Topic
Factorization lemma
About: Factorization lemma is a research topic. Over the lifetime, 16 publications have been published within this topic receiving 434 citations.
Papers
1 Jun 1986
TL;DR: In this article, the authors consider the problem of canonical and non-canonical Wiener-Hopf factorization and prove the following results: 1.4.1., 2.5.2, 3.6.3, 4.7.4, and 5.8.
Abstract: I: Canonical and Minimal Factorization.- Editorial introduction.- Left Versus Right Canonical Factorization.- 1. Introduction.- 2. Left and right canonical Wiener-Hopf factorization.- 3. Application to singular integral operators.- 4. Spectral and antispectral factorization on the unit circle.- 5. Symmetrized left and right canonical spectral factorization on the imaginary axis.- References.- Wiener-Hopf Equations With Symbols Analytic In A Strip.- 0. Introduction.- I. Realization.- 1. Preliminaries.- 2. Realization triples.- 3. The realization theorem.- 4. Construction of realization triples.- 5. Basic properties of realization triples.- II. Applications.- 1. Inverse Fourier transforms.- 2. Coupling.- 3. Inversion and Fredholm properties.- 4. Canonical Wiener-Hopf factorization.- 5. The Riemann-Hilbert boundary value problem.- References.- On Toeplitz and Wiener-Hopf Operators with Contour-Wise Rational Matrix and Operator Symbols.- 0. Introduction.- 1. Indicator.- 2. Toeplitz operators on compounded contours.- 3. Proof of the main theorems.- 4. The barrier problem.- 5. Canonical factorization.- 6. Unbounded domains.- 7. The pair equation.- 8. Wiener-Hopf equation with two kernels.- 9. The discrete case.- References.- Canonical Pseudo-Spectral Factorization and Wiener-Hopf Integral Equations.- 0. Introduction.- 1. Canonical pseudo-spectral factorizations.- 2. Pseudo-?-spectral subspaces.- 3. Description of all canonical pseudo-?-spectral factorizations.- 4. Non-negative rational matrix functions.- 5. Wiener-Hopf integral equations of non-normal type.- 6. Pairs of function spaces of unique solvability.- References.- Minimal Factorization of Integral operators and Cascade Decompositions of Systems.- 0. Introduction.- I. Main results.- 1. Minimal representation and degree.- 2. Minimal factorization (1).- 3. Minimal factorization of Volterra integral operators (1).- 4. Stationary causal operators and transfer functions.- 5. SB-minimal factorization (1).- 6. SB-minimal factorization in the class (USB)..- 7. Analytic semi-separable kernels.- 8. LU- and UL-factorizations (1).- II. Cascade decomposition of systems.- 1. Preliminaries about systems with boundary conditions.- 2. Cascade decompositions.- 3. Decomposing projections.- 4. Main decomposition theorems.- 5. Proof of Theorem II.4.1.- 6. Proof of Theorem II.4.2.- 7. Proof of Theorem II.4.3.- 8. Decomposing projections for inverse systems..- III. Proofs of the main theorems.- 1. A factorization lemma.- 2. Minimal factorization (2).- 3. SB-minimal factorization (2).- 4. Proof of Theorem I.6.1.- 5. Minimal factorization of Volterra integral operators (2).- 6. Proof of Theorem I.4.1.- 7. A remark about minimal factorization and inversion.- 8. LU- and UL-f actorizations (2).- 9. Causal/anticausal decompositions.- References.- II: Non-Canonical Wiener-Hopf Factorization.- Editorial introduction.- Explicit Wiener-Hopf Factorization and Realization.- 0. Introduction.- 1. Preliminaries.- 1. Peliminaries about transfer functions.- 2. Preliminaries about Wiener-Hopf factorization.- 3. Reduction of factorization to nodes with centralized singularities.- II. Incoming characteristics.- 1. Incoming bases.- 2. Feedback operators related to incoming bases.- 3. Factorization with non-negative indices.- III. Outgoing characteristics.- 1. Outgoing bases.- 2. Output injection operators related to outgoing bases.- 3. Factorization with non-positive indices.- IV. Main results.- 1. Intertwining relations for incoming and outgoing data.- 2. Dilation to a node with centralized singularities.- 3. Main theorem and corollaries.- References,.- Invariants for Wiener-Hopf Equivalence of Analytic Operator Functions.- 1. Introduction and main result.- 2. Simple nodes with centralized singularities.- 3. Multiplication by plus and minus terms.- 4. Dilation.- 5. Spectral characteristics of transfer functions: outgoing spaces.- 6. Spectral characteristics of transfer functions: incoming spaces.- 7. Spectral characteristics and Wiener-Hopf equivalence.- References.- Multiplication by Diagonals and Reduction to Canonical Factorization.- 1. Introduction.- 2. Spectral pairs associated with products of nodes.- 3. Multiplication by diagonals.- References.- Symmetric Wiener-Hopf Factorization of Self-Adjoint Rational Matrix Functions and Realization.- 0. Introduction and summary.- 1. Introduction.- 2. Summary.- I. Wiener-Hopf factorization.- 1. Realizations with centralized singularities..- 2. Incoming data and related feedback operators.- 3. Outgoing data and related output injection operators.- 4. Dilation to realizations with centralized singularities.- 5. The final formulas.- II. Symmetric Wiener-Hopf factorization.- 1. Duality between incoming and outgoing operators.- 2. The basis in (C and duality between the feedback operators and the output injection operators.- 3. Proof of the main theorems.- References.
69 citations
TL;DR: In this paper, the authors prove imbedding and multiplier theorems for discrete Littlewood-Paley spaces introduced by M. Prazier and B. Jawerth in their theory of wavelet-type decompositions of Ίriebel-Lizorkin spaces.
Abstract: We prove imbedding and multiplier theorems for discrete Littlewood—Paley spaces introduced by M. Prazier and B. Jawerth in their theory of wavelet-type decompositions of Ίriebel—Lizorkin spaces. The corresponding inequalities for discrete spaces defined in terms of characteristi c functions of dyadic cubes, with respect to an arbitrary positive locally finite measure on the Euclidean space, are useful in the theory of tent spaces, weighted inequalities, duality theorems, interpolation by analytic and harmonic functions, etc. Our main tools are vector-valued maximal inequalities, a dyadic version of the Carleson measure theorem, and Pisier's factorization lemma. We also consider more general inequalities, with an arbitrary family of measurable functions in place of characteristic functions of dyadic cubes, which occur in the factorization theory of operators.
55 citations
TL;DR: In this article, it was shown that given two arbitrary lattices of equal density in the Euclidean space Rn, a bounded quasi-periodic and piecewise affine vector field on Rn (a so-called "modulation field") can be built so that the second lattice is the image of the first one under the map x to x-v(x).
Abstract: It is shown that, given two arbitrary lattices of equal density in the Euclidean space Rn, a bounded quasi-periodic and piecewise affine vector field nu on Rn (a so-called 'modulation field') can be built so that the second lattice is the image of the first one under the map x to x-v(x). The proof relies on a factorization lemma for matrices with determinant equal to one. Each factor represents a shear-like transformation of Rn which, in turn, is closely approximated by a periodic set of 'slips' in the lattice.
27 citations
23 Apr 2018
TL;DR: In this paper, the authors provide a complete and explicit description of all solutions to the left tangential operator Nevanlinna-Pick interpolation problem assuming the associated Pick operator is strictly positive.
Abstract: The main results presented in this paper provide a complete and explicit description of all solutions to the left tangential operator Nevanlinna– Pick interpolation problem assuming the associated Pick operator is strictly positive. The complexity of the solutions is similar to that found in descriptions of the sub–optimal Nehari problem and variations on the Nevanlinna– Pick interpolation problem in the Wiener class that have been obtained through the band method. The main techniques used to derive the formulas are based on the theory of co-isometric realizations, and use the Douglas factorization lemma and state space calculations. A new feature is that we do not assume an additional stability assumption on our data, which allows us to view the Leech problem and a large class of commutant lifting problems as special cases. Although the paper has partly the character of a survey article, all results are proved in detail and some background material has been added to make the paper accessible to a large audience including engineers.
5 citations
TL;DR: In this article, the separable weak-bounded approximation properties of a dual Banach space were studied. But they were strictly stronger than the ap-proximation property and weaker than the bounded approximation property.
Abstract: . In this paper we introduce and study the separable weakbounded approximation properties which is strictly stronger than the ap-proximation property and but weaker than the bounded approximationproperty. It provides new sufficient conditions for the metric approxima-tion property for a dual Banach space. 1. IntroductionLet X and Y be Banach spaces. We denote by B(X,Y) the space of boundedlinear operators from X into Y, and by F(X,Y), K(X,Y ), W(X,Y), andB S (X,Y) its subspaces of finite rank operators, compact operators, weaklycompact operators, and separable-valued bounded linear operators.Recall that a Banach space X is said to have the approximation property(AP) if there exists a net (S α ) ⊂ F(X,Y ) such that S α → I X uniformly oncompact subsets of X. If (S α ) can be chosen with sup kS α k ≤ 1, then X issaid to have the metric approximation property (MAP). The following is a longstanding open problem [1].The Metric Approximation Problem. Does the approximation propertyof the dual space X
4 citations
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| Year | Papers |
|---|---|
| 2018 | 2 |
| 2015 | 2 |
| 2010 | 1 |
| 2004 | 1 |
| 2001 | 1 |
| 1998 | 1 |