Sets of matrices all infinite products of which converge

Daubechies, Ingrid; Lagarias, Jeffrey C.

doi:10.1016/0024-3795(92)90012-y

Cited by 331 publications

(261 citation statements)

References 22 publications

Supporting

Mentioning

256

Contrasting

Unclassified

Order By: Relevance

“…Then in (5) r − 1 inequalities of r are, in fact, equalities. Without loss of generality one can think that in this case the first r − 1 of inequalities (5) are equalities, and then from (8) and (9) it follows that…”

Section: Symbolic Sequencesmentioning

confidence: 99%

See 1 more Smart Citation

Matrix products with constraints on the sliding block relative frequencies of different factors

Kozyakin

2014

Linear Algebra and its Applications

View full text Add to dashboard Cite

One of fundamental results of the theory of joint/generalized spectral radius, the BergerWang theorem, establishes equality between the joint and generalized spectral radii of a set of matrices. Generalization of this theorem on products of matrices whose factors are applied not arbitrarily but are subjected to some constraints is connected with essential difficulties since known proofs of the Berger-Wang theorem rely on the arbitrariness of appearance of different matrices in the related matrix products. Recently, X. Dai [1] proved an analog of the Berger-Wang theorem for the case when factors in matrix products are formed by some Markov law.We extend the concepts of joint and generalized spectral radii to products of matrices with constraints on the sliding block relative frequencies of occurrences of different factors, and prove an analog of the Berger-Wang theorem for this case.

show abstract

Section: Symbolic Sequencesmentioning

confidence: 99%

“…, r} are considered. In the latter case the question about the rate of growth of all possible matrix products (1) can be answered in terms of the so-called joint or generalized spectral radii of the set of matrices M [7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Matrix products with constraints on the sliding block relative frequencies of different factors

Kozyakin

2014

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…These two pairs are important because with linear combinations of their tranlates we can get all solutions (f p) of (1). For any dyadic numberx: (2) Notice that both sums are nite as the supports of f i p i i = 0 1 lie in the set ;1 1]. Now, using relation (1) which is applied to the pair of functions (f(x) = f 0 (x=2) p (x) = p 0 (x=2)=2) and then to the pair (f 1 (x=2) p 1 (x=2)=2), after evaluations of the functions f 0 p 0 f 1 p 1 at the half-integers, we obtain a system of functional equations for f 0 p 0 f 1 …”

Section: Elementary Properties Of H S 21mentioning

confidence: 99%

“…Then it is easy to prove that kA(z=2 n )k 1 + O(1=2 n ) in the disk jzj R. We can complete the proof as in the previous section because the hypotheses of Theorem 5 are satis ed. 2 Corollary 14 If ;5=2 < 2 3=2 and if ;9=2 < 3 < 7=2, then each component of the limit matrix P(z) = lim P n (z) is the Fourier transform of a distribution with support in the interval ;1 1].…”

Section: A Characterization Of Fourier Transformst 0 Tmentioning

confidence: 99%

Fourier analysis of 2-point hermite interpolatory subdivision schemes

Dubuc

Lemire²,

Merrien

2001

The Journal of Fourier Analysis and Applications

View full text Add to dashboard Cite

Two subdivision schemes with Hermite data on Z are studied. These schemes use 2 or 7 parameters respectively depending on whether Hermite data involve only rst derivatives or include second derivatives. For a large region in the parameter space, the schemes are convergent in the space of Schwartz distributions. The Fourier transform of any i n terpolating function can be computed through products of matrices of order 2 or 3. The Fourier transform is related to a speci c system of functional equations whose analytic solution is unique except for a multiplicative constant. The main arguments for these results come from Paley-Wiener-Schwartz theorem on the characterization of the Fourier transforms of distributions with compact support and a theorem of Artzrouni about convergent products of matrices.Math Subject Classi cation: 40A20, 42A38, 46F12, 65D05, 65D10

show abstract

“…More speciÿcally, we are interested in the largest possible rate of growth of such products. Issues of this type arise naturally when considering linear time-varying systems of the form x t+1 = A t x t , as well as in many other contexts; see [22,8].…”

Section: Introductionmentioning

confidence: 99%

The boundedness of all products of a pair of matrices is undecidable

Blondel

Tsitsiklis

2000

Systems & Control Letters

184

139

View full text Add to dashboard Cite

We show that the boundedness of the set of all products of a given pair of rational matrices is undecidable. Furthermore, we show that the joint (or generalized) spectral radius ( ) is not computable because testing whether ( )61 is an undecidable problem. As a consequence, the robust stability of linear systems under time-varying perturbations is undecidable, and the same is true for the stability of a simple class of hybrid systems. We also discuss some connections with the so-called "ÿniteness conjecture". Our results are based on a simple reduction from the emptiness problem for probabilistic ÿnite automata, which is known to be undecidable.

show abstract

Sets of matrices all infinite products of which converge

Cited by 331 publications

References 22 publications

Matrix products with constraints on the sliding block relative frequencies of different factors

Matrix products with constraints on the sliding block relative frequencies of different factors

Fourier analysis of 2-point hermite interpolatory subdivision schemes

The boundedness of all products of a pair of matrices is undecidable

Contact Info

Product

Resources

About