Characterizations of rectangular (para)-unitary rational functions

Alpay, Daniel; Jørgensen, Palle E. T.; Lewkowicz, Izchak

doi:10.7494/opmath.2016.36.6.695

Cited by 6 publications

(13 citation statements)

References 33 publications

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“…We think the corresponding examples are sufficient for the comparison of our results with the ones in [2]. The case of N = 1 is of special interest as it directly establishes a link between our results and the results in [2]. It has been observed already in [28] , see also e.g.…”

Section: Potapov-blaschke Factorizations: Scalar Casesupporting

confidence: 71%

“…In this subsection, we consider only the cases K = M = 1 and N = 1, 2. We think the corresponding examples are sufficient for the comparison of our results with the ones in [2]. The case of N = 1 is of special interest as it directly establishes a link between our results and the results in [2].…”

Section: Potapov-blaschke Factorizations: Scalar Casesupporting

confidence: 61%

“…We are able to parametrize in a systematic way all possible masks of orthogonal scaling functions and the corresponding wavelet or multi-wavelet masks independently of their dimension. We compare our results with the recent characterization of orthogonal multi-wavelets in [2].…”

Section: Introduction and Notationmentioning

confidence: 82%

“…The constructions of several compactly supported scaling functions and multi-wavelets are given in Section 4. In subsection 3, we compare our results with the characterization in [2]. The characterization in [2] makes use of the so-called Potapov-Blaschke-products and is also valid for rational F .…”

Section: Introduction and Notationmentioning

confidence: 95%

“…with the same vectors v j are as in (2). In the literature, the casesM = 1 andM = M are called the 1-rank and the full rank cases, respectively [9,10,26].…”

Section: Introduction and Notationmentioning

confidence: 99%

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System theory and orthogonal multi-wavelets

Charina

Conti

Cotronei

et al. 2019

Journal of Approximation Theory

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In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer functionof a conservative linear system. The complex matrices A, B, C, D define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multi-wavelets. The structure of the unitary matrix defined by A, B, C, D allows us to parametrize in a systematic way all classes of possible wavelet and multiwavelet masks together with the masks of the corresponding refinable functions.

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Section: Potapov-blaschke Factorizations: Scalar Casesupporting

confidence: 71%

Section: Potapov-blaschke Factorizations: Scalar Casesupporting

confidence: 61%

Section: Introduction and Notationmentioning

confidence: 82%

Section: Introduction and Notationmentioning

confidence: 95%

“…with the same vectors v j are as in (2). In the literature, the casesM = 1 andM = M are called the 1-rank and the full rank cases, respectively [9,10,26].…”

Section: Introduction and Notationmentioning

confidence: 99%

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System theory and orthogonal multi-wavelets

Charina

Conti

Cotronei

et al. 2019

Journal of Approximation Theory

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Realizations of Infinite Products, Ruelle Operators and Wavelet Filters

Alpay

Jørgensen

Lewkowicz

2015

J Fourier Anal Appl

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Using the system theory notion of state-space realization of matrixvalued rational functions, we describe the Ruelle operator associated with wavelet filters. The resulting realization of infinite products of rational functions have the following four features: 1) It is defined in an infinite-dimensional complex domain.2) Starting with a realization of a single rational matrix-function M , we show that a resulting infinite product realization obtained from M takes the form of an (infinitedimensional) Toeplitz operator with the symbol that is a reflection of the initial realization for M . 3) Starting with a subclass of rational matrix functions, including scalar-valued ones corresponding to low-pass wavelet filters, we obtain the corresponding infinite products that realize the Fourier transforms of generators of L 2 (R) wavelets. 4) We use both the realizations for M and the corresponding infinite product to obtain a matrix representation of the Ruelle-transfer operators used in wavelet theory. By "matrix representation" we refer to the slanted (and sparse) matrix which realizes the Ruelle-transfer operator under consideration.

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Characterizations of families of rectangular, finite impulse response, para-unitary systems

Alpay

Jørgensen

Lewkowicz

2016

J. Appl. Math. Comput.

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We here study Finite Impulse Response (FIR) rectangular, not necessarily causal, systems which are (para)-unitary on the unit circle (=the class U ). First, we offer three characterizations of these systems. Then, introduce a description of all FIRs in U , as copies of a real polytope, parametrized by the dimensions and the McMillan degree of the FIRs.Finally, we present six simple ways (along with their combinations) to construct, from any FIR, a large family of FIRs, of various dimensions and McMillan degrees, so that whenever the original system is in U , so is the whole family.A key role is played by Hankel matrices.1 β=0,Hence, in Engineering terminology Φ(t) (and often F (z)) is referred to as Finite Impulse Response (i.e. the support of Φ(t) is finite).Moreover F (z) will be called causal whenever 1 ≥ q (i.e. Φ(t) ≡ 0 for all 0 > t) and strictly causal if 0 ≥ q. Similarly, (strictly) anti-causal when (q ≥ n + 1) q ≥ n.

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Characterizations of rectangular (para)-unitary rational functions

Cited by 6 publications

References 33 publications

System theory and orthogonal multi-wavelets

System theory and orthogonal multi-wavelets

Realizations of Infinite Products, Ruelle Operators and Wavelet Filters

Characterizations of families of rectangular, finite impulse response, para-unitary systems

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