Characterizations of orthonormal scale functions: A probabilistic approach

Dobric, Vladimir; Gundy, Richard F.; Hitczenko, Paweł

doi:10.1007/bf02921943

Cited by 15 publications

(20 citation statements)

References 9 publications

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“…In particular, it was Šikić and Weiss who, rather forcefully, focused the author's attention on this notion, in connection with a previous paper [5]. They identified this type of continuity in the case where g(ξ ) ≡ 1, and call it "almost everywhere dyadic continuity at zero."…”

Section: Lawton's Theorem: the General Casementioning

confidence: 93%

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Low-Pass Filters, Martingales, and Multiresolution Analyses

Gundy

2000

Applied and Computational Harmonic Analysis

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Necessary and sufficient conditions for a trigonometric polynomial to be a lowpass filter have been given by A. Cohen (Ann. Inst. H. Poincaré Anal. Nonlinéaire 7 (1990), 439-459) and W. Lawton (J. Math. Phys. 31 (1990), 1898-1901). We give necessary and sufficient conditions for an arbitrary periodic function to be a low-pass filter, following the approach of Lawton. The technique also applies to functions that give rise to a Riesz basis.

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Section: Lawton's Theorem: the General Casementioning

confidence: 93%

“…Papadakis et al [14,Theorem 3.16] have formulated one of them. The other is a probabilistic version, due to Dobrić et al [5]; for this work, we owe a debt of gratitude to H. Šikić and G. Weiss for their many critical comments. However, neither one of these papers contains results that resemble what is done in this paper.…”

Section: Introductionmentioning

confidence: 99%

Low-Pass Filters, Martingales, and Multiresolution Analyses

Gundy

2000

Applied and Computational Harmonic Analysis

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“…In the theorem below, which is known (see, for example, [9]), we will assume that the equation holds everywhere. This is not a restriction as proved in [4]. For completeness we include a short proof.…”

Section: Examplesmentioning

confidence: 83%

“…The graph of the density function is pictured in Figure 2. 4 . The graph of the density function is pictured in Figure 3.…”

Section: Examplesmentioning

confidence: 99%

Random variable dilation equation and multidimensional prescale functions

Belock

Dobric

2001

Trans. Amer. Math. Soc.

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Abstract. A random variable Z satisfying the random variable dilation equation MZ d = Z + G, where G is a discrete random variable independent of Z with values in a lattice Γ ⊂ R d and weights {c k } k∈Γ and M is an expanding and Γ-preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density ϕ which will satisfy a dilation equationWe have obtained necessary and sufficient conditions for the existence of the density ϕ and a simple sufficient condition for ϕ's existence in terms of the weights {c k } k∈Γ . Wavelets in R d can be generated in several ways. One is through a multiresolution analysis of L 2 R d generated by a compactly supported prescale function ϕ. The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of ϕ allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when ϕ is a prescale function.

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“…Characterizations of low pass filters for an MRA on L 2 (R) are already known: see the papers by M. Papadakis, H. Sikić and G. Weiss [23] and by V. Dobrić, R. F. Gundy and P. Hitczenko [9]. Afterwards, R. F. Gundy [12] addressed the same question when condition (iv) in the definition of MRA is relaxed by assuming that {φ(x − k) : k ∈ Z} is a Riesz basis for V 0 .…”

Section: −1mentioning

confidence: 99%

Characterization of low pass filters in a multiresolution analysis

Antolín

2009

Studia Math.

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Abstract. We characterize the low pass filters associated with scaling functions of a multiresolution analysis in a general context, where instead of the dyadic dilation one considers the dilation given by a fixed linear invertible map A : R n → R n such that A(Z n ) ⊂ Z n and all (complex) eigenvalues of A have modulus greater than 1. This characterization involves the notion of filter multiplier of such a multiresolution analysis. Moreover, the paper contains a characterization of the measurable functions which are filter multipliers.

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Characterizations of orthonormal scale functions: A probabilistic approach

Cited by 15 publications

References 9 publications

Low-Pass Filters, Martingales, and Multiresolution Analyses

Low-Pass Filters, Martingales, and Multiresolution Analyses

Random variable dilation equation and multidimensional prescale functions

Characterization of low pass filters in a multiresolution analysis

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