Unimodular Function Research Articles

Douglas-Rudin approximation theorem for operator-valued functions on the unit ball of [formula omitted]

Douglas and Rudin proved that any unimodular function on the unit circle T can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of Cd. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result (2009) [4] on the approximation of matrix-valued unimodular functions on T.

Conjugations on L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} Space on the Real Line

Influenced by PT operators (parity and time reversal operators) the general conjugations acting in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{L}}^\ extbf{2}$$\\end{document} space on the real line with the Lebesgue measure are studied. The behaviour of conjugations with respect to multiplication operators and translations is investigated. We consider conjugations defined as the convolution of the Fourier transform of a given unimodular (even) function and test functions. We characterize all conjugations commuting (intertwining) with all translation operators on the real line. The conjugations leaving kernels of Toeplitz operators (a generalization of model spaces) on H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{H}}^\ extbf{2}$$\\end{document} space on the upper half plane invariant are also characterized. Analogous results are obtained for conjugations and kernels of Wiener–Hopf operators.

Approximate local isometries of Banach algebras of differentiable functions

Let X and Y be compact subsets of R such that X and Y coincide with the closures of their interiors. For any n∈N, let C(n)(X) be the Banach algebra of all n-times continuously differentiable complex-valued functions f on X, with the norm ‖f‖C=maxx∈X⁡(∑k=0n(|f(k)(x)|/k!)). We prove that every approximate local isometry of C(n)(X) to C(n)(Y) is an isometric linear algebra monomorphism multiplied by a fixed n-times continuously differentiable unimodular function. This description allows us to establish the algebraic and 2-algebraic reflexivity of the set of linear isometries of C(n)(X) onto C(n)(Y). Furthermore, this algebraic reflexivity becomes topological whenever X and Y are compact intervals of R. Another application of our main result shows that the sets of isometric reflections and generalized bi-circular projections of C(n)(X) are topologically and 2-topologically reflexive.

Unimodular Hausdorff and Minkowski dimensions

This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced in this work, which provide a common generalization to stationary point processes under their Palm version and unimodular random rooted graphs. The use of unimodularity in the definitions of dimension is novel. Also, a toolbox of results is presented for the analysis of these dimensions. In particular, analogues of Billingsley’s lemma and Frostman’s lemma are presented. These last lemmas are instrumental in deriving upper bounds on dimensions, whereas lower bounds are obtained from specific coverings. The notions of unimodular Hausdorff size, which is a discrete analogue of the Hausdorff measure, and unimodular dimension function are also introduced. This toolbox allows one to connect the unimodular dimensions to other notions such as volume growth rate, discrete dimension and scaling limits. It is also used to analyze the dimensions of a set of examples pertaining to point processes, branching processes, random graphs, random walks, and self-similar discrete random spaces. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces.

On Gabor orthonormal bases over finite prime fields

We study Gabor orthonormal windows in $L^2({\Bbb Z}_p^d)$ for translation and modulation sets $A$ and $B$, respectively, where $p$ is prime and $d\geq 2$. We prove that for a set $E\subset \Bbb Z_p^d$, the indicator function $1_E$ is a Gabor window if and only if $E$ tiles and is spectral. Moreover, we prove that for any function $g:\Bbb Z_p^d\to \Bbb C$ with support $E$, if the size of $E$ coincides with the size of the modulation set $B$ or if $g$ is positive, then $g$ is a unimodular function, i.e., $|g|=c1_E$, for some constant $c>0$, and $E$ tiles and is spectral. We also prove the existence of a Gabor window $g$ with full support where neither $|g|$ nor $|\hat g|$ is an indicator function and $|B|<<p^d$. We conclude the paper with an example and open questions.

Locality of connective constants

The connective constant μ(G) of a quasi-transitive graph G is the exponential growth rate of the number of self-avoiding walks from a given origin. We prove a locality theorem for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin (and a further condition is satisfied). The proof is based on a generalized bridge decomposition of self-avoiding walks, which is valid subject to the assumption that the underlying graph is quasi-transitive and possesses a so-called unimodular graph height function.

Connective constants and height functions for Cayley graphs

The connective constant μ ( G ) \mu (G) of an infinite transitive graph G G is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called “unimodular graph height functions”. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph height function termed here a “group height function”. A necessary and sufficient condition for the existence of a group height function is presented, and may be applied in the context of the bridge constant, and of the locality of connective constants for Cayley graphs. Locality may thereby be established for a variety of infinite groups including those with strictly positive deficiency. It is proved that a large class of Cayley graphs support unimodular graph height functions, that are in addition harmonic on the graph. This implies, for example, the existence of unimodular graph height functions for the Cayley graphs of finitely generated solvable groups. It turns out that graphs with non-unimodular automorphism subgroups also possess graph height functions, but the resulting graph height functions need not be harmonic. Group height functions, as well as the graph height functions of the previous paragraph, are non-constant harmonic functions with linear growth and an additional property of having periodic differences. The existence of such functions on Cayley graphs is a topic of interest beyond their applications in the theory of self-avoiding walks.

Real-linear isometries between subspaces of continuous functions

Let X and Y be locally compact Hausdorff spaces. Let A and B be complex-linear subspaces of C0(X) and C0(Y), respectively. Suppose that for each triple of distinct points x,x′,x″∈X, there exists f∈A such that |f(x)|≠|f(x′)| and f(x″)=0. Also suppose that for each pair of distinct points y,y′∈Y, there exists g∈B such that |g(y)|≠|g(y′)|. For such A and B, we prove the following statement: If T is a real-linear isometry of A onto B, then there exist an open and closed subset E of ChB, a homeomorphism φ of ChB onto ChA and a unimodular continuous function ω on ChB such that Tf=ω(f∘φ) on E and Tf=ω(f∘φ¯) on ChB∖E for all f∈A, where ChA and ChB are the Choquet boundaries for A and B, respectively. Moreover, we remark that the separation condition on A cannot be omitted in the above result.

Non-vanishing functions and Toeplitz operators on tube-type domains

We prove an index theorem for Toeplitz operators on irreducible tube-type domains and we extend our results to Toeplitz operators with matrix symbols. In order to prove our index theorem, we proved a result asserting that a non-vanishing function on the Shilov boundary of a tube-type bounded symmetric domain, not necessarily irreducible, is equal to a unimodular function defined as the product of powers of generic norms times an exponential function.

Toeplitz operators and arguments of analytic functions

For a Toeplitz operator Tφ, we study the interrelationship between smoothness properties of the symbol φ and those of the functions annihilated by Tφ. For instance, it follows from our results that if φ is a unimodular function on the circle lying in some Lipschitz or Zygmund space Λα with 0 < α < ∞, and if f is an Hp-function (p ≥ 1) with Tφf = 0, then f ∈ Λα and $$\|f\|_{\Lambda^a} \le c\|\varphi\|^d_{\Lambda^a} \|f\|_p$$ for some c = c(α, p) and d = d(α, p); an explicit formula for the optimal exponent d is provided. Similar—and more general—results for various smoothness classes are obtained, and several approaches are discussed. Furthermore, since a given non-null function f ∈ Hp lies in the kernel of \({T_\psi}\) with \({\psi=\bar z\bar f/f}\) , we derive information on the smoothness of Hp-functions with smooth arguments. This can be viewed as a natural counterpart to the existing theory of analytic functions with smooth moduli.

Certain invariant subspace structure of $L^2(\Bbb T^2)$ II

Let M be an invariant subspace of L2(T2). Considering the largest z- invariant (resp. w-invariant) subspace Fz (resp. Fw) in the wandering subspace M zwM of M with respect to the shift operator zw. If Fw 6 f0g and Fz 6 f0g, then we consider the certain form of invariant subspaces M of L2(T2). Furthermore, we study certain classes of invariant subspaces of L 2 (T 2 ). 1. Introduction and preliminaries Let T 2 be the torus that is the cartesian product of 2 unit circles in C. Let L 2 (T 2 ) and H 2 (T 2 ) be the usual Lebesgue and Hardy space on the torus T 2 , respectively. A closed subspace M of L 2 (T 2 ) is said to be invariant if zM ‰ M and wM ‰ M. As is well known, the structure of invariant subspaces is much more complicated. In general, the invariant subspaces of L 2 (T 2 ) are not necessarily of the form φH 2 (T 2 ) with some unimodular function φ. The structure of Beurling-type invariant subspaces has been studied, and some necessary and sufficient conditions for invariant subspaces to be Beurling-type have been given (cf. (1, 2, 5), etc). Further, many authors had attempted to study the form of invariant subspaces of L 2 (T 2 ) (cf. (4, 6, 7), etc). In (4), we studied the structure of an invariant subspace M as a zw- invariant subspace. We gave an alternative approach of Beuring-type in- variant subspaces and a certain class of invariant subspace which contains the class of invariant subspaces of the form φH 2 0 (T 2 ), where H 2 0 (T 2 ) = ff 2 H 2 (T 2 ) : f (0, 0) = 0g and φ is a unimodular function in L 1 (T 2 ). For (m, n) 2 Z 2 and f 2 L 2 (T 2 ), the Fourier coefficient of f is defined

Characterization of analytic phase signals

In many cases, a real-valued signal χ( t) may be associated with a complex-valued signal a( t) e iθ( t) , the analytic signal associated with χ( t) with the characteristic properties χ( t) = a( t) cosθ( t) and H( a(·)cosθ(·))( t) = a( t)sinθ( t). Using such obtained amplitude-frequency modulation the instantaneous frequency of χ( t) at the time t 0 may be defined to be θ′( t 0), provided θ′( t 0) ≥ 0. The purpose of this note is to characterize, in terms of analytic functions, the unimodular functions F( t) = C( t) + iS( t), C 2( t) + S 2 ( t) = 1, a.e., that satisfy HC( t) = S( t). This corresponds to the case a( t) ≡ 1 in the above formulation. We show that a unimodular function satisfies the required condition if and only if it is the boundary value of a so called inner function in the upper-half complex plane. We also give, through an explicit formula, a large class of functions of which the parametrization C( t) = cosθ( t) is available and the extra condition θ′( t) ≥ 0, a.e. is enjoyed. This class of functions contains Blaschke products in the upper-half complex plane as a proper subclass studied by Picinbono in [1].

Unimodular functions and interpolating Blaschke products

The result by Bourgain that every unimodular function ψ \psi on the unit circle can be factored as ψ = e i v ~ B 1 B ¯ 2 \psi = e^{i \tilde v} B_1 \overline B_2 with B 1 B_1 and B 2 B_2 Blaschke products can be improved. We show that the same result holds with B 1 B_1 and B 2 B_2 interpolating Blaschke products. This will at the same time be a refinement of Jones’s result that every unimodular function can be approximated in the H ∞ H^\infty -norm by a ratio of interpolating Blaschke products.

Badly Approximable Unimodular Functions in Weighted L p Spaces

A function u on the unit circle T is said to be badly approximable in the weighted space L p(T,w)} if ¦¦u + f¦¦ L p(T,w) ≥¦L p(T,w) for all f ∊ H∞. We prove that if an unimodular function u is badly approximable in L p(T, w}) for all p ∊ (0, +∞) and some non-zero weight w, then \(\overline{u}\) is an inner function. We describe the inner functions Θ and the weights w on the unit circle T such that Θ is badly approximable in L p(T, w) for all p > 0. It turns out that, for given inner functions Θ, the class of all weights satisfying the above-mentioned condition depends only on the zero set of Θ. In other words, Θ is badly approximable in L p(T, w) for all p ∊ (0, +∞) if and only if \(\overline{B}\) is badly approximable in L p(T,w) for all p ∊ (0, +∞), where B is a Blaschke product with simple zeros and such that Θ−1(0) = B−1(0).

Sensitivity minimization of MIMO systems with a stable controller

In this paper, the sensitivity minimization problem of multivariable linear time-invariant, bi-proper or strictly proper, systems using a stable controller is considered. The problem is reduced to a 1-block or 2-block H∞ - optimization problem in the case of square or nonsquare systems, respectively. The solution of this optimization problem, if satisfying an upper bound constraint which depends on an initial choice of a unimodular transfer function in RH∞ would be the Youla parameter and hence a stable stabilizing controller to the original system. A search procedure for an initial choice of an unimodular function is discussed.

The spectra of Toeplitz operators with unimodular symbols

The spectrum σ(Tφ) of a Toeplitz operator Tφ on the open unit disc D for a unimodular symbol φ is studied and many sufficient conditions for σ(Tφ)⊆∂D or σ(Tφ) = are given. In particular if φ is a unimodular function in H∞ + C, then σ(Tφ)⊆∂D or σ(Tφ) =

Strong stabilization of MIMO systems via [formula omitted]∞ optimization

In this paper, the strong stabilization problem of multivariable linear time-invariant systems is considered. The problem is reduced to a 2-block or 1-block H ∞-optimization problem. The solution of this optimization problem, if satisfying an upper bound constraint which depends on an initial choice of a unimodular transfer function in RH ∞, would be a stable stabilizing controller to the original system.

Linear isometries between subspaces of continuous functions

We say that a linear subspace A A of C 0 ( X ) C_0 (X) is strongly separating if given any pair of distinct points x 1 , x 2 x_1, x_2 of the locally compact space X X , then there exists f ∈ A f \in A such that | f ( x 1 ) | ≠ | f ( x 2 ) | \left | f(x_1 ) \right | \neq \left | f(x_2 ) \right | . In this paper we prove that a linear isometry T T of A A onto such a subspace B B of C 0 ( Y ) C_0(Y) induces a homeomorphism h h between two certain singular subspaces of the Shilov boundaries of B B and A A , sending the Choquet boundary of B B onto the Choquet boundary of A A . We also provide an example which shows that the above result is no longer true if we do not assume A A to be strongly separating. Furthermore we obtain the following multiplicative representation of T T : ( T f ) ( y ) = a ( y ) f ( h ( y ) ) (Tf)(y)=a(y)f(h(y)) for all y ∈ ∂ B y \in \partial B and all f ∈ A f \in A , where a a is a unimodular scalar-valued continuous function on ∂ B \partial B . These results contain and extend some others by Amir and Arbel, Holsztyński, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.

Kernels of Toeplitz operators, smooth functions, and Bernstein type inequalities

Let ϕ be a unimodular function on the unit circle\(\mathbb{T}\) and let Kp(ϕ) denote the kernel of the Toeplitz operator Tϕ in the Hardy space Hp, p≥1;\(K_p (\varphi )\mathop = \limits^{def} \{ f \in H^p :T_\varphi f = 0\} \). Suppose Kp(ϕ)≠{0}. The problem is to find out how the smoothness of the symbol ϕ influences the boundary smoothness of functions in Kp(ϕ). One of the main results is as follows.Theorem 1Let 1<p, q<+∞, 1<r≤+∞, q−1=p−1+r−1. Suppose |ϕ|≡1 on\(\mathbb{T}\) and ϕ∈W 1r (i.e.,\(\varphi ' \in L^r (\mathbb{T})\)). Then Kp(ϕ)⊂W 1q . Moreover, for any f∈Kp(ϕ) we have ‖f′‖q≤c(p, r)‖ϕ′‖r ‖f‖. Bibliography: 19 titles.

Holomorphic functions, measures and BMO

We extend the above theorem to the general case of finite strictly positive continuous measures on T, under the supplementary restriction that f (zo)r ). In the particular case where /l is the Lebesgue measure, Theorem 1 implies that the hypothesis '3r(z0)r ' ' is not needed. However, this restriction is not superfluous in the general case; see w 4, prop. 19 for a relevant counterexample. The above extension is purely topological in nature. We prove that for any complex continuous func t ionfon T and any complex number w Cf(T), the following are equivalent: a) For every finite strictly positive continuous measure # on T, there is an interval I c T such that w= 1//z(I) fxfdl~. b) f has non-zero winding number with respect to w. This equivalence enables us to determine the range of the BMO norm of q~o U, where q~ is any given continuous unimodular function on T and U varies in the set of all homeomorphisms of T onto itself. In the case where r has non-zero winding number with respect to 0, we show that (0o U has BMO norm equal to 1 for all U. If q9 has zero winding number with respect to 0, then the BMO norm of q)oU can be made arbitrarily close to zero and does not exceed