Representation Formulas for Hardy Space Functions Through the Cuntz Relations and New Interpolation Problems

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

doi:10.1007/978-1-4614-4145-8_7

Cited by 15 publications

(16 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then we have the following interpolation/boundary formula:f (x) = y∈V \{o} K (x, y) (∆f ) (y) +ˆB (Qf ) (b) dµ x (b) (9.2)valid for all f ∈ H E , and all x ∈ V .Proof. From[AJLM13,Jor11], we have that the projection Q ⊥ = I H E − Q is given byQ ⊥ f = y∈V (∆f ) (y) v y = y∈V |v y δ y | ⊥ f (x) = y∈V \{o} K (x, y) (∆f ) (y) , ∀x ∈ V. (9.4)Since f = Q ⊥ f + (Qf ) with Qf ∈ Harm (⊂ H E ), the desired formula (9.2) follows from the Poisson-representation:(Qf ) (x) =ˆB (Qf ) (b) dµ x (b) .…”

mentioning

confidence: 99%

Infinite weighted graphs with bounded resistance metric

Jørgensen¹,

Tian²

2018

Math. Scand.

Self Cite

View full text Add to dashboard Cite

We consider infinite weighted graphs G, i.e., sets of vertices V , and edges E assumed countable infinite. An assignment of weights is a positive symmetric function c on E (the edge-set), conductance. From this, one naturally defines a reversible Markov process, and a corresponding Laplace operator acting on functions on V , voltage distributions. The harmonic functions are of special importance. We establish explicit boundary representations for the harmonic functions on G of finite energy.We compute a resistance metric d from a given conductance function. (The resistance distance d (x, y) between two vertices x and y is the voltage drop from x to y, which is induced by the given assignment of resistors when 1 amp is inserted at the vertex x, and then extracted again at y.)We study the class of models where this resistance metric is bounded. We show that then the finite-energy functions form an algebra of 1 2 -Lipschitz-continuous and bounded functions on V , relative to the metric d. We further show that , in this case, the metric completion M of (V, d) is automatically compact, and that the vertex-set V is open in M . We obtain a Poisson boundary-representation for the harmonic functions of finite energy, and an interpolation formula for every function on V of finite energy.We further compare M to other compactifications; e.g., to certain path-space models.

show abstract

mentioning

confidence: 99%

Infinite weighted graphs with bounded resistance metric

Jørgensen¹,

Tian²

2018

Math. Scand.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We present a result which encompasses both cases. The result is valid in particular for rational functions r with possibly less than N pairwise different zeros, but having limit infinity at infinity, and also for Blaschke products, corresponding to results in [17] and [16] respectively. These cases are considered after the proof of the lemma.…”

Section: A Representation Theorem For Analytic Functionmentioning

confidence: 81%

“…Corollary 2.13 leads to the following natural question: Which functions can be written in the form Z r (z)F (r(z))? When r is a polynomial the answer was given in [17,Theorem 2.1,p.44] and in [16] when r is a finite Blaschke product. We present a result which encompasses both cases.…”

Section: A Representation Theorem For Analytic Functionmentioning

confidence: 99%

See 1 more Smart Citation

Preliminaries

Alpay¹,

Colombo²,

Sabadini³

2020

SpringerBriefs in Mathematics

Self Cite

View full text Add to dashboard Cite

We prove various Beurling-Lax type theorems, when the classical backwardshift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given.

show abstract

“…The interest in Finite Impulse Response functions within U (=para-unitary, in signal processing "dialect") is vast, see e.g. the books [11], [25], [29,Section 7.3], [37,Section 5.2], [40,Section 6.5], the theses [23], [31] and the papers [4], [5], [7], [10], [16], [24], [26], [32], [34], [36] and [43].…”

Section: Introductionmentioning

confidence: 99%

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

Alpay

Jørgensen

Lewkowicz

2016

J. Appl. Math. Comput.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Representation Formulas for Hardy Space Functions Through the Cuntz Relations and New Interpolation Problems

Cited by 15 publications

References 26 publications

Infinite weighted graphs with bounded resistance metric

Infinite weighted graphs with bounded resistance metric

Preliminaries

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

Contact Info

Product

Resources

About