## Approximation error for quasi-interpolators and (multi-)wavelet expansions (1999)

Venue: | APPL. COMPUT. HARMON. ANAL |

Citations: | 56 - 21 self |

### BibTeX

@ARTICLE{Blu99approximationerror,

author = {Thierry Blu and Michael Unser},

title = {Approximation error for quasi-interpolators and (multi-)wavelet expansions},

journal = {APPL. COMPUT. HARMON. ANAL},

year = {1999},

volume = {6},

pages = {219--251}

}

### Years of Citing Articles

### OpenURL

### Abstract

We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L² via an appropriate sequence of inner products. In particular, we consider integer shift-invariant approximations such as those provided by splines and wavelets, as well as finite elements and multi-wavelets which use multiple generators. We estimate the approximation error as a function of the scale parameter T when the function to approximate is sufficiently regular. We then present a generalized sampling theorem, a result that is rich enough to provide tight bounds as well as asymptotic expansions of the approximation error as a function of the sampling step T. Another more theoretical consequence is the proof of a conjecture by Strang and Fix, which states the equivalence between the order of a multi-wavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the two-scale relation to obtain explicit formulae for the coefficients of the asymptotic development of the error. The leading constants are easily computable and can be the basis for the comparison of the approximation power of wavelet and multi-wavelet expansions of a given order L.

### Citations

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Citation Context ...riance led to the concept of multiresolution224 BLU AND UNSER analysis [35, 36]. In the usual case of interest where � is compactly supported and where V T is a wavelet-like space, it has been shown =-=[13]-=- that (4) must be replaced by �n � � f�a 0 �n � � � VT (integer scale invariance), (7) where a 0 � 2 is a positive integer (scale factor). Yet, noninteger scale factors (of particular interest is the ... |

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163 |
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Citation Context ... For our analysis, we have purposely chosen to consider a very broad class of linear approximation operators. An interesting subset of them includes the cases usually designated by quasi-interpolants =-=[11, 20, 24, 42]-=-, the various types of projectors encountered in the context of the wavelet transform [14, 45], but also more general polynomial preserving operators that have been studied recently [9, 29, 33]. A gen... |

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Citation Context ... This prevents us from giving the Nyquist function a local meaning. These remarks led to a first generalization of the notion of interpolation through the loss of a certain amount of shift invariance =-=[4, 42]-=-. More specifically, it was recognized that defining an approximation space VT t � span n�� ��� T � n�� � L 2 , where � is a function with acceptable frequency and time localization, can be more robus... |

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Citation Context ... functions q, so that our basic representation spaces are q-integer shift-invariant. We use a vector formalism well adapted to the study of multiwavelets, which have attracted much attention recently =-=[1, 15, 27, 28, 51]-=-. For our analysis, we have purposely chosen to consider a very broad class of linear approximation operators. An interesting subset of them includes the cases usually designated by quasi-interpolants... |

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Spline approximation by quasi-interpolants
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Citation Context ... For our analysis, we have purposely chosen to consider a very broad class of linear approximation operators. An interesting subset of them includes the cases usually designated by quasi-interpolants =-=[11, 20, 24, 42]-=-, the various types of projectors encountered in the context of the wavelet transform [14, 45], but also more general polynomial preserving operators that have been studied recently [9, 29, 33]. A gen... |

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Citation Context ...on on the left side of this equation, and with the use of (40), (43), and (44), and finally after some (tedious) rearrangements, we obtain the value of �; this validates (46). Note that, as proven in =-=[17]-=-, H0 has spectral radius 2 so that Iq � (1/2 L�1 )H0 is never singular. ■ With these constants, we can now derive the asymptotic value of the approximation error in the multi-wavelet case. L THEOREM 7... |

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- 1990
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Citation Context ...e context of the wavelet transform [14, 45], but also more general polynomial preserving operators that have been studied recently [9, 29, 33]. A general account of quasiinterpolation can be found in =-=[21]-=-. Here, we will see that the order constraint is translatedAPPROXIMATION ERRORS 221 into a simple moment condition for the analysis functions—a very weak form of biorthonormality, which we call “quas... |

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Ondelettes á localisation exponentielles
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Citation Context ... functions q, so that our basic representation spaces are q-integer shift-invariant. We use a vector formalism well adapted to the study of multiwavelets, which have attracted much attention recently =-=[1, 15, 27, 28, 51]-=-. For our analysis, we have purposely chosen to consider a very broad class of linear approximation operators. An interesting subset of them includes the cases usually designated by quasi-interpolants... |

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Approximation power of biorthogonal wavelet expansions
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Citation Context ...rators. An interesting subset of them includes the cases usually designated by quasi-interpolants [11, 20, 24, 42], the various types of projectors encountered in the context of the wavelet transform =-=[14, 45]-=-, but also more general polynomial preserving operators that have been studied recently [9, 29, 33]. A general account of quasiinterpolation can be found in [21]. Here, we will see that the order cons... |

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Citation Context ...S The concept of a multiresolution analysis (MRA) [35, 36] has proven to be the key for the construction of a whole variety of wavelet bases. These include orthogonal [6, 18, 34, 35], semi-orthogonal =-=[3, 12, 47]-=-, and biorthonormal wavelets [14, 49], as well as the more recent multi-wavelets [1, 15, 27]. MRA provides a simple geometrical interpretation of the decomposition process in terms of projections onto... |

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Citation Context ...ther involved) counterexample by Jia [30], discussed below. Several authors worked around the problem for q � 1 by adding constraints and introducing sophisticated notions of controlled approximation =-=[16, 25, 32]-=-. In our case, the situation appears to be more favorable and there is no major difficulty in extending the Strang–Fix equivalence for q � 1 without adding to our initial assumptions.232 BLU AND UNSE... |

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Multiresolution analysis of multiplicity d: applications to dyadic interpolation
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Citation Context ... functions q, so that our basic representation spaces are q-integer shift-invariant. We use a vector formalism well adapted to the study of multiwavelets, which have attracted much attention recently =-=[1, 15, 27, 28, 51]-=-. For our analysis, we have purposely chosen to consider a very broad class of linear approximation operators. An interesting subset of them includes the cases usually designated by quasi-interpolants... |

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- 1997
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Citation Context ...or the so-called Strang–Fix conditions in the Fourier domain) and the rate of decay of the minimum expansion error [10, 42] (initially least squares solution, later extended to the other L 2 measures =-=[31]-=-). In [42], Strang and Fix conjectured that such an equivalence would also hold for multiple generators but their initial claim was put into question by the construction of a (rather involved) counter... |

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The Fourier theory of the cardinal functions
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