Approximation of continuous and discontinuous functions by generalized sampling series
- Butzer, Ries, et al.
- 1987
(Show Context)
Citation Context ... the following interpolation property S(n) = δ(0 , n), n ∈ Z, 2 where δ(0 , n) is the Kronecker’s delta. Thus S(x) is a sampling function of V ; other properties of sampling functions can be found in =-=[3]-=- and [10]. The idea to derive sampling series from reproducing kernels on Hilbert spaces is not new; see [6], [9] and [13]. Moreover from the theory of frames (a notion more general than the notion of... |

5 |
Truncation error of sampling theorem expansion
- Helms, Thomas
- 1962
(Show Context)
Citation Context ... 0 outside a bounded interval), this error has been estimated by D. Jagerman in [8]. By imposing some extra conditions on f besides being band-limited, the truncation error can be decreased; see [4], =-=[7]-=-, [8], [14] and [15] for an overview of the results concerning this type of error. In this work we get Jagerman-type estimates for |RNf(x)|, in case where f is in a translation invariant space VW as a... |

1 |
sampling and frames, Wavelet-A Tutorial, (Ed. C.Chui
- Benedetto
- 1991
(Show Context)
Citation Context ...Moreover from the theory of frames (a notion more general than the notion of a basis in a Hilbert space), one can obtain general sampling series involving regular as well irregular sample points; see =-=[1]-=- and [2]. Notice that in case where the space V is generated by a Riesz basis (which is a special case of frames), one can apply the orthonormalization trick to obtain an orthonormal basis (see [5] p.... |

1 |
Sampling theorem for the Fourier transform of a distribution with bounded support
- Cambell
- 1968
(Show Context)
Citation Context ...(γ) = 0 outside a bounded interval), this error has been estimated by D. Jagerman in [8]. By imposing some extra conditions on f besides being band-limited, the truncation error can be decreased; see =-=[4]-=-, [7], [8], [14] and [15] for an overview of the results concerning this type of error. In this work we get Jagerman-type estimates for |RNf(x)|, in case where f is in a translation invariant space VW... |

1 |
Y.M.Liu, A class of band-limited cardinal wavelets, preprint
- Walter
(Show Context)
Citation Context ... φ̂(γ) has the interpolating property which Meyer’s function does not share; this is made possible by allowing φ̂(γ) to take complex values. An extensive analysis on this class is originally given in =-=[12]-=-. Let θ(γ) be a real valued function such that, for some k ∈ Z and 0 < ≤ pi/3 we have (a1) θ(−γ) = −θ(γ) + 2 k pi, k ∈ Z (b1) |θ(γ)| = (2 k + 1) pi, |γ| ≥ pi + (c1) θ(γ) + θ(2 pi − γ) = (2 k + 1) ... |

1 |
J.Thomas, On Truncation error for sampling representations of band-limited signals
- Yao
- 1966
(Show Context)
Citation Context ... a bounded interval), this error has been estimated by D. Jagerman in [8]. By imposing some extra conditions on f besides being band-limited, the truncation error can be decreased; see [4], [7], [8], =-=[14]-=- and [15] for an overview of the results concerning this type of error. In this work we get Jagerman-type estimates for |RNf(x)|, in case where f is in a translation invariant space VW as above. In or... |